What is repeated here, and what is true

The seven origami axioms are a conventional list that somebody could extend with an eighth.

What decides it: A single fold is specified by bringing existing points and lines into coincidence, and the combinations of those alignments are enumerable. The figure enumerates them, requires that every combination receive a verdict, and asserts that exactly seven of them specify a fold. Huzita catalogued six in 1991 and Hatori found the seventh in 2001; the completeness is a count, not a convention.

Tested in Why the list stops at seven.

Folding is strictly more powerful than straightedge and compass, so any regular polygon a compass misses can be folded instead.

What decides it: The first half is true and the second does not follow. A fold solves cubics, so its reach is the Pierpont primes rather than the Fermat primes — a strict enlargement, and still a restriction. The census computes the arithmetic rather than quoting a list, and the first counterexample is small: 11 is prime, 10 has a factor of 5, and no arrangement of cubics produces a five.

Tested in The eleven-sided one nobody can fold.

A sheet of paper cannot be folded in half more than seven times, whatever its size.

What decides it: The bound is not a count, it is a length. Halving in one direction spends paper at the closed end as the square of the layer count, so the minimum length for n folds goes as 2^n(2^n + 4)(2^n - 1)/6 times the thickness — computed here rather than quoted. At 0.1 mm an A4 sheet does run out at seven, which is where the folklore comes from; twelve folds need about 1.2 km, and Britney Gallivan folded twelve in 2002 having gone and got it.

Tested in How many times can it be halved.

The seven origami axioms are a discovery of the 1990s, and the fact that folding solves cubics is a modern result.

What decides it: Beloch published the construction in 1936 and it is exactly the sixth axiom: a single fold placing two points onto two lines simultaneously, which is a common tangent to two parabolas and therefore a root of a cubic. The generator solves the cubic by bracketing and bisection and then asserts, to a part in a thousand million, that every fold line it draws is tangent to both parabolas at once — the whole content of the 1936 paper, re-run.

Tested in Fifty years in the wrong language, at the figure it turns on.

Complex origami is a matter of skill and patience; a sufficiently good folder could have made modern models at any point in the tradition's history.

What decides it: Stack thickness is the layer count times the sheet thickness, and a fold stops working when the stack approaches the smallest feature being folded. At a 3 mm working limit, copier paper at 100 µm reaches sixteen layers and tissue at 18 µm reaches a hundred and twenty-eight — a sevenfold difference in what is foldable, set entirely by the substrate. The generator refuses to draw unless thinner paper reaches strictly more layers.

Tested in The paper had to arrive first, at the figure it turns on.

The number of waves on a ruffled edge is fixed by how much extra length the edge has.

What decides it: Every profile in the family carries the same excess to within 5e-15, solved for rather than assumed, and they have two, three, five and eight waves. The excess fixes the product of amplitude and wave count and nothing else, so it constrains a one-parameter family and picks no member of it.

Tested in The excess does not choose its waves, at the figure it turns on.

A corrugation can be tapered any way a design needs, since parallel folds have no constraints to satisfy.

What decides it: The vertex angles of a zigzag corrugation depend on the row heights and not on the column widths, so Kawasaki's two alternating sums come to 180° exactly when the rows above and below a vertex are equally deep. Tapering by row instead puts them at 186.4° and 173.6°, and `foldcheck` asserts the refusal.

Tested in A leaf packs by corrugating, at the figure it turns on.

A tighter packing is always better, so a folding pattern should use as many folds as it can.

What decides it: With a thickness, more folds means a thicker stack. The bundle radius falls and then rises, the generator refuses to draw unless the minimum is strictly inside the range of counts it was given, and at the stated parameters the tightest bundle is at twelve folds with both four and twenty-four failing to fit.

Tested in The bud chooses the pattern, at the figure it turns on.

A folding pattern with many creases needs many actuators, so a large deployable is necessarily a complicated mechanism.

What decides it: The Miura in the census has twelve interior vertices and one degree of freedom: fixing one fold angle determines every other panel position. The generator counts the vertices from the built pattern and the freedom from the kinematics, and the count of drivers follows from the freedom rather than from the crease count.

Tested in No motor in the fold, at the figure it turns on.

A finer fold pattern always packs better, so the limit on packing is how precisely the pattern can be made.

What decides it: Each fold consumes a fixed length of surface to get round its own radius, so the share of the sheet lost to hinges is exactly proportional to the fold count — the generator asserts the proportionality against the ratio of the counts it was given. At the stated radius the hinges pass half the sheet at sixty-four folds, and nothing finer buys anything.

Tested in Nothing in a body folds on a line, at the figure it turns on.

Folding a surface into a container increases the area it holds without limit, so an organ can always gain area by folding more finely.

What decides it: Each layer occupies its own thickness of the container's depth, so the depth available to the remaining layers falls as the count rises. The held area peaks and then falls to zero; the generator refuses to draw unless the peak is strictly inside the range of counts and unless at least one count fills the box with sheet alone.

Tested in How much surface fits in a body, at the figure it turns on.

Any connected shape can be built by threading a single scaffold strand through it.

What decides it: A square lattice is bipartite, so a route alternates between two colours of cell and the counts can differ by at most one. A plus of five helices is coloured one to four and admits no route; an exhaustive search that never looks at the colouring agrees, over every shape in the sweep.

Tested in A sheet that routes itself, at the figure it turns on.

Whether a crease pattern's panels can be painted in two colours is a fact about the drawing, decided by how the creases happen to be arranged.

What decides it: The panels two-colour exactly when every interior vertex carries an even number of creases, and nothing else about the arrangement matters. A star of three creases in a square sheet has one interior vertex, an odd count, and no colouring; a star of four has a colouring; five has none again.

Tested in The sheet has two sides, at the figure it turns on.

How much a crease pattern shrinks when it is folded and how thick it becomes are two separate properties, and a good pattern is one that does well at both.

What decides it: The layer count integrated over the folded footprint is the area of the sheet, because that is where the paper is. So the shrink factor and the average layer count are the same number, measured on five patterns whose folded states are computed rather than posed, and no pattern can improve one without paying in the other.

Tested in The paper is all still there, at the figure it turns on.

A crease pattern drawn carefully by hand will fold flat if its angles look about right, since the conditions are only approximate in real paper anyway.

What decides it: Kawasaki's condition is an exact equation, and the fraction of randomly drawn vertices that come within a tolerance of it falls in proportion to that tolerance — one power for each vertex that has to satisfy it. A Miura fold whose vertices are moved by two thousandths of a cell has none left that satisfy it at all.

Tested in Almost every pattern fails, at the figure it turns on.

If a small patch of a repeating pattern folds flat, the pattern folds flat — the patch is a fair sample of it.

What decides it: A corrugation tapered across its folds passes every condition as a single row and fails Kawasaki as four, because a one-row patch has no interior vertex for it to fail at. Of the waterbomb tessellation's 512 repeating rules, 56 pass on a two-by-two patch and 32 on every larger one.

Tested in Where the paper stops, at the figure it turns on.

A colour change is a matter of folding a flap back where the design wants a different colour, and costs about as much paper as the region it covers.

What decides it: The flap covers the paper underneath as well as being paper itself, so showing a region of the reverse removes twice that area from the front. Measured on the folded state across four flap widths: front and reverse are equal when the flap is a third of the sheet, and the whole face is reverse when it is a half.

Tested in Bringing the other side to the front, at the figure it turns on.

The best packing of equal circles into a square is symmetric, so a design search may safely be restricted to symmetric arrangements.

What decides it: With two discs the best arrangement lies on the diagonal, which is not symmetric about the vertical axis, and a search restricted to that mirror falls 14.6% short of the free one. The claim needs a group named before it means anything, and even then it is a heuristic rather than a theorem.

Tested in When symmetry costs, at the figure it turns on.

A square is the right shape for a sheet of origami paper, and the efficiency of a packing changes only gradually as the sheet is stretched.

What decides it: Efficiency against proportion is not smooth. It reaches π/4 at every aspect ratio that is a ratio of two factors of the flap count and drops well below it in between — 78.5% at 1.5 and at 6 for six flaps, 66.3% at the square. For seven flaps, which is prime, the only such proportion in the range is seven to one.

Tested in The square is a choice, at the figure it turns on.

A fold that is half a degree out is half a degree out, and a pattern of many such folds is simply a pattern that is half a degree out all over.

What decides it: A strip of thirty-two creases each half a degree out in the same direction lands its far end 0.28 panel widths from where it belongs — more than a quarter of a cell. The same size of error scattered at random lands it 0.05 adrift. One grows with the crease count and the other with its square root.

Tested in Error is folded too, at the figure it turns on.

The waterbomb tessellation is the preliminary base tiled — a grid of squares with both diagonals and both midlines in every cell.

What decides it: The vertical grid lines are absent. With them every grid corner would carry eight creases; without them it carries six, with sectors of 45°, 90°, 45°, 45°, 90° and 45°, measured off the pattern. That difference is what makes it a waterbomb rather than a preliminary base, and it is visible in the vertex census the generator computes.

Tested in The base that tiles, at the figure it turns on.

A repeating pattern can be checked on a small patch, because every vertex of the pattern is a copy of every other.

What decides it: The waterbomb's grid-corner vertex comes in four varieties according to the parities of its row and its column, and a two-by-two patch contains exactly one of them. Twenty-four of the 512 repeating rules pass on that patch and fail on any patch containing the other three, all of them on Maekawa.

Tested in A unit that folds is not a tessellation, at the figure it turns on.

A pattern that packs into a smaller space is a better pattern for a deployable.

What decides it: The Yoshimura and the waterbomb both fold about thirty-two times smaller on the same sheet, and the waterbomb needs 14.3 sheet-widths of creasing to do it against the Yoshimura's 11.8. The preliminary base folds eight times smaller for 4.8, which is a better rate than either — and it is a base rather than a corrugation, so it does not cover an area.

Tested in What a corrugation costs, at the figure it turns on.

Folding solves cubics and no more, so the hendecagon is out of reach of paper for the same permanent reason the heptagon is out of reach of a compass.

What decides it: The limit to cubics is a limit on folding one crease at a time. Two simultaneous alignments reach the quintic, and φ(11) is 10 = 2·5, so the hendecagon becomes constructible. The first polygon that defeats two simultaneous folds is the twenty-three-sided one, because φ(23) is 22 = 2·11.

Tested in Two creases at once, at the figure it turns on.

The largest equilateral triangle in a square is the one whose base is the square's side.

What decides it: That triangle has side 1 and area 0.4330. Tilting it by 15° gives side 1.03528 and area 0.46410, which is 2√3 − 3 — obtained here twice, once by maximising the support width over rotations and once by solving a quadratic for a corner construction, with no code in common.

Tested in The largest triangle in a square, at the figure it turns on.

Folding's ability to reach polygons a compass cannot is what makes it a better tool for inscribing a shape in a sheet.

What decides it: Ranked by how much of a square they use, the four best regular polygons up to sixteen sides are the eight, twelve, sixteen and fifteen-sided ones, and every one of them is compass-constructible. The best polygon a compass cannot build is the thirteen-sided one, in fifth place at 76.91%.

Tested in The biggest one that can also be folded, at the figure it turns on.

A curved-crease pattern is a smooth object whose behaviour at a junction depends on how sharply the creases curve there.

What decides it: Three vertices with identical tangent directions and quite different curvatures give identical sector angles and identical sums. Moving one tangent by seven degrees breaks the alternating sum by fourteen, and no curvature repairs it, because the curvature is not in the condition.

Tested in Where curved creases meet, at the figure it turns on.

Moving the end of a crease changes the crease pattern, so it changes whether that pattern folds. A drawing is the pattern, and everything in the drawing is part of it.

What decides it: Four sets of crease lengths on one set of sector angles, and thirteen more along a sweep of a single crease. The number of assignments that fold stays at four throughout, and the worst disagreement between Kawasaki's two alternating sums anywhere in the sweep is 1.8 × 10⁻¹⁵ radians. The folded footprint spreads by a factor of 2.86 across the four sets and 2.03 along the sweep.

Tested in The lengths are free, at the figure it turns on.

A crease pattern with every crease marked mountain or valley describes one folded object. The only question left is whether that object exists.

What decides it: A strip creased at 0.25, 0.50 and 0.75 and marked MMM admits three legal stackings, all of them the same shape on the table and all of them different piles. Across the eight assignments of those creases the counts are 3, 2, 1, 2, 2, 1, 2, 3 — every one of the eight folds, and only two of them name a single object.

Tested in More than one way to lie flat, at the figure it turns on.

A crease pattern is a drawing, so a crease may be drawn wherever it is wanted — stopping in the middle of the sheet, or meeting two others at a point, like any other line in any other diagram.

What decides it: Ninety-two interior vertices across the eight patterns in the library, at degrees four, six and eight only — fifty-nine, thirty-two and one. Nothing odd and nothing below four. All eight assignments of a three-crease vertex fail Maekawa, differing by one or by three and never by two.

Tested in Nothing meets at three, at the figure it turns on.

Cutting is what lets a sheet curve. Put a cut into a flat sheet and the paper is released, so it can take shapes it could not take while it was whole.

What decides it: A slit that removes no paper leaves the turn at the point it passes through at exactly 360°, and the sheet stays flat. A wedge of 60° leaves 300° — five sixths of a turn — and the sheet closes into a cone of half-angle 56.44°, whose sine is that same five sixths. What buys the curvature is the missing wedge and not the cut.

Tested in What one cut buys, at the figure it turns on.

A sheet that grows in both directions at once is doing something folding does. The behaviour comes out of the crease pattern, so a sheet that has only been cut — never creased anywhere — cannot have it.

What decides it: A sheet cut into squares joined at their corners measures exactly minus one at every opening sampled between 0° and 45°, to machine zero, with no crease in it anywhere. Over the same span of travel a Miura fold of comparable size runs from −0.12 to past −49, and is never the same number twice.

Tested in Bought with holes, at the figure it turns on.

A curved crease is a practical difficulty rather than a barrier. Approximate it finely enough with straight creases and the difference stops mattering, exactly as a curve drawn on a screen is a polygon nobody complains about.

What decides it: At 4, 8, 16, 32, 64 and 128 segments the total turning collected at the joints of one crease is 128.34° every time, identical to within a millionth of a millionth of a degree. Over the same range the greatest distance from the true curve falls as the segment count to the power −1.998 and the turning at each joint as the power −1.000. The picture converges; the kink does not.

Tested in A curve has no panels, at the figure it turns on.

A crease pattern that satisfies every rigid-folding condition can be built out of solid panels and folded. That is what the conditions are for — they are the definition of what a panelled assembly is allowed to do.

What decides it: A strip creased along parallel lines has no interior vertex anywhere in it, so every local condition holds with nothing to evaluate — and rolled far enough it drives one panel through another. On strips of six, ten and sixteen panels it happens at 60°, 36° and 22.5° a crease, which is 360°/n to within the half-degree the scan resolves, and the two panels then share a chord of two panel widths.

Tested in Paper through paper, at the figure it turns on.

Clearance at a hinge is a manufacturing allowance. The crease pattern says where the fold goes; how much slack the joint is built with is a decision for whoever makes the thing, and has nothing to do with the geometry.

What decides it: The travel a gap buys is fixed by the gap and by nothing else. Between panels 22 per cent of a panel length thick, a membrane hinge reaches 40.0° at a gap of 0.08, 72.1° at 0.16, 107.5° at 0.30 and 140.9° at 0.62 — measured by contact test on the outlines, and matching 180° − 2·arctan(t/g) at every value to the tenth of a degree the label carries. The error the gap has to receive is fixed by the pattern in the same units: a 0.4° bias at every crease puts the far end of a strip 0.056 panel widths out after eight creases and 0.223 after thirty-two.

Tested in Nowhere to put the error, at the figure it turns on.

A pattern very close to the Miura fold will fold very nearly like a Miura. The equations are continuous, the displacement is tiny, and a small error in the drawing means a small error in the motion.

What decides it: The exact pattern satisfies its own three equations to 6.7 × 10⁻¹⁶, which is where the arithmetic stops. The same pattern with every vertex moved by a hundredth of a panel misses them by 1.7 × 10⁻², fourteen orders of magnitude larger, after being fitted the best single panel its own edge lengths allow. Across five displacements from a millionth to a hundredth the shortfall rises as the displacement to the power 0.9999, fitted rather than assumed.

Tested in The only pattern that moves, at the figure it turns on.

A third is an awkward thing to fold, and reaching one needs an exotic operation — the fold that solves a cubic, or a halving method that only ever gets close.

What decides it: The closure of the axioms over a bare square reaches 133 references in two folds using only the three alignments that are linear in the coordinates, and every coordinate among them is a fraction, with denominators 1, 2, 3, 4, 5, 6, 8, 10, 12, 16 and 20. Thirds are in that list. They arrive from folding through two points, folding a point onto a point and folding square to a line, with no bisector and no cubic anywhere.

Tested in A fold needs something to align, at the figure it turns on.

The proportion of A-series paper is what repeated halving settles down to. Start from any rectangle, fold it in half enough times, and the shape drifts towards the familiar one.

What decides it: Halving sends a proportion of r to 2/r, which is its own inverse, so nothing drifts anywhere. A sheet at 1.3 halves to 1.538461538461538 and back to 1.300000000000000 — the return is exact to better than a part in a thousand million million — and it does that for ever. Every starting proportion but one alternates between exactly two shapes, whose geometric mean is √2 to the last digit the arithmetic carries.

Tested in The rectangle that keeps its shape, at the figure it turns on.

Adding a feature to a finished origami design means going back to the packing. The circles have to be re-laid, so the existing flaps end up somewhere else and the whole crease pattern is different.

What decides it: A strip 0.45 cells wide slid across a cut 4 cells long adds 1.8000 of paper to a sheet of 20.0000 and changes nothing else: all 44 creases away from the cut come out at the length they went in, to better than a part in a thousand million million, and the 5 that cross it are longer by 0.45 and by nothing else. Swept over seven strip widths, no untouched crease moves at all.

Tested in Paying in paper, at the figure it turns on.

A flap of a given length costs the same paper wherever it is put on the sheet. Where the flaps go is a matter of arranging them so their circles do not overlap, and the edge of the sheet is just where the arranging has to stop.

What decides it: The paper three flaps of length 0.28 actually cover, integrated over the sheet rather than taken from a fraction, comes to 0.2463 in the middle, 0.1232 on an edge and 0.0616 in a corner — the whole circle, its half and its quarter, agreeing to 2.1 × 10⁻⁸. Four flaps allowed onto the boundary reach a length of exactly 0.5000 against 0.2500 for whole discs inside the sheet.

Tested in The corner is worth four times the middle, at the figure it turns on.

The folded Yoshimura is a cylinder, so its radius is the circumference divided by 2π — and the circumference is the width of the sheet, since the course goes round exactly once.

What decides it: The course closes into a polygon of as many sides as the pattern has columns, and a polygon is shorter than the circle through its corners. On six columns the circle is 4.72% longer than the ring, so the sheet's width over 2π gives 0.159155 where the radius is 0.166667. On ten columns the error is 1.66%, and the closed form for the shortfall — one sixth of (π/N)² — accounts for both.

Tested in The cylinder the pattern chooses, at the figure it turns on.

A folded sheet shrinks by one factor, so quoting how many times smaller it gets says everything there is to say about the folded size.

What decides it: Measured along each axis, the accordion draws in 8.000 times across the sheet and 1.000 times down it — one direction untouched — while the Miura draws in 2.728 and 1.748. Both numbers are needed and neither is recoverable from the product, and a structure that has to enter a cylindrical opening is constrained by the larger of the two rather than by their product.

Tested in A shrink is two numbers, at the figure it turns on.

A twist tessellation is a design: somebody chose the polygon, chose how far to rotate it, and drew the pleats to match, which is why only a few of them exist.

What decides it: The construction takes a tiling and returns the pattern. Over four tilings and four twist angles the polygon's corner is the supplement of the tiling's own angle beneath it at all 7,396 corners tested, to 3.1 × 10⁻¹⁵ radians, and Kawasaki's alternating sums agree at every one of 630 vertices of the finished patterns to 8.9 × 10⁻¹⁵. Nothing about the polygon's shape was chosen; only its size and its rotation are free.

Tested in Any tiling makes a twist, at the figure it turns on.

A twist tessellation's twists can be any size, since the twist is a free parameter and the pleats simply take up whatever is left over.

What decides it: On a tiling with two kinds of vertex the pleat is between two unlike polygons and it demands that the two sides facing each other be the same length. On the rhombille that fixes the hexagon's sides at exactly three times the triangle's — tan 60° ÷ tan 30°, reached by a walk that never forms a tangent — and giving every twist the same size instead fails Kawasaki at all 90 interior vertices, worst residual 103.18°.

Tested in Where two twists share a pleat, at the figure it turns on.

A twist tessellation can be built at any twist angle, since the flat-folding conditions are about the angles and the angles satisfy them identically.

What decides it: Above 55.52° the pleats between neighbouring twists have negative width and there is no paper for them. Below 12.37° the angles go on satisfying every condition and the pattern has no mountain-valley assignment at all — searched exhaustively on patches of seven, eleven and twenty-three twists, at every angle down to a degree. A single twist, at the same angle, has one.

Tested in Fenced at both ends, at the figure it turns on.

Flat-foldable crease patterns are vanishingly rare, so a crumpled sheet — which is disorder itself — is a mess of vertices that could not possibly fold flat.

What decides it: A square folded along randomly chosen lines and unfolded satisfies Kawasaki at every interior vertex it has, on all five seeds tested and at every fold count from two to twelve, with the letters the folding itself wrote passing Maekawa and the big-little-big lemma too. The same square with the same number of creases drawn on it at random passes at 0 of 797 vertices.

Tested in The creases a sheet gives itself, at the figure it turns on.

How many layers deep a crumpled sheet ends up is a property of the folding and cannot be recovered from the flattened pattern, which has lost the order the creases were made in.

What decides it: The layer count of the stack and the facet count of the flattened pattern are equal on every sheet tested — seven seeds, fold counts from two to thirteen — and both equal the number of creases plus one. A sheet with 138 recorded creases has 139 facets and folds to 139 layers. Nothing about the order the folds were made in is needed.

Tested in Every facet is a layer, at the figure it turns on.

A program that reports no valid mountain-valley assignment has established that none exists, the same way a program reporting one has established that it does.

What decides it: The two are not the same kind of claim. A yes is checkable in one pass over the vertices by somebody who did not run the search. A no rests entirely on the search having been exhaustive, and this site's own solver produced false negatives for two hours — reporting no assignment for crease patterns that had demonstrably been folded flat, because a step budget shared across the whole recursion was exhausted and the exhaustion was returned as a contradiction.

Tested in A no costs more than a yes, at the figure it turns on.

A folded model's silhouette is the outline of the sheet it was folded from, rearranged — so the shape a folded object presents is the shape of the paper's edge.

What decides it: Testing every panel edge of a flat folded state for whether it has paper on one side and nothing on the other, the sheet's own boundary accounts for 29.3% of the exposed edge on a preliminary base, 26.7% on a square twist, 20.8% on a Miura and 6.7% on a Yoshimura. On a waterbomb tessellation it is 0.0%: the raw edge never reaches the outside.

Tested in The outline is mostly crease, at the figure it turns on.

Dividing a sheet into an arbitrary number of equal parts by folding is done by successive approximation, so the result is always close and never exact.

What decides it: The crossing construction reaches 1/n exactly for every n from 2 to 24, checked by comparing two integers rather than against a tolerance, in n folds. Fujimoto's halving method run alongside it is still 4.1 × 10⁻⁶ from a third after fourteen folds, with its error falling by a factor of exactly 0.5 at every one.

Tested in One crossing, and then another, at the figure it turns on.

The √2 rectangle is the unique shape that divides into copies of itself, which is why the paper standard is built on it.

What decides it: A 1 × √n rectangle cut across its long side into n strips gives n rectangles of ratio √n, for every n — checked to 4.4 × 10⁻¹⁶ over n from 2 to 8. What is unique to n = 2 is not the property but the operation: halving is the one division a folder performs without thinking, and every other member of the family needs a construction first.

Tested in One member of a family, at the figure it turns on.

Folding a map is folding its rows and then folding its columns, so the number of ways to fold an m by n map should be the number of ways to fold a strip of m by the number of ways to fold a strip of n.

What decides it: The products are 4, 12, 32, 100 and 36 for the 2×2, 2×3, 2×4, 2×5 and 3×3 maps. The actual counts are 8, 60, 320, 1980 and 1368 — ratios of 2, 5, 10, 19.8 and 38. The two directions interact, the interaction grows, and nothing in the one-dimensional answer forecasts it.

Tested in Two directions that will not separate, at the figure it turns on.

A one-degree-of-freedom vertex means one crease drives the others, so the relationship between two fold angles changes as the vertex closes — nearly flat they move together, and near the end one runs away from the other.

What decides it: The ratio of the tangents of the half fold angles is constant to 8 × 10⁻¹³ across 199 points of the motion, on four different vertices, and equals cos((α+β)/2) ÷ cos((α−β)/2) — a form the solver never evaluates, since it intersects two cones and reads the angles off the result. The fold angles themselves are wildly non-linear in one another; the tangents of their halves are not.

Tested in The vertex is geared, at the figure it turns on.

A crease pattern whose interior vertices all carry an even number of creases has a two-colouring of its panels, and therefore passes the condition every flat folded state must pass.

What decides it: A square of paper with a square hole cut out of it and three creases running from the hole to the rim has no interior vertices whatever, so every vertex carries an even number of creases vacuously. Its three panels form a cycle of odd length and take no two colours, and composing the reflections round that cycle leaves one panel 1.87 sheet-widths from where the other route puts it. The equivalence is a theorem about a disc of paper.

Tested in Even is not enough, at the figure it turns on.

Because the flat-folding conditions at a vertex are continuous statements about its angles, the number of markings that satisfy them varies smoothly as the angles are moved.

What decides it: A four-crease vertex whose two smallest sectors are equal admits eight assignments that satisfy every local condition, at every angle from 30° to 89.9°. Break the tie by a tenth of a degree and it admits four. The count is a step function of the angles, and the step is at a set of angles of measure zero which happens to contain every vertex in the standard bases.

Tested in Where the lemma says nothing, at the figure it turns on.

A folded model determines the crease pattern that made it: the creases are where the paper turns, so unfolding is a matter of reading them off.

What decides it: An exhaustive census of strips with four creases on a twelfths grid finds 233 distinct folded profiles — outline together with the number of layers over every point of it — of which 71 are produced by more than one crease pattern, and one is produced by four. Unfolding recovers the pattern; a photograph of the folded object does not.

Tested in The shadow does not name the pattern, at the figure it turns on.

The local conditions are weak because they are only necessary; given one crease's letter, what they do determine spreads across the sheet.

What decides it: Fixing one crease and propagating developability, Kawasaki, Maekawa and the big-little-big lemma to a fixed point settles three creases of a twist tessellation's hundred and fifty-eight, one of a Miura's thirty-eight, and one of a waterbomb patch's seventy-six. Between sixty-six and twenty-five interior vertices are left holding more than one labelling in each case. Nothing spreads.

Tested in How little the conditions decide, at the figure it turns on.

A small cut is a small change: what a cut does to a sheet's behaviour goes to nothing as the cut goes to nothing.

What decides it: A square of paper with three creases running from a square hole to the rim cannot be folded flat, and the two routes round the sheet leave a panel 1.8654 sheet-widths apart. Shrinking the hole from eleven per cent of the sheet's area to one part in ten thousand changes that number in no decimal place. A wedge cut of half the size takes away half the angle; a hole of half the size takes away exactly what it took away before.

Tested in A cut that removes no paper, at the figure it turns on.

The record a folding leaves on a sheet is a crease pattern, because creases are what folding makes.

What decides it: Nine random folds of the whole stack leave a pattern with no odd-degree interior vertex, at every seed tried, and it passes every condition the subject has. Nine folds that leave one layer of the stack behind leave between four and fourteen odd-degree interior vertices, and the same checker refuses the result — correctly, because a crease that stops inside the paper is not something a flat folded sheet can carry.

Tested in The crease that stops in the middle, at the figure it turns on.

A quadrilateral mesh that is developable and flat-foldable at every vertex folds rigidly, since the vertex conditions are what rigid folding needs.

What decides it: Moving one row of such a mesh sideways leaves the worst departure from Kawasaki at 5.1 × 10⁻¹⁴ degrees — exactly where it was — and takes the table of cosines at its vertices off rank one by 0.005. The sheet then folds with two panels leaving a shared corner 6.6 × 10⁻⁴ of a sheet-width apart, and the gap grows with the displacement.

Tested in The family the Miura belongs to, at the figure it turns on.

A misplaced crease is a local fault: its consequences are worst near it and fall away with distance, so a local repair is a repair.

What decides it: Moving one row of a quadrilateral mesh sideways changes every entry of that row's cosine table and leaves the flat-folding conditions untouched at 5.1 × 10⁻¹⁴ degrees. What fails is the agreement between *columns* — a quantity with no location — and the panels that come apart are not the ones next to the displacement. No change confined to the row that moved restores it.

Tested in Where an error goes, at the figure it turns on.

An optimal packing determines where every flap goes: that is what makes it the answer to the design problem.

What decides it: The optimal packing of seven equal discs in a square — reproduced here to fourteen decimal places without consulting the published value — holds six of them rigidly and leaves the seventh free to move 0.1127 of the sheet's width in some direction with nothing overlapping and the radius unchanged. Eleven discs leave two such, and fourteen leave one.

Tested in The flap nobody holds, at the figure it turns on.

Putting a design on a grid is a rounding, so the right whole-number version is the one that rounds each limb to its own nearest grid multiple, and a finer grid always rounds better.

What decides it: For limbs in the ratio 1 : 1.37 : 2.15 : 1.62, rounding each to its own nearest on a twelve-unit grid leaves the worst limb 5.9 per cent wrong; the best whole-number version at that resolution leaves it 2.4 per cent wrong, and it uses 31 units against 35. Going from six units to eight makes the per-limb rounding worse rather than better.

Tested in Spelling a tree on a grid, at the figure it turns on.

The universal molecule depends continuously on the polygon it fills, so a small change to a design gives a slightly different crease pattern.

What decides it: Sliding one corner of a pentagon along its edge, the molecule holds six creases, and then at one shape — measured at 0.2872 of the way along — three of the polygon's edges vanish at the same instant and the count jumps to seven. It does not pass through six and a half. Either side of that shape the molecule is combinatorially different, and no continuous deformation takes one to the other.

Tested in The skeleton changes its mind, at the figure it turns on.

A corrugation whose creases are not parallel folds to a curved surface, and which curved surface is a matter of how the pattern is drawn.

What decides it: For a fan corrugation, the folded straight creases pass through a single point to within 1.8 × 10⁻¹⁵ of a sheet-width, at every stage of the motion tested. The folded surface is not a general curved one and it is not a matter of drawing: it is a cone, and the only freedom is where the apex is — which moves from the sheet's own plane out to two thirds of a sheet-width as the corrugation closes.

Tested in The corrugation that curves, at the figure it turns on.

The seven axioms are seven distinct capabilities, so each of them contributes folds the others cannot make.

What decides it: On a bare square the four elementary axioms specify 38 folds between them and draw 12 distinct lines, and only the bisector of two lines contributes any line the others do not — the other three contribute nothing whatever. At the next round the perpendicular-through-a-point contributes nothing again. A round later it contributes 1,661 lines and the bisector, which carried the first two rounds, contributes 4,994 against the other two axioms' 263,703.

Tested in What each axiom is worth, at the figure it turns on.

Folding reaches cube roots, so it reaches anything a root can be written with — a fifth root is a root and enough folds will get there.

What decides it: A number a fold reaches satisfies a rational equation whose degree is a product of twos and threes and nothing else. The fifth root of two satisfies x⁵ − 2 and nothing of lower degree with small coefficients — checked here by exhausting every integer polynomial of degree below five — and five is neither a two nor a three, so no number of folds arrives at it.

Tested in The numbers a fold reaches, at the figure it turns on.

A map with a square missing has fewer squares and therefore fewer foldings than the rectangle it came from.

What decides it: A three-by-three map folds 1,368 ways. Remove a corner square and the eight that remain fold 848 ways; remove the *centre* square instead and the same eight fold 8,016 ways — nearly six times the full rectangle's count. A missing square takes its creases with it, and creases are what forbid a stacking.

Tested in The map that is not a rectangle, at the figure it turns on.

Developability, Kawasaki, Maekawa and the big-little-big lemma decide whether a single vertex folds flat.

What decides it: Over 329 random degree-six vertices, 3,208 labellings satisfy all four conditions and 2,632 of them have a flat folded state — 576 do not, on 72 of the vertices. At degree eight the shortfall is 512 of 1,440. It is zero over 373 random degree-four vertices, and it is two at every degree-four vertex whose two smallest sectors are equal.

Tested in Crimp it away and ask again, at the figure it turns on.

A crease pattern with its mountains and valleys marked determines the folded object, so two folders working from the same marked pattern produce the same model.

What decides it: At the degree-six vertex a waterbomb tessellation repeats, six of the eighteen foldable markings admit six distinct stackings each. At the degree-eight vertex at the centre of a preliminary base, 96 of the 112 foldable markings admit three or four. At every degree-four vertex tested, every foldable marking admits exactly one.

Tested in One marking, many objects, at the figure it turns on.

A folded object's outline and layer count under-determine its crease pattern only because a photograph is a poor record; a complete description of the folded object determines it.

What decides it: Adding the full stacking to the observation resolves 69 of the 71 ambiguous profiles that four creases on a twelfths grid produce, and 12 of the 47 that three creases on sixteenths produce. The rest are pairs of distinct crease patterns whose folded objects agree band for band and fold for fold.

Tested in The order does not name it either, at the figure it turns on.

A developable quadrilateral mesh that is flat-foldable at every interior vertex folds rigidly, since flat-foldability at the vertices is what rigid folding needs.

What decides it: A one-parameter family of meshes, every member developable and Kawasaki-exact at every interior vertex, closes its loop only at one value of the parameter. Five per cent of a panel away the four vertices round one face disagree by 0.041 radians about the crease they share, and at forty per cent by 0.356 — about 0.82 radians per panel length, first order.

Tested in The condition that is not flat-foldability, at the figure it turns on.

A self-folding sheet has to be biased at every vertex, because the local conditions leave each vertex free to choose and nothing carries a choice from one vertex to the next.

What decides it: On a quadrilateral mesh that folds rigidly, fixing one crease's fold angle determines all twelve creases with exactly one consistent set of angles, and on a four-by-four Miura all twenty-four. The same fixing under the flat-folding conditions determines one crease of thirty-eight on a Miura and three of eighty-four on a twist tessellation.

Tested in One crease decides the sheet, at the figure it turns on.

Panel thickness is a correction to a zero-thickness pattern: the geometry is unchanged and what is added is clearance.

What decides it: A hinge on the mid-surface, where a crease pattern draws it, gives a fold no travel at all in either direction. A hinge at a face gives it a straight angle in one direction and nothing in the other, and moving it to the opposite face exchanges the two exactly — so which face a hinge is on is set by the crease's letter, and flipping every letter is a different machine.

Tested in Thickness has a sign, at the figure it turns on.

The pleat conditions of a twist tessellation are a real constraint on any tiling, so a construction that satisfies them has done work.

What decides it: On the square, triangular, hexagonal, elongated triangular and truncated square tilings the ratio each edge demands between the twists at its two ends is exactly one, so the propagation carries the same number everywhere and every loop closes for free. Only the rhombille, whose tiles are not regular, has ratios other than one — three and a third — and they still multiply to one round every loop.

Tested in The propagation that never had to work, at the figure it turns on.

A twist tessellation's twist angle changes the pattern, so it changes how many ways the pattern can be creased.

What decides it: On a square twist patch the count of flat-foldable assignments is sixteen at every angle from 0.002 radians to 1.54, while the smallest sector falls from 89.7° to 0.7° and the paper the pleats consume falls from 24 per cent to 0.02 per cent. On a triangular patch it is 128 below 0.216 radians and 64 above, and 0.216 is exactly where two sectors tie.

Tested in The dial that decides nothing, at the figure it turns on.

A grid restricts where creases can be placed and not what kinds of vertex a design can contain, so a box-pleated design has the same vertex vocabulary as any other.

What decides it: Enumerating every developable, Kawasaki-satisfying sector sequence whose entries are whole multiples of 45° gives, up to rotation and reflection and to degree eight, exactly six vertices. At 30° there are thirty; at 60°, two; at 90°, one.

Tested in The whole alphabet of a grid, at the figure it turns on.

Grafting is additive: two features cost the sum of the two strips slid in for them, because a graft costs its width times the length of its cut and nothing else.

What decides it: Two perpendicular grafts of width 0.3 into a unit grid add 0.69 of paper against 0.60 for the two bills computed separately — an excess of 0.09, which is exactly the 0.3 by 0.3 rectangle where the strips cross. Measured at widths from 0.1 to 1.2, and at unequal widths, the excess is the product of the two widths every time.

Tested in The second term, at the figure it turns on.

A curved fold can use as much paper as the sheet provides: the rulings run off from the crease and the surface continues until it reaches the edge.

What decides it: The rulings of a curved crease meet at a distance sin β / (β′ + κ) from it. On a circle of radius 0.3 with rulings at 51.6° that is 0.235; on an ellipse, a parabola and a wave the reach is the same fraction of each curve's own tightest radius, 0.783, to within 9 × 10⁻⁴ against a neighbouring-ruling intersection computed independently.

Tested in Where the rulings run out, at the figure it turns on.

A sheet that has been folded and opened out can be folded into a second model; the old creases are a nuisance rather than an obstruction.

What decides it: Where a crease of one pattern crosses a crease of another, the vertex they make has opposite sectors equal, and Kawasaki holds there exactly when the crossing is a right angle. Over the 654 crossings the site's own printed patterns make with one another, 80 are right angles and 574 are not — and every one of the 574 is a vertex with no flat folded state.

Tested in The sheet remembers, at the figure it turns on.

What a folder can construct is bounded by what the axioms reach, so the reachable set is the right description of a folder's power.

What decides it: At three folds — with one axiom rather than seven — the axioms reach 553,823 marks on the sheet, and 94.4 per cent of them have another mark within a fifth of a millimetre on a 150 mm square. The median gap between a mark and its nearest neighbour is 0.058 mm, which is under the width of a crease.

Tested in Closer than a crease is wide, at the figure it turns on.

Seven is a property of paper — a count of the useful things one fold can do, arrived at by collecting them.

What decides it: Enumerating the ways to spend two degrees of freedom with five kinds of alignment, discarding the one combination that determines nothing, gives exactly seven without any of the seven being named. The same enumeration at two folds gives twenty-two, at three fifty, at four ninety-five and at five a hundred and sixty-one.

Tested in Seven, and then twenty-two, at the figure it turns on.

A folding question that comes back no comes back with nothing to show: the only evidence is that a search finished without finding anything.

What decides it: At a single vertex a refusal is exhibited rather than asserted. Every one of the 576 degree-six refusals in the census arrives after exactly one crimp, and every one of the 512 degree-eight refusals after one or two — and what is exhibited is a smaller vertex whose smallest sector has the same letter on both sides, which a reader checks in a moment.

Tested in A short reason to say no, at the figure it turns on.

Where the big-little-big lemma is silent the reduction has to guess, so which crimp is taken can change whether a vertex is found to fold.

What decides it: Every pair of crimps offered at once was taken in both orders and the resulting vertices compared: 1,648 such pairs in the 45° catalogue, and not one pair left different vertices. Over 428,928 vertex-and-letter pairs with sectors deliberately tied, taking the first available crimp and taking the last gave the same verdict every time.

Tested in A tie is not a decision, at the figure it turns on.

How many mountain-and-valley letterings a vertex admits depends on the sizes of its sectors, so two vertices with different angles admit different letterings.

What decides it: 759 degree-six vertices drawn at random were sorted by which of their sectors are strictly smaller than both neighbours. Inside each of the fifteen groups the admissible letterings were identical as a list, not merely equal in number, and a count computed from the arrangement alone — with no vertex present anywhere in it — reproduced every group's total.

Tested in The order decides the count, at the figure it turns on.

Folding the model from paper coloured on one side and photographing both faces of it records enough to tell it from the other patterns that fold to the same outline.

What decides it: Every crease pattern of four creases on a grid of twelfths that folds flat was grouped by the outline it folds to, and every group holding more than one genuinely different pattern was asked whether the colour separates it. Of 71 such groups, the colour separated none, and the complete layer order separated 69.

Tested in Which side is showing, at the figure it turns on.

Driving each face's closure to zero, one at a time, is what it means for a quadrilateral mesh to fold rigidly.

What decides it: A mesh whose four face residuals were each driven to 10⁻¹⁴ was put through a propagation that solves every vertex from one fold angle and requires them all to agree: it had no consistent folded state at all, because the four faces had closed on four different sets of vertex configurations. The condition is one propagation closing round every loop, and solving that is a different system.

Tested in Solving every face at once, at the figure it turns on.

A quadrilateral mesh that has been solved to fold rigidly can be manufactured to the same accuracy as any other folded sheet, because the geometry is exact either way.

What decides it: Every length of a solved mesh was changed by the same small amount, repeatedly and at four sizes. An error of a fiftieth of a millimetre on a 150 mm sheet leaves a closure mismatch of 0.002 radians; a fifth of a millimetre leaves 0.02; half a millimetre leaves no closure at all. The same experiment on a Miura, at five per cent, leaves 4 × 10⁻¹⁵.

Tested in Solved is not built, at the figure it turns on.

Since fixing any one fold angle determines every other one, an engineer putting a single actuator on a rigidly folding sheet has no reason to prefer one crease to another.

What decides it: Each of a mesh's twenty-four creases was driven in turn and the derivative of every other crease's angle with respect to it measured by central difference on the propagation itself. The worst amplification runs from 1.00 to 1.77 depending on which crease is chosen, and on a Miura the number of consistent foldings the choice leaves runs from one to eight.

Tested in Which crease to push, at the figure it turns on.

A Miura has one degree of freedom, so fixing one of its fold angles fixes the shape of the folded sheet.

What decides it: Every choice of vertex configuration consistent with a fold angle of 0.9 radians at one crease was enumerated on a four-by-four Miura. Two complete foldings survive: one 3.95 units wide with ten mountains, one 1.975 wide with thirteen, both closing at every vertex to 2 × 10⁻¹⁵. The same enumeration on a mesh with no two vertices alike returns one.

Tested in The Miura folds two ways, at the figure it turns on.

A folded sheet that opens in both directions at once has a Poisson's ratio, in the way a material does, so quoting one number for the sheet describes what any part of it does.

What decides it: The ratio was measured cell by cell off a placed folding. On a Miura the nine cells agree to two parts in ten million million and the sheet reports their number. On a mesh with no repeating cell the nine run from −3.532 to +0.438 — opposite signs — and the sheet's own figure of −0.175 is an average that no cell of it reports.

Tested in Nothing to average over, at the figure it turns on.

What limits a colour change is which panels of the crease pattern belong to the class that shows the reverse side, so a design has to be arranged to put one of them where the colour is wanted.

What decides it: Every printed pattern was folded and its footprint sampled. On six of the eight, panels of both classes lie over more than sixty per cent of the folded footprint, and on four of them over ninety-nine per cent. What varies is not whether a panel of the other class is present but how deep in the stack it is — a mean of 8 layers on the preliminary base and 60 on the Yoshimura.

Tested in Decided before the design, at the figure it turns on.

A concentric-circle pleat can be drawn with as many creases as the sheet has room for, since each circle is a valid curved crease and they nest.

What decides it: The reach of a circular crease of radius R is R sin β, measured off the envelope of its own rulings at nine radius-and-angle pairs. Two circles a distance w apart need w of surface between them, so the inner one must satisfy R ≥ w / sin β — solved by bisection off the same machinery and agreeing with the formula to five parts in a thousand.

Tested in The gap between two curves, at the figure it turns on.

A vertex where creases run off the edge of the paper is unconstrained: the theorems are silent there, so any assignment of mountains and valleys will do.

What decides it: The theorems are silent because they are about a different object, not because the object is free. A fan of paper at a boundary vertex is a one-dimensional crease pattern, and the condition that decides one is exact. A fan of three creases spanning a straight angle with sectors of 40°, 60°, 20° and 60° folds in four of its eight letterings and not in the other four.

Tested in The vertices nobody checks, at the figure it turns on.

If every vertex of a pattern can be re-lettered locally, the pattern can be: a sheet is its vertices, and what holds at each of them holds for the sheet.

What decides it: A square twist has four vertices, each of which is connected on its own, and its 256 admissible letterings fall into sixteen pieces of sixteen. A hexagon twist has six and falls into sixty-four. The count of pieces is two to the number of creases whose two ends are both interior vertices, on every pattern small enough to enumerate.

Tested in The creases that cannot move, at the figure it turns on.

A rigid folding that satisfies every closure condition describes an object that could be built out of panels and hinges.

What decides it: Six meshes solved to closure residuals between 8×10⁻¹⁴ and 4×10⁻¹² radians, each followed through its motion one branch at a time. Five are solid at every angle sampled. The sixth has two panels sharing 1.18 panel widths of the same space at every angle from 0.08 to 2.7 radians, and the two are two steps apart in the sheet.

Tested in Closing is not building, at the figure it turns on.

Designing at twenty-two and a half degrees is box pleating on a finer grid: the same vertices, spelled with smaller squares.

What decides it: The catalogues are disjoint in size and in kind. Up to degree six a forty-five-degree grid admits five kinds of vertex, a thirty-degree grid nineteen and a twenty-two-and-a-half-degree grid fifty-six — and the share of letterings whose decision needs a search falls from 60 per cent to 35 to 22 as the grid gets finer.

Tested in The other grid, at the figure it turns on.

A circle packing is specified by its radius: two packings of the same number of flaps with the same radius are the same design.

What decides it: At five and six flaps, independent runs of the same search that agree about the best radius to four decimal places return arrangements whose contact graphs carry different numbers of contacts and cannot be relabelled into one another. At seven, eight and nine they do not, which is the control.

Tested in Two packings, one radius, at the figure it turns on.

What a fold can locate is a property of the axioms: the reference points two folds reach are the same set whatever sheet they are folded on.

What decides it: At the same area and the same depth, five proportions locate different numbers of distinguishable marks. One fold: nine on a square, twenty-three on a double square, twenty-nine on each of the other three. Two folds: 565 on a square against 45,705 on the A-series rectangle.

Tested in The sheet decides which points exist, at the figure it turns on.

The shape of a crease pattern's folding set can only be found by enumerating the letterings, so it is unknowable for any pattern worth folding.

What decides it: The component count is two to the number of creases with an interior vertex at each end, and that number is one pass over the pattern's own edge list. On the printed shelf it runs from zero to forty-eight, and the three checks below confirm the prediction on patterns with 2^86 letterings.

Tested in The pieces without the list, at the figure it turns on.

The several folded states of one marking are variants of one another, so a folder who has made the wrong one can rearrange the layers into the right one.

What decides it: The smallest rearrangement a folded strip admits — slide one layer past the layer above it — is legal nowhere at all on a strip with an even number of segments, over 768 tried, and 560 of the 672 stackings measured have no move out of them by any route.

Tested in Nothing slides past anything, at the figure it turns on.

A symmetric crease pattern folds into a symmetric object, so a designer who wants a symmetric model draws a symmetric pattern.

What decides it: The preliminary base is carried to itself by all eight symmetries of the square, and not one of its 112 flat-foldable letterings survives a rotation. The square twist, drawn with the same eight, keeps both diagonal mirrors in thirty-two letterings each and neither of the other two in any.

Tested in The symmetry the letters cannot keep, at the figure it turns on.

A cut changes the pattern where the cut is: the creases it does not touch are unaffected.

What decides it: Cutting one crease of a square twist takes three creases out of the settled set rather than one, because the two vertices it releases were the far ends of two others — sixteen pieces to two. A cut along a crease that was never settled takes two out and quarters the count.

Tested in A cut is not local, at the figure it turns on.

Since the surviving repeating rules fold to one object, a folder holding one of them can be re-crimped into any of the others.

What decides it: The thirty-two rules occupy thirty-two different pieces of the patch's folding set. The buried letters differ in every pair, no move that survives the conditions changes a buried letter, and each rule's own lettering has legal changes available to it.

Tested in Thirty-two rules, thirty-two pieces, at the figure it turns on.

Cutting a hole in a sheet can only reduce what can be constructed on it, since it removes paper a crossing could have landed on.

What decides it: One round of the linear alignments reaches 9 references on a plain square and 212 on a square of the same area with a hole in it — and 133 on the plain square after two rounds. Crossings that land inside the hole are discarded, and the count is still an order of magnitude larger.

Tested in A hole is an edge, at the figure it turns on.

A folded sheet's stackings are its folded strip's stackings with more segments, so the counts and the rearrangements behave the same way and only get bigger.

What decides it: Evenly creased strips of four to seven segments admit 672 stackings between them, and 3,740 swaps tried on those find 112 legal ones. Every printed pattern here whose panels the search can order has one stacking or two, thirty-nine swaps are available across the four of them, and not one is legal.

Tested in No height to swap, at the figure it turns on.

Where the paper stops is a fact about the crease pattern and about which vertices the theorems apply to. It has no reading in the folded object, which is just a pile.

What decides it: Sort the panels of each printed pattern by whether one of their own sides is a raw edge of the sheet, and count how many other panels each lies over. The rim averages lower on all six patterns that have panels of both kinds, and equal on the seventh; it is never higher.

Tested in The rim lies over less, at the figure it turns on.

A pattern whose every vertex satisfies every condition in the subject is a pattern that almost certainly folds; the gap between the local conditions and the global question is a theoretical one that shows up in constructed counterexamples.

What decides it: Every lettering of four printed patterns, sieved twice. On the preliminary base all 112 admitted letterings fold. On the fold-and-cut triangle 18 of 30 do. On the square twist 8 of 256, and on one stated slice of the hexagon twist 1 of 64 — and the letterings this site had chosen for both twists were not among them.

Tested in The lettering that folds nowhere, at the figure it turns on.

Nothing useful can be said about whether a large crease pattern folds, because deciding it is NP-hard and the pattern has more letterings than there are atoms.

What decides it: A crease settles which of its two panels is higher, so the letters are a directed graph on the panels and a loop in it is a proof of failure obtainable in one pass. The 49-panel square twist tiling drawn here carried a loop of twenty-eight, and only seven of forty independent redraws of the same patch avoid one.

Tested in The tiling the unit could not promise, at the figure it turns on.

A crumpled sheet is the most constrained thing in the subject: it has hundreds of creases pushed together at random and is jammed into the one shape they allow.

What decides it: Run the ordering search over populations of randomly creased sheets and over the printed patterns. Every printed pattern has exactly one folded state and none of its thirty-nine available swaps is legal; crumples of four folds average 2.16 states with a maximum of nine, and six of their three hundred and thirty-five swaps are legal.

Tested in The crumple keeps its options, at the figure it turns on.

A colour change is a technique: the designer decides where the reverse of the sheet should appear and arranges the folding to put it there.

What decides it: On a pattern with one ordering of its panels, which side shows at each point is fixed by that ordering and nothing about it is free. The preliminary base shows the front over 0.1% of its footprint, the square twist over 48.6%, the hexagon twist over 90.5% — three values spread across the whole range, none of them chosen.

Tested in Which side arrives, at the figure it turns on.

A count reproduced by a second program is a count confirmed. Two routes to one number is a formality once the first one is written and tested.

What decides it: The second route here agreed with the published counts on eight of nine sizes and gave 384 against 320 on the ninth. The disagreement was a real defect — two folds wrapping one edge of the folded square, drawn in opposite directions, were being treated as facing opposite ways — and nothing else in this collection would have found it.

Tested in The map counted from the layers, at the figure it turns on.

A folded leaf's thickness follows from its area: divide the lamina by the packed footprint and that is how many layers deep the bud has to be.

What decides it: That quotient is the average depth over the footprint, and the deepest point is 1.926 times it on every leaf patch measured — eight layers against 4.15, twelve against 6.23, sixteen against 8.31, twenty against 10.39. The bud has to accommodate the deepest point.

Tested in Twice as thick where it is thickest, at the figure it turns on.

A hole helps a packing by making paper cheaper: a flap at its rim claims half a disc instead of a whole one, so the saving is the discount and the design gets that much more flap.

What decides it: The discount is not what a search finds. Two flap tips six tenths of a sheet apart carry flaps of 0.300 on solid paper — half the gap, exactly — and 0.500 across a hole four tenths wide, because their discs may overlap where there is no paper to share. The gain rises with the hole's width and has nothing to do with the rim price.

Tested in Spending the cheap paper, at the figure it turns on.

The size of a folding job is the number of creases in it. A pattern with six creases is a short job and one with eighty is a long one.

What decides it: Measure the length instead. The preliminary base has eight creases and 724 mm of folding; the square twist has twelve and 704; the Yoshimura has eighty-six and 2,380 while the Miura has thirty-eight and 1,049. Ordering the eight printed patterns by count and by length gives two different orders.

Tested in How much line is on the paper, at the figure it turns on.

A population of crease patterns that satisfy every condition in the subject is a population of crease patterns, and a measurement over it is a measurement over the subject.

What decides it: Sieve the four populations this site uses by whether their members have a flat folded state. Five of thirty-three cannot be placed at all, six place and admit no ordering of their panels, and only thirteen are known to fold. On the mesh population it is two of six.

Tested in The patterns a checker is tested on, at the figure it turns on.

Self-intersection is found by simulating a fold and watching for it. There is no way to know a pattern must pass through itself without following its motion.

What decides it: At the flat state, two panels passing through one another is exactly the failure of the non-crossing rules, and those are decidable by an exhaustive search over orderings of the panels. Four of six quadrilateral meshes here are refused by it — a proof, from a search that follows no motion and computes no fold angle.

Tested in A collision is an order, at the figure it turns on.

A deployable pattern is chosen because it packs well, so the patterns used in hardware are the ones that convert folding into compaction most efficiently.

What decides it: Divide each printed pattern's total crease length by the compaction it achieves. The Yoshimura gives sixty layers for fourteen sheet-widths of crease, the waterbomb thirty-one for fourteen, and the Miura — the one that flew — gives nine for six, which is fourth of eight. The three patterns above it in the ranking and the four below it are all patterns nobody deploys.

Tested in Fourth of eight, and still not chosen for it, at the figure it turns on.

Dividing a square into an n-by-n grid is elementary. The construction is exact, the answer is a rational number, and there is nothing left to say about the result.

What decides it: The result is an n-by-n map of stamps. How many ways it folds flat is 8 at two, 1,368 at three, 300,608 at four and unknown at five, and there is no formula for any of them. The construction is closed and the object it makes is open.

Tested in The grid a division makes, at the figure it turns on.

A rule of thumb that gets the right answer has the right reason behind it, or near enough — a wrong explanation would show up as a wrong result.

What decides it: Three explanations in circulation here get their conclusions right and their mechanisms wrong, and the third put two unfoldable patterns on this site's own printed shelf for eleven years. Every gate stayed green throughout, because a gate checks the conclusion and nothing checks the reason.

Tested in Taught with a wrong reason, at the figure it turns on.

A crease pattern is the list of its vertices, edges and assignments. Checking the conditions at every vertex in that list checks the pattern.

What decides it: Subdivide every pattern this collection draws wherever two of its segments meet, and count the vertices the subdivision has to invent. The eight printed patterns gain none. Four of the six tessellation patches gain 12, 18, 12 and 5 — and those four are exactly the four whose panels cannot be placed in the plane at all, while every vertex in their lists passes every condition.

Tested in The vertex the list does not have, at the figure it turns on.

Two creases drawn across each other are just four creases meeting at a point, which is the commonest kind of vertex in the subject and folds perfectly well.

What decides it: Sweep a crossing through eight angles and every way of lettering its two lines: thirty-two vertices, of which four satisfy Kawasaki's condition, eighteen satisfy the big-little-big lemma, all thirty-two are developable, and none satisfies Maekawa's. Letter the four spokes independently instead and eight of a hundred and twenty-eight fold — every one of them a right angle whose letters change across the point, which is four creases meeting rather than two crossing.

Tested in Two creases that cross, at the figure it turns on.

A patch of a tessellation is the tessellation on a smaller piece of paper. Which units are kept and how the outermost pleats are finished is presentation rather than mathematics.

What decides it: The same five tilings, cut to the same square two ways. Keeping whole units and running the outstanding pleats to the rim gives 0, 12, 18, 12 and 5 places where two creases cross, and panels that cannot be placed at all on four of them. Generating over a larger region and clipping gives no crossing on any of them and panels that place to a part in 10^14.

Tested in Cutting a patch out of a plane, at the figure it turns on.

The edge of the sheet is a detail at the margin of a crease pattern. What a pattern is about happens in the middle of it, and the boundary is a small correction.

What decides it: Count the vertices of every pattern drawn here and sort them by whether they sit on the paper's edge. The share on the edge runs from 39% on the waterbomb tessellation to 91% on the fold-and-cut triangle, and on the twist tessellation patches — the largest patterns here — it is between 34% and 44%. On the same patches, between a quarter and nine tenths of the repeating units that appear on the paper are cut by its edge rather than held whole.

Tested in Most of a patch is edge, at the figure it turns on.

A patch of a corrugation is a sample of the material. Measure its packing ratio or its Poisson's ratio and the number is the tessellation's number, up to the noise of a small sample.

What decides it: Fold one tiling at six unit sizes on the same square and measure the average layer count over the folded footprint. It climbs from 3.89 to 4.79 on the square grid, 3.92 to 4.80 on the triangular, and 4.05 to 4.88 on the honeycomb — a rise of a quarter, in one direction, tracking the share of units the paper's edge cuts rather than any noise.

Tested in The property a patch does not have, at the figure it turns on.

A tessellation is drawn on whatever paper is available. Which sheet it goes on is a presentation decision, like how large to print it.

What decides it: The rectangle a Miura of c columns and r rows at slant α occupies has proportion (c + tan α) / r, checked against the bounding box of the drawn pattern on six sizes and agreeing to a part in 10^16. A square therefore requires tan α to be a whole number — 45°, 63.43°, 71.57° — and at the 20° slant this collection draws, the six sizes come out between 0.79 and 2.18.

Tested in The paper a pattern asks for, at the figure it turns on.

The fold-and-cut theorem is about any straight-line drawing, so a construction that implements it handles any straight-line drawing.

What decides it: The construction here handled convex outlines only, because a reflex corner needs a split event in the shrinking wavefront. With split events implemented it reaches an L and a star — and it still refuses a five-cornered dart, where four of the nine perpendiculars a node wants fall off the ends of the edges they were dropped onto and leave vertices of odd degree, which no assignment of mountains and valleys can fold.

Tested in One cut for a star, at the figure it turns on.

A pattern that cannot fold flat cannot be folded at all, so a crossing is simply an impossible piece of drawing with nothing more to say about it.

What decides it: Compose the rotations round a crossing and ask which directions out of the flat state keep the composition at the identity. Two survive, and each of them moves two of the four creases and leaves the other two at zero — the two simple folds along the two straight lines. Both are exact rigid motions of four flat panels; neither can reach a flat folded state, and the two cannot run at once, which the closure reports as a residual of 0.095.

Tested in Two mechanisms at one point, at the figure it turns on.

The universal molecule fails at a reflex corner because the straight skeleton cannot be computed there. Compute the skeleton and the molecule follows.

What decides it: The skeleton of a non-convex outline is computed here now, with split events, and it passes the same equidistance assertion the convex case does on every shape tried. The fold-and-cut construction uses it and reaches an L and a star. The molecule still does not exist for those outlines, because a molecule is a crease pattern for one piece of paper and a split is the moment the shrinking region becomes two.

Tested in The corner that splits the shrink, at the figure it turns on.

What matters about removing paper from a sheet is how much is removed. A hole and a bite out of the edge of the same area leave a designer in the same position.

What decides it: Price both against plain squares of the same area, by integrating the paper a flap claims at every point. The hole saves 1.91%, 4.71% and 9.84% at flap lengths of 0.06, 0.15 and 0.30 of a sheet; the notch of identical area saves 1.11%, 2.58% and 4.73%. And four bites of equal area but different shape spread over a factor of 4.6, from 1.38% for a wide shallow one to 6.31% for a deep narrow one.

Tested in A notch is not a hole, at the figure it turns on.

A checker that passes every member of a large and varied test population has been tested against the faults it might meet.

What decides it: A fault absent from every member is a fault untested. Random drawings of two segments carry a crossing 23% of the time and of twelve segments effectively always, with fifteen crossings on average; the thirty-three patterns in the four populations here carried none until a construction produced one, at which point the count went straight to thirty-two on a single patch.

Tested in Drawn by the same hand, at the figure it turns on.

A cut removes paper, so what it does to a pattern is whatever removing that paper does. A slit that removes no paper at all removes nothing.

What decides it: A ring with three, five, seven, nine or eleven creases across it places its panels 1.75, 1.59, 1.43, 1.32 and 1.24 sheet widths apart — no folded state at all. Cut it open with a slit of zero width, from the hole to the rim, and every one of them places to rounding and two-colours. No paper was removed, no crease was moved, and no letter was changed.

Tested in A cut that reaches the edge, at the figure it turns on.

Each of the seven operations describes a fold. Given the points and lines it asks for, the fold exists and can be made.

What decides it: Over four thousand alignments drawn at random inside a square: placing a point on a line through a second point names no fold at all 23.8% of the time and two folds 76.2% of the time; placing a line on a line always names two, of which 13.3% are creases that never touch the paper. Only three of the five tested name exactly one fold and land it on the sheet every time.

Tested in An axiom may name no fold, at the figure it turns on.

A crease pattern file is the pattern. If a file is well-formed and its vertices satisfy the conditions, the pattern it describes can be folded.

What decides it: Four of this collection's own tessellation patches exported as valid files, with every vertex satisfying every condition — and carried 12, 18, 12 and 5 places where two edges cross at a point neither of them names. Nothing in a list of vertices, edges and letters can forbid that, because the crossing is not in the list.

Tested in A file has no paper, at the figure it turns on.

Reading a crease pattern is reading which lines are there and which way each one folds. Everything a reader needs is on the page.

What decides it: At every place one crease ends on another — forty-six of them across the five printed patterns that have any — the drawing is identical to two creases crossing, and the two readings differ by whether the pattern folds at all. The reader decides, correctly and without noticing, and no notation records which reading was meant.

Tested in The reader decides the junction, at the figure it turns on.

A repeating fold pattern on a finite object has to stop somewhere, and stopping is stopping — the edge of a leaf is where its corrugation was cut off, in the same way a patch of a tessellation is where the drawing ran out of paper.

What decides it: A corrugation's vertex angles depend on the row heights and not on the column widths, so the widths are free — every one of them, independently. Tapered by a factor of eighty-two from the middle column to the outermost, the pattern still folds with a worst Kawasaki residual of 4×10⁻¹⁶ and no crossing or stub anywhere. A tessellation patch cut to a square has neither property.

Tested in A leaf ends its pattern, at the figure it turns on.

Whether a crease pattern has a flat folded state can only be settled by searching the orders its panels might take, so a pattern too large to search is a pattern nothing can be said about.

What decides it: A crease fixes which of its two panels is above the other before any search begins. Collect that one statement per crease and look for a circle in what they demand: a circle is a proof that no order exists, and finding one costs a single pass. On a rhombille patch of a hundred and fifty-seven panels — where the search refuses to start — the pattern's own lettering closes a circle of sixteen.

Tested in A proof in one pass, at the figure it turns on.

A pattern whose letters demand an impossible order of the layers has a fault somewhere in it, and the fault can be found by looking hard enough at one vertex.

What decides it: Enumerate every labelling of a single interior vertex at degrees four, six and eight. A hundred and fifty of the three hundred and thirty-six satisfy every condition, and none of them puts its panels in a circle — because the only labelling that could is the strict alternation, and Maekawa's count refuses it at every degree while Kawasaki and the big-little-big lemma both hold.

Tested in The loop a vertex cannot close, at the figure it turns on.

A contradiction in a pattern's letters is a small local fault: a search reports a circle of eight or ten panels, so eight or ten panels are what is wrong.

What decides it: The circle a depth-first search reports is the first one it happens to meet. The panels that lie on some circle are the non-trivial strongly connected parts of the same graph, and on a square patch that is thirty-five of forty-nine panels with fifty-two of its eighty-four arrows inside — against a reported circle of eight.

Tested in The loop is not the tangle, at the figure it turns on.

The circles a layer relation can contain are whatever the pattern's panels happen to allow, so their lengths are a property of the drawing and carry no information.

What decides it: Over 1,149 circles measured across every population here, not one has an odd number of panels and not one has four. Odd is closed by the two-colouring of the panels, which follows from a crease turning the sheet over; four is closed by Maekawa's count. The observed shortest is six, on every family.

Tested in A contradiction is even, at the figure it turns on.

A test that proves a crease pattern cannot fold, and proves it in one pass, makes the exhaustive search over orderings unnecessary for anything but confirmation.

What decides it: Enumerate every labelling of the square twist. 256 pass every vertex condition, 8 have a folded state, and of the 248 failures the one-pass test catches four. On the fold-and-cut triangle it catches none of the twelve. Where it is silent the search is not a confirmation; it is the only thing that answers.

Tested in Consistent is not foldable, at the figure it turns on.

A tessellation patch that will not fold is one that has been drawn too large or twisted too far, and pulling either dial back brings it into range.

What decides it: The turn angle changes nothing at all: thirteen consistent letterings of a hundred and twenty at every turn from 0.15 radians to 1.0. Size changes a great deal but is not the quantity — a Yoshimura of sixty-five panels is consistent in ninety-five per cent of its draws and a square patch of forty-nine in thirteen. What separates them is twenty-two independent chains of panels against thirty-six.

Tested in Letters that agree get rarer, at the figure it turns on.

Maekawa's count and the layer ordering are separate questions about a pattern: one is about the letters at a point and the other about what the layers can do, and neither has anything to say about the other.

What decides it: Over all 512 waterbomb rules, 120 close a circle of exactly four panels round one interior vertex, and every one of those 120 fails Maekawa at that vertex. None of the 32 rules that satisfy the count closes a circle of any length. The count is what makes the shortest layer contradiction unavailable.

Tested in The rule that breaks the count, at the figure it turns on.

Ranking tests by cost means timing them, and a test that is faster on the cases anybody runs is the cheaper test.

What decides it: Counted in operations rather than seconds, the crossing sweep is quadratic in the creases and costs 39,621 comparisons on the largest patch here, while the layer refusal reads 282 creases once. The sweep is the cheapest by wall clock on small patterns and the most expensive of the four polynomial tests on large ones, and a timing would have hidden the crossing entirely.

Tested in The refusal that reads the list once, at the figure it turns on.

A test that fires on no member of a well-assembled population of crease patterns is a test with nothing to do.

What decides it: The layer refusal fires on none of the thirty-three, because every member was produced by a construction that returns a lettering and a returned lettering is one that works. Reletter them and the same population runs from a hundred per cent down to seven, with the hexagonal twist consistent in four redraws of sixty.

Tested in A population that cannot fail, at the figure it turns on.

A crease pattern that came from an actual folding is a pattern that folds, so its drawing carries the guarantee.

What decides it: The guarantee is carried by the letters and not by the drawing. Every crumple here arrives with letters that agree, and relettering the identical creases takes the share that agree from a hundred per cent at eight panels to twenty-eight at thirty-nine. The lines are unchanged; only what was written on them moved.

Tested in The letters a crumple was given, at the figure it turns on.

Cutting one crease is the smallest possible change to a crease pattern's layer relation: it deletes one constraint and leaves everything else exactly as it was.

What decides it: It deletes one constraint and one determination. The two panels the crease joined are no longer held together by a fold, so the composition of reflections stops deciding where the far one goes. Of 474 single cuts across four patches, 16 leave a sheet whose panels place, and not one cut of a crease with an interior vertex at each end does.

Tested in One cut removes one arc, at the figure it turns on.

A cheap test that proves a crease pattern cannot fold is a first pass at the expensive one — it catches the easy cases and the search mops up.

What decides it: On six developable quadrilateral meshes the search refuses four and the cheap test refuses none. The four it misses are not hard cases of what it looks for; they are patterns whose letters agree completely and whose panels cannot be stacked for geometric reasons the letters never mention.

Tested in Two refusals that refuse differently, at the figure it turns on.

An axiom of paper folding specifies a fold, so writing one down is writing down a crease.

What decides it: The third axiom names two folds — the two angle bisectors of the lines it is given, always perpendicular to each other — and over four thousand random pairs on a square, 2,965 leave both of them on the paper. The fifth names two whose angle to each other is anything from half a degree to ninety. Neither statement chooses.

Tested in The axiom that names two folds, at the figure it turns on.

A fold is worth what it adds to the reachable points, so counting them measures what a construction buys.

What decides it: From one fold to two the count of reference points rises from nine to five hundred and sixty-five — sixty-three times — while the largest distance from anywhere on the sheet to the nearest reference falls from 0.354 of a sheet to 0.089, which is four times. The two quantities do not move together and only the second is about reach.

Tested in How far from the nearest reference, at the figure it turns on.

A crease pattern file that validates is a record of a foldable object — the format would not accept it otherwise.

What decides it: All 256 admissible letterings of the square twist make valid files and 248 of them describe an object with no folded state. The format's optional face-ordering field is the only place a verdict could live, it is almost never written, and nothing about a file with it missing is invalid.

Tested in The file records no verdict, at the figure it turns on.

What a folder needs to know about a crease pattern is the conditions at its vertices; everything else follows from folding carefully.

What decides it: On the square twist, four labellings of 256 are refused by their letters and 244 more by the layers. The vertex conditions account for none of the 244, and no teachable rule has ever existed for them — the only test is a search over orderings, which does not finish past about twenty panels.

Tested in The first thing about layers, at the figure it turns on.

A leaf's corrugation is shaped by its taper, so the taper is what its folding properties depend on.

What decides it: Four width profiles — even, the printed taper, a strong taper and a one-sided ramp — give 174 consistent letterings of 200, the same number in every case, on a pattern with eighteen chains. Growing the rows takes it 200, 181, 174, 158, 126 as the chains go 6, 12, 18, 24, 30. The taper is invisible to the question.

Tested in The taper decides nothing, at the figure it turns on.

Two thousand independent letterings of a crease pattern, none of which agrees with itself, is good evidence that the pattern has no consistent lettering.

What decides it: The rhombille patch is exactly that case, and a search that applies the same test during the choice rather than after it produces a consistent lettering in five hundred and sixty-one steps — twenty different ones from twenty different starting orders, each checked afterwards by the machinery that could not find it.

Tested in The lettering nobody could draw, at the figure it turns on.

The four conditions at a vertex are what makes finding a flat-foldable assignment difficult, so a search for one spends its effort satisfying them.

What decides it: Over five tessellation patches carrying between thirty-six and a hundred and twenty-six interior vertices, the four conditions dead-ended a search zero times. Every one of the two hundred and fifty-nine backtracks recorded was a circle in the arcs the letters force, which is a condition no vertex can see.

Tested in Which condition does the refusing, at the figure it turns on.

A crease pattern's mountain-valley assignment is a property of the pattern, so two correct letterings of the same drawing agree about most of it.

What decides it: On five tessellation patches, the lettering a search returns differs from the one the construction produced on forty-five of eighty-four creases, sixty-six of a hundred and six, sixty-seven of a hundred and forty-two, eighty-seven of a hundred and forty-two, and a hundred and fifty-five of two hundred and eighty-two — and on four of the five, both letterings pass every condition and neither is wrong.

Tested in One solution of a search nobody ran, at the figure it turns on.

A pattern whose letters contradict themselves can be repaired by folding: push the right vertices through and the contradiction goes away.

What decides it: Every move the five patches admit was tried, from the lettering each patch arrives with and from a lettering found by search. Nineteen moves in two thousand nine hundred and sixty-four candidates survive the conditions at a vertex, and every one of the nineteen leaves the lettering on the same side of the contradiction question it was already on.

Tested in Every move leaves the verdict, at the figure it turns on.

A crease pattern's crease count is a count of creases, so it can be quoted about the pattern without further checking.

What decides it: The hexagonal tessellation patch reports a hundred and forty-two creases and produces a hundred and thirty arcs. The twelve that carry no arc are between eight millionths and six hundred-thousandths of a sheet long — a micrometre or so on paper — and they appear at one turn angle of four, which is the one the collection draws.

Tested in Twelve creases a micrometre long, at the figure it turns on.

The Miura fold's assignment is a single rule that has to be learnt as a whole: rows alternating one way, columns the other.

What decides it: Sixty-four repeating rules exist for the pattern and sixteen of them fold. All sixteen make a column crease change letter at every row; all four ways of writing the row letters appear among them, in every combination. The rule has one bit in it, and the row half of the usual recipe is decoration.

Tested in Sixty-four rules, sixteen fold, at the figure it turns on.

How hard it is to letter a pattern is a property of the pattern, so a well-defined search on a fixed pattern costs what it costs.

What decides it: On the rhombille patch the same search costs eighty-four nodes with one order of decisions and more than twenty thousand with another, over a hundred and twenty runs that differ in nothing else. On the four other patches of the same family the same hundred and twenty runs vary by less than a factor of two.

Tested in Four easy patches and one that is not, at the figure it turns on.

Whether a repeating rule can produce a layer contradiction depends on the rule, so a large enough rule space will contain some that do.

What decides it: The Yoshimura's sixty-four rules produce thirty-eight failures and not one closes a circle, because every interior vertex has degree six and a repeating rule gives the two halves of the course through it one letter. The alternation a circle needs would require those two halves to differ, and no rule in the family can write it.

Tested in Where a rule can close a loop, at the figure it turns on.

A search that has already run for a long time is closer to finishing than one that has just started, so the sensible thing is to let it run.

What decides it: Over a hundred and twenty runs on the same pattern, a fifth finish inside a hundred nodes and two fifths do not finish inside twenty thousand. Stopping every attempt at a hundred and restarting costs five hundred and twelve nodes in expectation; allowing twenty thousand costs sixteen thousand two hundred and ninety-one.

Tested in Stopping is cheaper than finishing, at the figure it turns on.

A test population assembled to be representative will show up the difference between two methods that answer the same question.

What decides it: Across four populations and twenty-eight patterns, sampling forty letterings and searching for one give the same answer on every member, and the search's worst case is sixty nodes. The patterns where the two differ — two of which the sampler cannot settle at all — belong to no population, because they are what the construction produces when it is asked to clip.

Tested in Four populations with nothing to separate, at the figure it turns on.

Whether a crease pattern folds flat is decided by its combinatorics — how its panels and creases are joined — with the angles supplying detail rather than verdicts.

What decides it: One tessellation patch at two turn angles a thousandth of a radian apart has the same panels, the same creases, the same chains and the same number of labellings at every vertex. At 0.2155 radians it has no labelling at all, proved by exhausting the search; at 0.216 one is found in forty-five steps.

Tested in Where a sector crosses sixty, at the figure it turns on.

Map folding is hard because a map's creases make demands about the layers that contradict each other, so the cheap test that finds such contradictions should be informative about it.

What decides it: Every labelling of every map up to three by three was enumerated — four hundred and fifty-four labellings that pass every condition at every vertex — and four of them close a circle in the arcs, all four on the largest map. On the eight smaller maps the test fires zero times.

Tested in The test that never fires on a map, at the figure it turns on.

A pattern whose consistent labellings are rare is a pattern where one is hard to find, so the share that agree is a measure of difficulty.

What decides it: A sheet crumpled to eight folds has eleven of forty random labellings agreeing with themselves, against thirty-four of forty at four folds — and the search finds one in seventy-two steps on a pattern with seventy-one panels, having taken back two letters. The share falls by two-thirds and the cost per panel does not move.

Tested in Rare is not hard, at the figure it turns on.

The set of points a folder can reach is a well-defined object, so it can be computed to whatever depth is wanted.

What decides it: The third round from a bare square specifies three hundred and seventy-eight thousand folds, two hundred and seventy-four thousand of them distinct, and their pairwise crossings number in the tens of billions. The closure is refused rather than computed, and what replaces it is a sample and a bound.

Tested in The third fold cannot be listed, at the figure it turns on.

The Miura fold's assignment is learnt as a pair of rules — one for the rows, one for the columns — and both have to be got right.

What decides it: Of the sixty-four repeating rules, sixteen fold flat. All four ways of writing the row letters appear among them, paired with all four ways of alternating the columns. The row half of the recipe rules out nothing at all.

Tested in Half the recipe is decoration, at the figure it turns on.

A box-pleated design's mountain-valley assignment is a detail to be settled once the grid and the flaps are decided, and a designer can fill it in by working across the sheet.

What decides it: On a sixteen-by-sixteen grid — the scale a box-pleated base is drawn at — one labelling in a hundred drawn under the vertex conditions has no circle in the arcs it forces. The share falls from a hundred in a hundred at two divisions, through seventy-three at six, to one at sixteen.

Tested in What a grid costs in circuits, at the figure it turns on.

A leaf's corrugation is a biological pattern, so its mountain-valley assignment is something the plant's own geometry arrives at.

What decides it: The corrugation's sixty-four repeating rules were built and checked at six different geometries — the printed proportions, even columns, a one-sided ramp, a violent taper, taller rows and a steeper zigzag. The same sixteen rules survive every time, by number, and they are the Miura fold's sixteen.

Tested in The leaf's rules are the Miura's, at the figure it turns on.

A backtracking search whose cost varies by three orders of magnitude between starting seeds has found a hard instance.

What decides it: The same search on the same patch, with the choice of which letter to try first replaced by a constant, costs eighty nodes on every seed — against a median of two hundred and fifty and a worst run of fifteen thousand eight hundred and seventy-two. The spread was the search's own coin.

Tested in The difficulty was in the coin, at the figure it turns on.

Two searches that differ in which letter they try first will generally cost different amounts, so measuring one of them is measuring an arbitrary choice.

What decides it: Swapping every letter on the sheet at once leaves Maekawa, big-little-big and Kawasaki exactly as they were and reverses every arc, so the admissible letterings are closed under the swap and the two searches walk mirror trees. Measured on a hundred and forty-two patterns: identical step counts on every one.

Tested in The order that is its own mirror, at the figure it turns on.

A backtracking search is made faster by choosing better variables to branch on, and the crease pattern's own structure — which creases lie on many independent circuits — is the right thing to choose them by.

What decides it: Branching on circuit participation is never cheaper than the standard rule on any of the eighty-seven patches that have a lettering, and is worse on fifty-two of them, turning one patch's worst case of fifty-one steps into two thousand three hundred and ninety-four.

Tested in Which choice the cost lives in, at the figure it turns on.

Branching on the creases that lie on many independent circuits is a better rule for proving that a crease pattern has no consistent lettering.

What decides it: It proves three patches in sixty-three steps against a quarter and a half million for the standard rule, and on nine others it needs 879 to 524,287 steps against the standard rule's fifteen. Neither rule dominates; the pattern decides.

Tested in The order that proves nothing exists, at the figure it turns on.

A search made deterministic can keep most of its variety by randomising only the decisions that are genuinely free — the creases no vertex constrains.

What decides it: Doing exactly that returns four distinct letterings of twenty on two of the five patches and one on the other three, against forty of forty for a randomised search. The variety lives in the branch decisions, which is where it is expensive.

Tested in One witness or forty, at the figure it turns on.

A crease too short to see is a blemish that can be removed by deleting it, leaving the pattern otherwise unchanged.

What decides it: Deleting the hexagonal patch's twelve gives exactly the hundred and thirty creases and fifty-four interior vertices that were predicted, and a sheet whose panels no longer close: two routes to one panel disagree by more than two sheet widths.

Tested in The crease the drawing cannot show, at the figure it turns on.

A pattern with a blemish in it is a pattern drawn slightly wrongly, and the repair is to correct the drawing.

What decides it: The drawing is correct at every pitch. The blemish appears only at the one pitch where a ring of twists is halfway out of the sheet — 0.335 gives a hundred and forty-two creases and no fragment, 0.345 gives a hundred and thirty and no fragment, and the printed 0.34 gives a hundred and forty-two with twelve of them in pieces.

Tested in A patch on a knife edge, at the figure it turns on.

A pattern family that loses its consistent letterings does so at a threshold in one parameter, so one sweep of that parameter finds the whole of the failure.

What decides it: Sweeping turn angle alone finds one threshold. Sweeping turn angle against pleat width finds nine patches with no lettering across three tilings, in a corner of the plane rather than along a line, and the pleat width moves the threshold as much as the turn angle does.

Tested in A region with no lettering, at the figure it turns on.

Measuring five representative crease patterns from a family tells you what the family does.

What decides it: Ninety-six patterns from a stated grid over the same family contain nine with no consistent lettering — a case none of the five is anywhere near — and show that the heavy search cost the five exhibited belongs entirely to one of them.

Tested in A population nobody chose, at the figure it turns on.

The five tilings a twist tessellation can be built on are five examples of one construction, so a measurement across them is a measurement of the construction.

What decides it: Four of them have vertices that are all alike and one does not. On that one the side distances have to be propagated rather than assumed, and it is the only tiling whose search has a tail, the only one whose shallow patches take minutes to draw, and the only one where the drawing depends on where the propagation started.

Tested in The dial and the tiling that is not alike, at the figure it turns on.

A crease pattern whose consistent letterings are rare is a crease pattern on which one is hard to find.

What decides it: On the orthogonal grid the share of drawn letterings that agree falls from a hundred of a hundred to one of a hundred as the grid goes from two divisions to sixteen, and the cost of finding one stays at exactly one step per panel — two hundred and fifty-six steps on two hundred and fifty-six panels, with no backtracking at all.

Tested in A corrugation never backtracks, at the figure it turns on.

A disordered crease pattern is a harder instance for a search than a regular one.

What decides it: Six crumples of deepening severity cost one step per panel apiece with no backtracking, which is exactly what the orthogonal grid and the tapered leaf cost. The only patterns in this collection whose search cost is not linear are the most regular ones it draws.

Tested in A crumple has no tail, at the figure it turns on.

Replacing a search's discrete choices with a continuous solve removes the difficulty rather than moving it.

What decides it: The ordering questions vanish and a new failure arrives in their place: a solve can fail to converge, can converge to a state that is not the wanted one, and returns a number rather than a verdict — so a refusal has to be distinguished from a non-convergence, which a discrete search never has to do.

Tested in The motion has no letters to choose, at the figure it turns on.

A box-pleated crease pattern whose every vertex satisfies the flat-folding conditions is a pattern that folds flat.

What decides it: At sixteen divisions, every one of a hundred letterings drawn to satisfy those conditions everywhere does so, and one of the hundred has no circle in the arcs its letters force. Ninety-nine pass the check and cannot be folded.

Tested in Ninety-nine in a hundred pass, at the figure it turns on.

A crease pattern is a hard instance for a search when it is large, irregular or disordered.

What decides it: Two hundred and fifty-six panels of orthogonal grid, seventy-one panels of random crumple and sixty-five of Yoshimura all search at one step per panel with no backtracking. The only patterns here that behave otherwise are tessellation patches clipped to a square sheet, which are neither the largest nor the least regular.

Tested in The edge is what makes it hard, at the figure it turns on.

A biological folding pattern is impressive because the organism has solved a hard combinatorial problem.

What decides it: The leaf's corrugation costs one step per panel with no backtracking at any geometry tried, its rule table is identical to the Miura's at six deliberately awkward geometries, and its letters are forced by its shape. The plant's difficulty is entirely in growing the geometry, and none of it is in choosing the letters.

Tested in The plant's pattern is not a hard case, at the figure it turns on.

A phenomenon measured on this subject's own objects, with this subject's own machinery, is a finding of this subject.

What decides it: The heavy tail, the restart arithmetic and the reason restarts work were all imported whole from the study of randomised search. What a crease pattern contributed was an instance — and the instance turned out to show the borrowing had brought the disease along with the cure.

Tested in The cure was named first, at the figure it turns on.

A tessellation patch is expensive to search because it has a rim: the vertices near the edge have creases running off the paper, so the propagation stalls there.

What decides it: The same rectangle of the same drawing, with the rim removed by identifying opposite edges, costs fifty-six thousand seven hundred and seventy-two steps against the cut version's forty-eight — on sixty-four panels, with the same sixty-four vertices asked the same conditions.

Tested in What the rim was doing, at the figure it turns on.

A patch cut out of a tessellation has fewer constraints than the tessellation because its rim vertices are short of creases and are not asked their conditions.

What decides it: On every cell measured here the cut rectangle and the glued one ask the identical number of vertices — sixteen and sixteen, sixty-four and sixty-four — because the cut is placed between the vertices. What the cut adds is letters, not missing questions.

Tested in The rim is four letters a cell, at the figure it turns on.

A vertex whose sectors are unequal has a smallest sector, so the big-little-big lemma applies to it.

What decides it: A Yoshimura vertex at a row height of 1.2 half-columns has sectors of 50.2° and 79.6°, which are unequal, and the lemma says nothing at all — because the four small sectors are arranged in adjacent pairs and none is strictly smaller than both its neighbours.

Tested in A knife edge nine decimals wide, at the figure it turns on.

A patch's difficulty depends on which creases its rim happens to cut, so a patch cut in a different place is a different problem.

What decides it: Twelve cuts across one period of the same square tessellation divide different creases, leave between forty-nine and sixty-one panels, and cost between twenty-five and thirty-three steps — a factor of 1.3, against a factor of more than a thousand for removing the cut.

Tested in Where you cut hardly matters, at the figure it turns on.

A tessellation patch is a difficult crease pattern because its structure is intricate.

What decides it: Every interior vertex of every patch here admits exactly four labellings of its creases, which is half what a grid, a leaf, a Miura or a crumple admits — the most constrained vertex in the collection, on all five tilings.

Tested in The most decided vertex here, at the figure it turns on.

A tessellation patch is the one family off the one-step-per-panel line, and it is off it because a rim makes a search work hard.

What decides it: Under a fixed letter order the five patches cost 0.53, 0.47, 0.51, 0.52 and 0.51 steps per panel against the grid's, the leaf's, the Miura's and the crumple's 1.00 — off the line on the cheap side, and joining their edges so the rim is gone costs a thousand times more.

Tested in The edge was not what made it hard, at the figure it turns on.

The orthogonal grid is the simplest crease pattern here, so it is the cheapest to reason about.

What decides it: The grid costs exactly one search step per panel at nine sizes, which is the highest constant in the collection: a twist patch costs 0.5 and a tilted Yoshimura 0.23 on the same measure.

Tested in The designer's grid is the dearest thing here, at the figure it turns on.

A crease pattern's boundary is a frame around the interesting part, so what happens at it does not affect what the pattern is.

What decides it: A leaf's pattern, whose creases end where the plant put them, costs one search step per panel at four geometries; a tessellation patch, whose rim was cut through the middle of its own structure, costs half that on five tilings.

Tested in Nothing grown was cut out of anything, at the figure it turns on.

A Möbius band folds flat when it has an odd number of creases across it, in the same way that a loop of paper folds flat when it has an even number.

What decides it: Three creases square across a Möbius band, and five, and seven, all have the right parity and all miss closing by two widths of the strip. The composition of an odd number of reflections in parallel lines is a reflection in a parallel line, and the gluing map is a reflection in the band's own axis.

Tested in Parity is not enough, at the figure it turns on.

A construction that identifies a cell's opposite edges is right when the counts it produces look sensible.

What decides it: A crease running exactly through a corner of the cell leaves the panel and letter counts looking entirely sensible and additive, and takes Euler's number to minus one on the torus. Nothing else in the collection notices.

Tested in Euler counts the gluing, at the figure it turns on.

A square grid folds flat, so any piece of one does, and so does any sheet made by joining a piece of one to itself.

What decides it: A cell of the grid holding an odd number of squares in each direction, with its opposite edges identified, has no two-colouring and no flat folded state. Measured at one, three and five squares; two, four and six fold.

Tested in A grid that will not close, at the figure it turns on.

The rim's contribution to a search is proportional to how much rim there is, since the free letters it supplies are.

What decides it: On a three-period square twist cell the search costs thirty nodes cut out, twenty-four and eighty-five with one pair of edges glued, and six hundred and twenty-five with both. The letters fall in equal steps and the cost does not.

Tested in Half the slack, at the figure it turns on.

Gluing one pair of a cell's edges gives a cylinder, and it does not matter which pair, since the result is the same kind of sheet either way.

What decides it: On the elongated tiling's twist cell the two cylinders have forty-two and fifty letters and twenty-two and thirty panels. On the symmetric square cell, where the counts are equal, the searches cost twenty-four nodes and eighty-five.

Tested in Which pair is glued, at the figure it turns on.

A sheet with holes in it folds flat when a loop drawn round the holes crosses an even number of creases.

What decides it: With two creases out to the left, one out to the right and one between the holes, a loop round both holes crosses three creases, which is odd, and a loop round each individual hole crosses three and two. Both individual loops have to be even, and the loop round both is their sum.

Tested in Two holes are two conditions, at the figure it turns on.

A population of crease patterns is fixed by saying how the patterns are drawn.

What decides it: The same three drawings on four sheets give twelve objects with different verdicts, different letter counts and search costs a factor of twenty-six apart. Which sheet the drawing sits on is a parameter of the population and it has never been named.

Tested in One population, four sheets, at the figure it turns on.

Any rectangle of a tessellation can be rolled into a cylinder, since the pattern continues everywhere.

What decides it: A rectangle that is not a whole number of periods does not glue: creases running off one edge arrive at the other at the wrong heights, and identifying them joins pieces that are not the same crease. The construction refuses rather than producing a sheet.

Tested in The seam that is not a symmetry, at the figure it turns on.

A duplicate entry in a derived structure is a tidiness problem.

What decides it: One crease contributing two arcs is harmless on five twist patches and fatal on a grid cell whose folded state comes back turned over. The same duplicate is a redundancy on one sheet and a contradiction on another, and which it is diagnoses the sheet.

Tested in The arc that arrived twice, at the figure it turns on.

A crease pattern is a complete description of a folding problem.

What decides it: The same rectangle of grid folds cut out of the plane and has no flat folded state at all with its edges identified, at every odd size. The drawing is identical in both cases and carries no field in which the difference could be written.

Tested in The drawing does not say what is glued, at the figure it turns on.

A lettering found on a patch of a tessellation is a lettering of the tessellation, since the patch is a piece of it.

What decides it: A patch lettering need not agree with itself across a translation, so it does not extend. A lettering of the glued sheet is invariant by construction and writes back onto patches of one, four and nine cells, passing every vertex condition at up to 1,512 creases.

Tested in The symmetry a gluing adds, at the figure it turns on.

A crease pattern's difficulty is concentrated at its vertices, so a pattern with none is trivial.

What decides it: A band with an odd number of creases has no interior vertices, satisfies every vertex condition in the subject vacuously, and has no flat folded state. Six of the twelve bands measured are in that position.

Tested in A crease with no vertex to belong to, at the figure it turns on.

The idealisations in this subject are about the material — thickness, stretch, crease width, memory, grain — and the geometry is exact.

What decides it: The geometry assumes the sheet is a disc. Drop it and the two-colouring becomes a real condition, the closure condition changes its right-hand side, and mountain and valley stop being globally defined. None of those is about the material.

Tested in The sixth thing that is not true, at the figure it turns on.

A cut is a cut, and the more of them there are the more the sheet is changed.

What decides it: A field of a hundred slits leaves a sheet that is still a disc, carrying no condition beyond the ones its vertices carry. One closed cut of a centimetre changes the sheet and adds a parity condition on every loop round it.

Tested in The cut that changes nothing, at the figure it turns on.

A construction starts from the sheet, so any sheet will do.

What decides it: A construction starts from marked points, and the only marks a fresh sheet has are where two of its edges meet. A cylinder's boundary is two smooth circles with no distinguished point anywhere, so there is nothing for the first fold to align.

Tested in A reference on a sheet with no corner, at the figure it turns on.

A crease-pattern file is a complete description of a folding problem.

What decides it: The same file describes a rectangle, two cylinders and a torus, which have different panel counts, different free-letter counts and different verdicts. No field in any format distinguishes them.

Tested in No format has a gluing, at the figure it turns on.

A corrugation reaching a stated packing ratio is one design among many, so four lineages arriving at the same one is evidence that it was selected.

What decides it: A corrugation of panels of lengths L1 to Ln folded flat piles them over a footprint as wide as its longest panel, so the ratio it reaches is the total length divided by the longest. That equals n exactly when every panel is the same length and is strictly smaller otherwise, which makes the uniform corrugation the unique member of the family at the top ratio. Five spacing rules are run and every non-uniform one falls short.

Tested in The census returns one, at the figure it turns on.

Dividing a volume between several folded surfaces costs nothing overall, since each gets a share and the shares add back up to the whole.

What decides it: The area a corrugation can hold peaks at S times D squared over 4t, which is quadratic in the depth D it has. Giving m surfaces a depth of S/m each caps every one of them at one m-squared-th of the single-surface ceiling, so the total across all m is one m-th of it. The shares add to less than the whole, and the shortfall is a factor of m.

Tested in Two surfaces in one box, at the figure it turns on.

A rippled edge settles into the wave number that costs it the least bending.

What decides it: Total squared curvature across a family of profiles carrying one excess of length rises as the square of the wave count, monotonically, at every step. So a least-bending rule has its minimum at the coarsest member offered and never selects any other, whatever the excess. A ripple with more than two waves in it is not being chosen by bending.

Tested in The container picks the member, at the figure it turns on.

The octagon uses a sheet better than the other polygons because eight sides is a rounder shape.

What decides it: It uses a SQUARE better, at 82.8% against the twelve-gon's 80.4%. Run the same maximisation over four rectangles and the octagon never leads any of them: the hexagon does, on every one. The even-sided ranking is 8, 12, 10, 6 on a square and 6, 8, 10, 12 on every rectangle tried, so the advantage belongs to the fourfold symmetry the square and the octagon share.

Tested in The square is in the answer, at the figure it turns on.

A folding construction is a sequence of alignments, so it does what it does on whatever paper it is performed on.

What decides it: The corner-to-midpoint fold puts the crease at 3/8 and 7/8 of a square's vertical edges and crosses the far edge at exactly 2/3. On an A-series sheet held tall the same alignment gives 7/16, 11/16 and 2/7; held wide, the crease leaves the paper entirely. Both are computed from the reflection on that sheet rather than scaled from the square's answer.

Tested in A construction assumes its sheet, at the figure it turns on.

The connected-crane arrangements are one design among the many a slit grid allows.

What decides it: Every subset of the interior lattice points is tried and tested for whether the piece stays in one object. On a three-by-three exactly one of the sixteen subsets works and it is the full set; on a four-by-four, twenty-one of five hundred and twelve, with a unique minimum of five joins of nine; on a five-by-five, seven hundred and eighty-five of sixty-five thousand five hundred and thirty-six.

Tested in Which cranes can stay joined, at the figure it turns on.

Folding a surface inside the folds of another multiplies what each level holds, so a nest of corrugations escapes the ceiling a single corrugation has.

What decides it: A packed level folded into a depth d from a composite of thickness τ holds at most d⁄4τ times its flat area, and the composite it hands upward is as thick as d. Multiplied down a nest, every intermediate depth cancels, so L levels reach at most D⁄4ᴸt. In a box a thousand sheets deep one level reaches 250, two reach 62.5 and three 15.6, with whole fold counts searched at every level.

Tested in A nest pays four a level, at the figure it turns on.

A block of helices that routes on the square lattice routes on any lattice a bundle is designed on, because a solid rectangle has no branch points to trap a route.

What decides it: On the honeycomb lattice twenty of the eighty blocks from one by one to ten by eight have no route, against none on the square lattice. Every block odd in both directions from three by three up passes the colour count and is refused by an exhaustive search, and the reason is a row: along its top edge every second helix has no neighbour off the row and the two corners have one neighbour each.

Tested in A row the route cannot leave, at the figure it turns on.

The waterbomb tessellation lacks a lettering recipe only because nobody has written one down; its thirty-two folding rules out of five hundred and twelve are a power of two and could be stated the way the Miura's are.

What decides it: A recipe of same-and-differ clauses allows an affine set of rules. The Miura's sixteen are affine: two clauses allow exactly those sixteen. The waterbomb's thirty-two obey three independent clauses, which together allow sixty-four rules, and exactly half of those sixty-four fold — three folding rules can be added bit by bit to give one that does not. At a degree-six vertex with its opposite creases tied, the counting theorem keeps six of eight letter patterns, which no parity picks out.

Tested in A recipe needs degree four, at the figure it turns on.

Whether a folding construction still works on a rectangle has to be checked construction by construction, since each one depends on the sheet in its own way.

What decides it: A rectangle is the image of a square under a stretch along its edges, which keeps lines, crossings, midpoints and ratios along a line. Seven constructions run on five sheets split exactly on that line: halving, a third from two crossing lines, a fifth by repeated crossings and Fujimoto's halvings return the square's fraction on all five, and Haga's fold, a corner halved and a corner brought to its opposite each return a different point on at least one — each of them folds across a slanted line.

Tested in A stretch keeps crossings, at the figure it turns on.

Among the cuttings of a slit grid of cranes that hold every crane by at least one join, about half hold the whole piece together, so the look at each crane settles roughly half of the question whatever the size of the grid.

What decides it: Counted exactly to twelve by twelve, the share falls at every size from five on — 64.6, 54.3, 51.1, 46.7, 43.6, 40.5, 37.7 and 35.1 per cent — and from eight by eight it loses the same fraction, about seven per cent, each time a row and a column are added. At six by six, 118,379 of the 119,004 sound cuttings that come apart do so only through stray pieces touching the border of the grid.

Tested in The border is where the cranes come apart, at the figure it turns on.

A pattern with six-crease vertices can only be taught as an object, because a correct list of instructions for its letters would need a separate prohibition for every kind of vertex it has and would be no shorter than checking the pattern.

What decides it: The Yoshimura pattern's twenty-six folding rules are exactly the rules obeying one parity — an even number of its four zigzag classes are mountains — and two prohibitions saying no course carries the letter all four zigzags share: 64, then 32, 28 and 26. The waterbomb tessellation has four kinds of six-crease vertex and needs only two prohibitions on top of its three parities: 512, 256, 128, 64, 48 and 32.

Tested in Two sentences for the Yoshimura, at the figure it turns on.

With a good enough set of cheap tests — the colour count, the count of ends, the cut conditions and the steps a route's ends force — every shape a strand cannot route is refused before any search is run.

What decides it: Every connected shape up to eleven helices on the square lattice and sixteen on the honeycomb was listed, and every shape the tests let through was searched. On the square lattice 32 shapes of eleven helices pass the colour count, the ends, the cuts, the forced steps and the colour of the forced ends, and have no route; on the honeycomb, six of sixteen. Every shape the tests turned away was searched as well, and none of those has a route.

Tested in Every cheap test misses a shape, at the figure it turns on.

The best regular polygon and the best foldable one agree on a square because of the square's symmetry, and on rectangles folding's extra reach — the heptagon, the nonagon, the fourteen-gon — should start to win.

What decides it: On seven proportions from a square to three to one, the best polygon other than the square is the octagon or the hexagon, both of which a compass builds, so the best a fold builds is the same polygon. The best polygon only a fold can build is the fourteen-gon, fourth on an eleven-by-ten sheet and fifth on every sheet from six by five to three by one, and on a square it is the twenty-one-gon, seventh.

Tested in Every even polygon beats every odd one, at the figure it turns on.

The even-sided slit grids of connected cranes meet the counting floor for the fewest joins exactly, and meet it in one way only, as four and six by six do.

What decides it: Counted exactly, eight by eight needs twenty-one joins, its floor, in one way. Ten by ten needs thirty-four against a floor of thirty-three, in 7,076 ways; twelve by twelve meets its floor of forty-eight in 689 ways; fourteen by fourteen needs sixty-six against sixty-five. Among the grids to fourteen whose floor is a whole number, only four and eight by eight meet it with no merge to spare.

Tested in The prediction held at eight and broke at ten, at the figure it turns on.

A folding wing made of several vertices can switch one region between states while the rest stays put, so a many-vertex fold is easier to switch than a single vertex with the same springs.

What decides it: In a chain of degree-four vertices sharing creases, every choice of branch at every vertex is a curve of configurations through the flat sheet, and no two of those curves meet anywhere else. So any switch — even of one vertex's branch — passes through the whole chain flat. On chains of two, three and four vertices the road between the two deepest resting states tops out at the flat sheet on 200 of 200, 199 of 200 and 200 of 200 settings, and that wall is the sum of every crease's flat energy: about three units a crease.

Tested in A chain of vertices switches all at once, at the figure it turns on.

Hinges that rest at different angles are springs that disagree, and a set of springs that disagree can have more than one position in which nothing pushes.

What decides it: A corrugation has one freedom, so every hinge's turn is the same number φ up to its sense, and the energy is the sum of cᵢ(φ − aᵢ)² ⁄ 2 over the hinges. That is a quadratic in φ whatever the aᵢ are, so it has exactly one stationary point, at the stiffness-weighted mean of them. Scanned at 601 states across the whole range, four hinges remembering 0.4, 0.8, 1.6 and 2.4 radians produce one turning point, at 1.300.

Tested in Springs that disagree do not offer a choice, at the figure it turns on.

Supply is charged the same way whatever the architecture, so a comb that beats a stack by eight unsupplied beats it by eight supplied.

What decides it: A wall of height c carries 2c of surface and its channel is sized for what it serves, so the channel is βc and is charged against the pitch: the comb reaches 2c ⁄ (τ + βc), which tends to 2 ⁄ β. A ply serves a fixed area however deep the stack is, so its channel is a constant β charged against the clearance: the stack reaches c ⁄ 4(τ + β), which does not saturate. The two are equal at c = 7τ ⁄ β + 8, checked against both curves at four values of β.

Tested in The channel grows with what it feeds, at the figure it turns on.

Enumerating the ways to spend two simultaneous fold lines' four degrees of freedom on alignments to the paper gives twenty-two operations.

What decides it: That enumeration takes four constraints from one pool, which admits three on one fold line and one on the other — determining neither. A fold line has two freedoms and needs two constraints naming it, so two folds determined by the paper alone are two axioms chosen independently: the multisets of size two from seven, which is 28. Enumerating with every alignment attached returns exactly C(m+6, m) at one to five folds — 7, 28, 84, 210, 462.

Tested in Each fold needs its own two, at the figure it turns on.

A long flap is what costs a design its scale, so shortening the longest limb is where to look when a base will not fit.

What decides it: Lengthening each edge in turn and re-solving gives the cost per unit. On a bird whose wings are 1.6 and whose head is 0.9, the head costs 0.0793 of scale a unit and each wing 0.0264 — three times as much for the shorter edge. On a lizard, the two front legs cost nothing measurable at all while the tail costs 0.0451.

Tested in The price of a limb is not its length, at the figure it turns on.

Finding earlier evidence for these claims would close the gap between what the field says and what it can show.

What decides it: The statistic is a mean over the claims whose gap is positive. A source two centuries earlier lowers it by 25 years for each of the four claims with very large gaps, and RAISES it for five claims whose gaps are small — by 51 for the kindergarten entry, 40 for fold-and-cut, 37 for the one-cut star and 35 for paper in Japan — because a small gap leaving the mean leaves a mean of larger ones.

Tested in Some discoveries would make it worse, at the figure it turns on.

A dome pattern needs the same number of divisions all the way out, because the excess has to be disposed of wherever it is.

What decides it: The excess is 1 − sin(s) ⁄ s, which is nothing at the pole and 36.3 per cent at a hemisphere's rim. A material giving 5 per cent absorbs the whole excess out to 0.55 radians, needs two divisions by 0.78, three by 0.97, and seven by the rim — so the count is a step function of the radius and a pattern with seven everywhere is over-divided for the first third of its own area.

Tested in Where a ring of divisions belongs, at the figure it turns on.

A cheap test for routing is a statement about routes, so adding one to the list improves the list by the same amount whichever lattice the shapes are drawn on.

What decides it: The colour count applied to a stretch a cut and an end have fenced off was added to the census and run on every connected shape to twelve helices on the square lattice and sixteen on the honeycomb. On the square lattice it refuses all thirty-two of the eleven-helix survivors and moves the smallest survivor to twelve. On the honeycomb it refuses none of the six at sixteen, and the smallest survivor does not move.

Tested in A test that only knows one lattice, at the figure it turns on.

The some-layers machine reaches every folded state because it may take the whole pile when it needs to, so the unrestricted move is the one its completeness rests on.

What decides it: The machine was restricted to blocks of at most one layer, at most two, and so on, and its reach enumerated at each depth on nine strips. Every one of them is complete at a depth of at most one less than its segment count, so on no strip measured is the whole pile the block that completes it. Four strips are complete at two or three layers.

Tested in The easiest strip needs the deepest reach, at the figure it turns on.

The regular polygons split into the ones a compass reaches and the ones only a fold reaches, and the second group is the harder group — a folder does the compass's polygons the easy way and the others with effort.

What decides it: The shortest tower to every regular polygon up to forty sides was computed from the factorisation of Euler's totient halved. The heptagon and the nonagon, which no compass reaches, take one extension step. The seventeen-sided polygon, which a compass does reach and which is the famous one, takes three — as many as any polygon on the list. Seven of the polygons a fold reaches in one step are polygons a compass cannot draw at all.

Tested in Gauss's polygon is the expensive one, at the figure it turns on.

A design's extensibility falls with how much of it is diagonal, so two designs for the same subject with the same number of diagonal creases are about equally easy to add a feature to later.

What decides it: The same k diagonal creases were placed three ways on a six-by-six grid and the admissible cuts counted each time with the graft's own cut finder. Six diagonals with no two sharing a row or a column leave no clear line in either direction. The same six gathered into one row leave five. The count spent is the count of distinct rows and columns occupied, not the count of creases.

Tested in A design that keeps its lines clear, at the figure it turns on.

Thinning the paper raises every ceiling a design meets, so a tradition with thinner paper can always reach more layers than one with thicker paper.

What decides it: A crease is a plastic hinge in the fibres at the fold, and a fibre is about twenty-five microns across. Of the six papers measured here, the four folded unbacked are between 1.6 and 4 fibres thick and the two below one and a half fibres are tissues — one of which is used only foil-backed and one of which is not folded at all. The stack ceiling keeps rising as the paper thins and stops being reachable.

Tested in The paper that will not hold a crease, at the figure it turns on.

The folding length of a crease pattern is the sum of its crease lengths, so summing the coordinates and multiplying by the size the sheet is printed at gives the millimetres a folder actually travels.

What decides it: The sum is in the pattern's own coordinates and the multiplication assumes those are sheet widths. Six of the eight printed patterns are built on a unit square and the two readings agree; the Miura is built as six cells of unit width and spans 6.37, so the reported 6,679 millimetres is 6.37 times the 1,049 the sheet has. The printable version of the same pattern, which divides by the pattern's own width, has reported 1,049 since it was written.

Tested in A length needs a scale, at the figure it turns on.

The proportion 1.1284 at which the hexagon overtakes the octagon is a measurement with no closed form, because the hexagon's best rotation on a nearly square sheet is not the obvious one.

What decides it: Held by both widths, a hexagon of circumradius R needs a sheet √3⁄2 + ½√(4R² − 1) long, so its share is a closed form and its crossing with the octagon is √3⁄2 + ½√(16(√2 − 1)⁄3√3 − 1) = 1.128441. The search and the formula agree to better than a ten-millionth on every polygon from seven sides to sixteen.

Tested in The crossing is as hard as the polygon, at the figure it turns on.

A sheet whose creases are a band w wide carries at most 1⁄w of crease a square metre, because a pattern in two directions at that spacing is the same length divided between two families at twice the spacing each.

What decides it: Two families crossing at spacing w carry 2⁄w and no two of their creases come closer than w except where they meet. Measured on the Miura and waterbomb families at up to twenty-four cells a side, density times the closest approach of two creases with no vertex in common settles at 1.80 and 1.89, and the finest pattern the spacing allows is 1.88 to 1.92 times the finest the density bound allows on every paper.

Tested in A paper limits spacing, not density, at the figure it turns on.

A machine restricted to shallow blocks of layers that still reaches a folded state must reach it by a much longer sequence, because each of its moves does less.

What decides it: Every state's shortest sequence was found by a breadth-first walk at each depth and compared with the shortest at full depth. On five uneven strips every state takes exactly its crease count at every depth. On six equal stamps the worst a shallower machine pays is one extra fold, on 8 states at depth two and 32 at depth three; on seven, two extra folds on 8 of 924.

Tested in A shallow machine pays in states, not folds, at the figure it turns on.

The all-layers machine's completeness on equal stamps ends at seven because the strip becomes too long, so its misses at seven and eight are scattered states with nothing in common but their size.

What decides it: Every pile of seven equal stamps was grouped by turning its bottom stamp to the top and by reading it downward. Every group has exactly fourteen piles, and the fourteen the machine misses are one group. At eight stamps the 64 missed piles are exactly the piles that reduce to that group when a stamp at either end of the strip is removed.

Tested in Fourteen states are one pile, at the figure it turns on.

A restart schedule can estimate the scale of a search's running times from the attempts it has already made, and so recover most of the gap between the universal schedule and the cutoff chosen by hindsight.

What decides it: A failed attempt at cutoff c costs c and reports only that the run needed more than c, so a rule that sets each cutoff from its failures sets it from the attempt number alone. On the 120 measured runs of the rhombille patch, three such rules cost at least 1,820, 1,805 and 1,204 nodes against 512 for the hindsight cutoff; the universal schedule with a unit taken from four other patches' searches costs 872.

Tested in A failure teaches a schedule nothing, at the figure it turns on.

To balance two actuators on geared creases, their stiffnesses should be matched through the gearing, so the product of stiffness and gearing is the quantity to equalise.

What decides it: The settled position minimises k₁x² + k₂(gx − δ)², which gives x = k₂gδ ⁄ (k₁ + k₂g²) and a residual and energy that depend on the stiffnesses only through k₂g² ⁄ k₁. Checked by minimising the energy directly at four gearings and stiffness ratios, and by comparing a pair at gearing one half and stiffness ratio four with a pair at gearing one and ratio one, which leave identical residuals.

Tested in A gearing reflects stiffness squared, at the figure it turns on.

Which creases of a driven Miura decide its folded state depends on how far it is folded — four of twenty-four at 0.6 radians and fourteen at 0.8 — so an actuator placement that removes the ambiguity at one angle may not at another.

What decides it: Every crease of three-by-three, four-by-four and five-by-five Miuras was driven at twenty angles from 0.1 to 3.0 radians with both configurations found at every vertex. The census is identical at every angle: four deciding creases of twelve, four of twenty-four and six of forty, the same creases each time. The fourteen at 0.8 came from 138 of 8,640 vertex solves returning one configuration of two.

Tested in The deciding set does not move, at the figure it turns on.

A folding wing meshed with loops of vertices can switch one region at a time, because a loop creates junctions where more than one combination of branches meets away from the flat sheet.

What decides it: Four vertices round one panel were driven at five angles from 0.3 to 1.8 radians and every consistent assignment with all creases folded was enumerated: one combination survives on a face with no two vertices alike and four on a Miura face, at every angle. Sampled from −2.4 to 2.4 radians, two different assignments at the same driven angle are never closer than 1.53 times it, so they meet only at the flat face.

Tested in A loop takes choices away, at the figure it turns on.

A twist tessellation whose edge weights are unequal collapses anisotropically, and one whose weights are equal collapses by a similarity, because the displacement each pleat contributes carries its own weight.

What decides it: Both directions fail on measured patterns. The stretched triangular grid writes weights spread six to one and collapses by a similarity to ten figures; the stretched honeycomb writes the same weight on every edge and collapses with principal factors 0.809 and 0.150, a ratio of 5.386. Across the twenty patterns the weight spread and the collapse ratio agree on neither the cases that are similarities nor their order.

Tested in The sheet draws in crooked, at the figure it turns on.

Folding a corrugation makes the sheet smaller in both directions, so each of the two directional factors is at least one and their product is how much smaller it gets overall.

What decides it: The leaf corrugation at a zigzag angle of 0.78 radians and four rows folds to a state whose extent across the pattern is larger than the flat sheet's, giving a cross factor of 0.98555 — measured on the two extents, not inferred. The folded extent is 0.6900 at every row count from two to ten, so the factor is 0.2464 a row and is below one for any sheet of four rows or fewer, while the areal factor stays above 1.47 throughout.

Tested in The direction that gets longer, at the figure it turns on.

A colour change is expensive because the other side of the paper is buried under the pile: the deeper a pattern stacks, the more layers stand between the visible face and the reverse colour.

What decides it: Ordered at every sample of the footprint, over every folded state and from both faces, the other side lies exactly one layer down wherever it is present on every pattern with one folded state — the preliminary base, both twists and a two-by-two Miura, with piles of three to eight layers. On the four patterns with more than one state it lies two down over at most 5.7 per cent of the samples, and nowhere three down.

Tested in One sheet down, at the figure it turns on.

Small Miura patches up to three by three have exactly one folded state, so the layer-order field would record nothing a reader holding the crease pattern could not compute.

What decides it: The overlap test that found those counts missed pairs of panels two columns apart: folded, they are congruent parallelograms slid along a shared slanted edge, with no proper crossing of edges and a shared strip that misses every probe point. Measured by the area two panels share, the three-by-two Miura has three folded states, the three-by-three six and the four-by-three eleven, and a strip two rows high has 2c − 3 for c columns.

Tested in One choice with eleven answers, at the figure it turns on.

Tessellations fold to a near-uniform depth and bases pile their layers in a few deep places, so the ratio of a design's deepest point to its mean depth separates the two families.

What decides it: Measured on the folded state of every printed pattern, the ratio is 1.00 on the preliminary base — a base — and on the Yoshimura, and 1.02 on the waterbomb tessellation; it is 1.73 on the Miura, 1.92 on the tapered corrugation, 2.15 and 2.98 on the two twists, and 5.89 on the fold-and-cut triangle. It is one exactly where every panel lies over every point, and large where the folded footprint keeps regions of different depth.

Tested in The deepest point pays for the paper, at the figure it turns on.

A crease pattern folds flat when every one of its vertices satisfies Kawasaki's and Maekawa's conditions.

What decides it: Both theorems are exact and both are about a single vertex. The figure draws a pattern whose every interior vertex passes every local test and whose layers still cannot be ordered — the obstruction is global and no vertex can see it. Deciding the general case is NP-hard (Bern and Hayes, 1996), which is the strongest possible statement that no strengthening of a local rule will close the gap.

Tested in Local is not global.

A program that runs the flat-folding theorems over a crease pattern decides whether the pattern folds flat.

What decides it: This site owns such a program and it is a filter rather than a decision procedure. The figure exhibits three patterns, each failing exactly one of the four local conditions and no other — the generator refuses to draw itself if that stops being true — and then a fourth that passes all four and is not thereby known to fold. The gap is not a defect to be closed: closing it would decide an NP-hard problem.

Tested in What a checker cannot check.

The universal molecule fills any polygon, which is what makes the tree method free of special cases.

What decides it: Convexity is a hypothesis, not a formality. The generator runs the construction on convex and reflex polygons alike and asserts that it succeeds on exactly the first set and refuses on exactly the second — the inward shrink is singular at a reflex corner, and what comes back is nothing rather than a worse pattern. The difficulty does not disappear; it moves to whoever chose the polygons.

Tested in The molecule that does not exist.

A theorem named for somebody is a theorem that person proved, and its name is a reasonable guide to when the subject learned it.

What decides it: Six results, each drawn from the year of the earliest proof anybody can point at to the year of the publication its name comes from. The mean gap is twenty-two years and the longest is fifty-five; every gap has the same sign. The generator refuses any entry whose eponym year precedes its proof year, since a name cannot come before the thing it names.

Tested in The name is not the date, at the figure it turns on.

A historical claim supported by a dated primary source is settled, in the way a computed result is settled.

What decides it: The record here is checked for shape, not for truth. Every entry must name a dated source, say what kind of thing it is, and count how many survive; the checker refuses an entry missing any of that and refuses a popular date earlier than its evidence with nothing said about the gap. It fired on two entries on its first run. What it cannot do — and no gate on this site can — is tell whether any date is right. Five of the fifteen claims rest on exactly one surviving source.

Tested in A record is not a proof, at the figure it turns on.

The dashed-and-dotted diagram convention is a presentational nicety; the real content of a model is the folds themselves.

What decides it: The basic symbols encode exactly one operation — fold this crease, this way, now — which is a simple fold. Over seeded random spacings with every assignment of each enumerated exhaustively, the share of a strip's flat foldings that any sequence of simple folds can reach falls from 71% at three creases to 13% at six. What the notation can say is therefore a strictly shrinking fraction of what can be folded, which is why it acquired named symbols rather than more lines.

Tested in What a dashed line can say, at the figure it turns on.

A growing leaf ruffles because its edge is floppy — a stiffer leaf with the same growth would stay flat.

What decides it: Stiffness does not appear anywhere in the computation. Given how much each part of the sheet grew, the curvature of the resulting metric is determined, and the generator computes it from the growth field alone. A stiffer sheet with the same growth has the same curvature and the same impossibility; what stiffness changes is the shape it settles into, not whether it has to leave the plane.

Tested in A sheet that grows cannot lie flat, at the figure it turns on.

More growth means more curvature, so the parts of an organism that grow fastest are the parts that end up most strongly curved.

What decides it: The curvature depends on the second derivative of the log of the growth factor, not on its size. A sheet grown by a factor of three uniformly is exactly flat, and one grown by a few percent unevenly is not. The figure runs a single parameter through zero and the curvature changes sign while the total growth barely moves.

Tested in Which way the disc curves, at the figure it turns on.

A crease is a line of very high curvature, so a heavily creased sheet is a strongly curved surface.

What decides it: Folding changes no distance measured along the sheet, so it changes no intrinsic curvature. The developability condition the checker runs at every interior vertex — sectors summing to 360° — is precisely the statement that the curvature there is zero, and it has held at every vertex of every pattern published here since the site's first phase.

Tested in A crease carries no curvature, at the figure it turns on.

Any pattern that folds flat can be used as a deployable, since the folded and open states both exist.

What decides it: A deployable needs a path between the two states that whatever drives it can traverse. The generator samples the corrugation's motion and asserts that the exposed span rises at every one of its steps; a pattern whose span fell anywhere would require something to pull it the wrong way, and growth cannot pull.

Tested in Opening with nothing to pull, at the figure it turns on.

Insect wings fold so compactly because of the material — a stiff sheet could not do it.

What decides it: The packed fraction of each geometry here is computed from the pattern's own parameters with no material property anywhere in the calculation, and the four geometries differ by a factor of three. What the material decides is the minimum bend radius, which is a separate bound taken up on the third rung of this ladder.

Tested in A wing that folds into nothing, at the figure it turns on.

Protein folding and paper folding are the same problem at different scales, so results about one transfer to the other.

What decides it: A crease pattern's states are filtered by conditions evaluated at one vertex with no reference to the rest of the sheet — four of sixteen survive at the worked vertex, a filter that removes 75%. A chain's states have no analogous test: whether two residues are close depends on every torsion between them, so nothing local constrains the space at all.

Tested in Two things called folding, at the figure it turns on.

A cut sheet gets progressively weaker as the cuts get longer, so its ability to hold together falls off gradually and there is a length at which it becomes unreliable.

What decides it: The count of pieces is 1 at every cut length short of the pitch — at 50%, at 90%, at 99%, at 99.9% — and jumps to one strip per row at exactly 100%. The quantity that falls gradually is the ligament, which is a length; the quantity that decides whether the sheet is one thing is a count, and it does not move until the ligament is gone.

Tested in One cut short of falling apart, at the figure it turns on.

Restricting a design to a grid is a pure loss: the packing gets worse and nothing is gained except neatness.

What decides it: On a four-division lattice the best packing of four flaps is 0.2500 against the free search's 0.2500 — no loss at all — and of six flaps 0.1250 against 0.1876, a loss of a third. But the lattice figures are optima, settled by enumerating every arrangement with a bound, while the free figures are the best a search has found and are not known to be optimal for any of these counts above four.

Tested in What the grid settles, at the figure it turns on.

Real paper tolerates a degree or two of error, so the measure-zero result is a mathematical nicety: plenty of patterns are close enough to flat-foldable for a real sheet.

What decides it: Over 40,000 random four-crease vertices the median distance to a flat-foldable one is 31.3° per sector and the mean 33.9°. Only 1.6% are within one degree, 8.2% within five and 16.4% within ten. A Miura whose vertices are moved by a fortieth of a cell, by contrast, sits 0.73° per sector away — so tolerance rescues perturbed patterns and does nothing whatever for drawn ones.

Tested in A near miss is nearly as rare, at the figure it turns on.

A construction that reaches every fraction by a single rule is the efficient way to divide a sheet, since it needs no searching and no special cases.

What decides it: The rule takes n folds to reach 1/n. The closure of the axioms over a bare square reaches 1/3, 1/4, 1/5, 1/6, 1/8 and 1/12 in two folds — against the rule's three, four, five, six, eight and twelve — and reaches no seventh, ninth, tenth or eleventh at all. The rule is worse wherever the search arrives and it is the only thing available where the search does not.

Tested in Cheap where it reaches, at the figure it turns on.

Requiring a design's circle packing to be symmetric costs efficiency, because a constrained optimum cannot be better than an unconstrained one.

What decides it: True of the optima and false of the searches. Constrained to a mirror, a search of the same effort found a packing 0.04 per cent better than the free search at five discs and 0.26 per cent better under a half-turn at eight — because halving the coordinates halves the space the same effort has to cover.

Tested in The shapes the optimum has, at the figure it turns on.

A crease pattern can be laid on the sheet any way round, because none of the flat-folding conditions changes when the paper is turned.

What decides it: True of the conditions and false of the paper. Turning the tapered corrugation a quarter turn moves the share of its crease length lying within 22.5° of the sheet's grain from 42.5 per cent to none at all; on the Yoshimura it moves from 28.6 to none. The theorems are silent about a choice that decides where nearly half a pattern's folding happens.

Tested in The fifth thing that is not true, at the figure it turns on.

An exact folding construction is the better one to use, because it lands on the answer and a convergent method only approaches it.

What decides it: Both methods were run four thousand times each by a hand that places every crease half a millimetre out on a 150 mm sheet. The exact ladder's final crease lands 0.31 mm from the mark at three parts and 0.56 mm at sixteen; Fujimoto's lands 0.35 mm out at three and 0.29 mm out at sixteen. The crossover is at four parts and is in the same place at every hand steadiness tried.

Tested in Exact is not accurate, at the figure it turns on.

At a vertex of degree four the four local conditions are the whole answer — an assignment passes them exactly when it folds.

What decides it: True of vertices whose angles are drawn at random and false of vertices drawn on a grid. Over degree-four vertices produced by cutting half a turn into random pieces, no lettering passes and fails to fold. Over degree-four vertices whose sectors are whole multiples of forty-five degrees, 42 per cent of the vertices carry at least one.

Tested in Which vertices are the random ones, at the figure it turns on.

Which crease patterns a site can reproduce is decided by a rule about ownership applied after the patterns have been chosen.

What decides it: Every pattern on this site's printed shelf was classified from the provenance it already carries. All eight fall into three classes; every one carrying a date was published as mathematics and every undated one is traditional or generated from a rule here. The class that would be excluded by a rule — a designer's model — is also the class the record dates and attributes, so the rule and the record select the same set.

Tested in The patterns nobody owns, at the figure it turns on.

A search for flat-foldable letterings can start from one that works and change it a little at a time: the good letterings are near one another.

What decides it: True of the smallest change a program can make and false of the smallest change a folder can make. Under a change of any two creases every folding of a vertex is reachable from every other, on every vertex measured. Under a change of two neighbouring creases, seven of sixteen generic degree-six vertices come apart, and at degree eight only four of sixteen stay in one piece.

Tested in Walking between two foldings, at the figure it turns on.

A rigid-origami mechanism has a manufacturing tolerance: a number of millimetres, below which the panels may be cut wrong and the mechanism still folds.

What decides it: The same error of a fifth of a millimetre destroys the closure when it points across the surface of solutions and costs five thousand times less when it points along it. Sixteen of the twenty directions in the mesh's length space are of the second kind, and no single number of millimetres describes both.

Tested in A tolerance is a direction, at the figure it turns on.

A thick-panel technique is characterised by what it does at a fold: choose the technique, and the thickness problem is solved for the pattern.

What decides it: Every published technique is analysed for two panels meeting at one crease. On the eight patterns printed here the deepest pile is eight, nine, seven, sixteen, sixteen, thirty-two and sixty layers, and the length the outside of a pile falls short of the inside is proportional to that count: 18.5 mm on a Yoshimura in copier paper, against 0.31 mm for the two-layer case every technique is drawn at.

Tested in The pile, not the panel, at the figure it turns on.

Counting the repeating rules a tessellation admits counts the tessellations it can make.

What decides it: Thirty-two rules survive every condition on a patch containing all four kinds of vertex. Folded and compared centroid by centroid and area by area, all thirty-two produce exactly one set of folded panels. They differ in which of their creases are mountains — twenty-two in half of them, twenty in the other half — which is the difference between the two sides of the paper.

Tested in Thirty-two rules, one object, at the figure it turns on.

A cut helps because it removes material: what kirigami buys is bought out of the paper that is no longer there.

What decides it: A slit removes no paper at all and buys a great deal. Cutting a single crease of a square twist doubles the share of letterings the pattern admits for each interior vertex the cut turns into a boundary vertex — two for a crease buried in the pattern, one for a crease that already reached the edge — and the paper's area is unchanged.

Tested in A cut is a licence, at the figure it turns on.

A measurement made over crease patterns is a measurement about crease patterns.

What decides it: Four constructions that each produce patterns satisfying every vertex condition — the printed shelf, twist tessellations, solved quadrilateral meshes and fold-and-cut patterns — disagree about how much a pattern shrinks when folded by a factor of 12.4, about crease length per unit of paper by 5.9, about the deepest pile by 2.2 and about the share of vertices on the paper's edge by 1.7.

Tested in Four ways to draw a pattern, at the figure it turns on.

The size of an instance is what makes it expensive: a bigger vertex is a harder vertex.

What decides it: With the degree held at six, vertices whose sectors coincide seven times cost 5.00 nodes per lettering to decide and vertices with no coincidence cost 1.75 — a factor of 2.9. Doubling the number of creases from four to six, over the same populations, raises the cost per lettering by a factor of 1.9.

Tested in The cost is in the coincidences, at the figure it turns on.

A crease pattern is a complete record of a model: publish it and the model is preserved.

What decides it: A crease pattern records the creases and their letters. Which of the orderings of its panels the folded object is takes 302 bits to say for a Yoshimura and 226 for a waterbomb tessellation, and no notation in the subject writes any of them down — including the files published here.

Tested in The half no notation records, at the figure it turns on.

Four unrelated lineages folding the same way is strong evidence that the fold is the best solution to their shared problem.

What decides it: The packing ratio a corrugation reaches is exactly its average layer count, by conservation, so every way of packing a sheet into a given fraction of its area has the same layer count and there is nothing to choose between them on that axis. The quantity that is free — crease length per unit of paper — varies six-fold across the patterns measured and is only weakly related to the ratio, and it is the quantity nobody compares.

Tested in Four finders, one option, at the figure it turns on.

A vertex on the edge of the paper is a weakened interior vertex: the same conditions apply, with less to check.

What decides it: Held at the same sectors, the two are different objects with different alphabets. The ring has 2^d letterings and folds in a quarter of them at degree four; the line has 2^(d−1) and folds in seven-tenths. Over 234 vertices the line never admitted fewer than the ring.

Tested in A ring and a line, at the figure it turns on.

A rigidly folding mesh has a manufacturing tolerance, which can be quoted as a length and applies to the whole motion.

What decides it: At a budget of 0.02 radians on a 150 mm sheet the same mesh allows 0.425 mm of cutting error at a third of a radian and 0.033 mm at two and a half — a factor of 12.9 — and the sensitivity tracks tan(ρ/2) to within two per cent over the range a builder uses.

Tested in The allowance is spent at the end, at the figure it turns on.

A self-folding sheet needs one actuator, and how much authority that actuator needs can be read off the mesh at a convenient fold angle.

What decides it: Every crease reports an amplification of exactly one somewhere in its motion and at least twice that somewhere else, and on the solved mesh twenty of twenty-four are worst within a sixth of a radian of the flat sheet — the configuration a self-folding sheet starts in.

Tested in The hardest instant, at the figure it turns on.

A hole in the sheet is a loss: it removes paper a design could have used, so a designer should never start from one.

What decides it: A flap against the edge of a hole claims half its disc, exactly as one against the edge of the sheet does. Against a solid square of the same area, the holed sheet's mean claim is lower at every flap length tried — 0.863 against 0.897 at a flap of 0.12 of the side.

Tested in A hole is cheap paper, at the figure it turns on.

The number of ways a strip of n stamps folds is the number of distinct folded objects it has.

What decides it: Reading a folding from the other end and turning it over both map foldings to foldings and neither ever fixes one, at any size up to eight stamps. The orbit count is 1, 2, 5, 14, 38, 120, 353 against 2, 6, 16, 50, 144, 462, 1392 — and 1392/4 is 348, not 353.

Tested in The count counts labels, at the figure it turns on.

A machine restricted to folding the whole pile at once is a weak model of folding, so it reaches only a small share of the folded states a strip has.

What decides it: On an evenly creased strip it reaches every state up to six stamps: 12, 32, 100 and 288 of 12, 32, 100 and 288, and never one the layer solver calls illegal. At seven it reaches 896 of 924. On five unevenly creased strips it reaches none of the 108 states they have between them.

Tested in Where the machine catches up, at the figure it turns on.

A crumpled sheet is disordered, so its mountains and valleys are arbitrary and could just as well have been otherwise.

What decides it: They could have been otherwise before the folding and cannot be afterwards. A six-fold crumple carries 16 to 39 creases with an interior vertex at each end, and over 294 changes the sheet admits, not one alters any of them.

Tested in The decision a crumple has taken, at the figure it turns on.

The record of this subject has a characteristic overrun, which can be quoted as one number.

What decides it: The median gap over the whole record is one year, which describes nothing: the six claims about practices run +1 to +980 with a median of 347, and the six about results run −55 to +76 with a median of −14. Not one practice runs the other way.

Tested in Two kinds of claim, at the figure it turns on.

What limits the size of a DNA origami is the length of the scaffold strand.

What decides it: It is the limit for two of the five families grown here — a square block at 100 helices and a single row at 113 — and it is never reached by the other two, which are stopped by the routing at 5 and 8 helices, a factor of twenty earlier.

Tested in Two ceilings, at the figure it turns on.

Whether two creases cross is a fact about the pattern, decided exactly by the coordinates, so it needs no tolerance and admits no doubt.

What decides it: Measure how far each crossing sits from the nearest end of the two creases making it. Across the forty-seven crossings on the patches that have them, the depths run from 0.00105 to 0.112 of a sheet width — 0.16 mm to 17 mm at printed size. Five are under half a millimetre, which is below the width of a drawn line, the accuracy of a hand fold, and the distance at which this collection's own reference points are called indistinguishable.

Tested in How deep is a crossing, at the figure it turns on.

A cheap test on a crease pattern is a screening step that saves time before the real decision, and the real decision is what actually settles the cases.

What decides it: Over the thirty-three patterns in the four test populations, the cheapest of the five refusals here catches five of them and the most expensive catches six. The cheap one is not screening for the expensive one: every pattern it refuses has more panels than the search will ever accept, so those five would have been reported undecided for ever.

Tested in The order the refusals come in, at the figure it turns on.

How much folding a pattern costs is its total crease length, and a pattern with twice the length is twice the work.

What decides it: Bin each pattern's crease length by distance from the sheet's edge, against each band's share of the paper. The fold-and-cut triangle puts 6.80 times its share into the middle 4% of the sheet and nothing at all into the outer 36%; the waterbomb tessellation runs between 0.89 and 1.35 across every band. Same measure, two patterns whose totals are within 2% of each other, and two entirely different demands on a hand.

Tested in Where the length sits, at the figure it turns on.

References accumulate through the paper as folds are made, and the sheet's edge is simply where the first ones happen to be.

What decides it: Of the five references the first fold adds to a bare square, four are on the sheet's edge. Of the five hundred and fifty-six the second adds, forty-eight are — a share falling from four fifths to one in twelve. The edge yields early because it is four lines that existed before any fold, and it saturates because a line meets it twice while two lines inside the paper cross once.

Tested in The edge was there first, at the figure it turns on.

A twist whose central ring reads as one letter fails for a reason that only a search over the orderings of its panels can find.

What decides it: Four of the thirty-two such letterings are refused by the letters alone, in one pass, and the circle they close is the ring itself — eight panels round the central square. The other twenty-eight have letters that agree perfectly and still have no folded state, so the search remains necessary for most of the group.

Tested in The ring is the loop, at the figure it turns on.

How often a pattern's letters can be made to agree with themselves is decided by how much pattern there is — the panels, the creases, the vertices.

What decides it: Grow three families along their own size parameters and plot them against the number of independent chains their panels form. All three fall, and at about thirty-four chains they sit at ninety-three per cent, sixty-four and thirteen. The count sets the scale and does not decide the answer.

Tested in A corrugation agrees with itself, at the figure it turns on.

Every crease pattern is exposed to the same failures, so a design method's output has to be checked for them like anything else.

What decides it: A pattern whose creases come from a straight skeleton has one to three independent closed chains of panels, because a skeleton is a tree — against thirty-six on the smallest tessellation patch. Two hundred and eighty letterings across seven such outlines contain no contradiction, and the structural reason says why there is almost nowhere for one to sit.

Tested in A tree cannot argue, at the figure it turns on.

Whether a lettering's arcs close a circle is a global property, so deciding it needs the folded sheet.

What decides it: For a repeating rule on a grid corrugation it needs three bits. A four-panel circle is available exactly where a column crease keeps its letter across a row and the row disagrees with it, and that test — which folds nothing — matches the arrow walk on all sixty-four rules of the Miura family and all sixty-four of the leaf's.

Tested in The loop is in the rule, at the figure it turns on.

A result measured on crease patterns, about crease patterns, using this collection's own machinery, is a result about folding.

What decides it: The heavy-tailed cost distribution and the restart strategy that answers it are properties of randomised backtracking search, described in another field before this one had the instrument to notice them. What the folding measurement adds is an instance and its parameters, not the phenomenon.

Tested in The tail was named somewhere else, at the figure it turns on.

A crease pattern whose panels cannot be stacked is a pattern that cannot be folded flat, so the verdict belongs to the pattern.

What decides it: Six quadrilateral meshes were refused or accepted at the labelling each was built with. Enumerating all sixteen or thirty-two admissible labellings of each and searching every one's orderings, two of the four refused meshes have labellings that stack — eight and four of them respectively — and two have none at any labelling.

Tested in Refused at one lettering, at the figure it turns on.

A search with a heavy-tailed run-time distribution should be cut off and restarted, and the right cutoff is the one that minimises the expected total.

What decides it: The arithmetic is correct and the distribution is the search's own: replacing its randomised choice with a constant costs eighty steps deterministically, against five hundred and twelve for the best restart strategy — and a deterministic search cannot be restarted, because a fresh seed produces the identical run.

Tested in Restarting what cannot be restarted, at the figure it turns on.

Crease density — total crease length per unit of paper — captures how much folding a pattern asks for.

What decides it: Two patterns with the same total can differ by any amount in what a hand has to do. On one printed patch the twelve shortest creases are 7.9 × 10⁻⁶ of a sheet and the thirteenth is 2.9 × 10⁻², a factor of five hundred, and the total is blind to all of it.

Tested in The shortest crease is not a crease, at the figure it turns on.

Choosing a better order for a backtracking search's decisions is generally worth doing.

What decides it: On the search over panel orderings it is worth exactly nothing: permuting the panels twelve ways leaves the node count identical on all seven patterns tried, from 1,636 to 1,188,571 nodes, because the set of partial orderings that survive the checks does not depend on what anything is called.

Tested in A search with nothing to reorder, at the figure it turns on.

A cycle in the relations a lettering forces between panels proves that the pattern has no flat folded state.

What decides it: On a square twist tessellation with its opposite edges identified, every lettering that satisfies the vertex conditions has a cycle in those relations, and one of them folds: written onto ordinary patches of one, four and nine periods it passes every vertex condition and forces no loop at all.

Tested in A loop that goes somewhere, at the figure it turns on.

A search that exhausts its tree without finding a consistent lettering has proved that the pattern has none.

What decides it: On a glued square twist tessellation the search exhausts in thirty-five steps, and the lettering it says does not exist passes all four vertex conditions and forces no layer loop on clipped patches of forty, a hundred and forty-four and three hundred and twelve creases.

Tested in The lettering that was proved impossible, at the figure it turns on.

A patch is what a tessellation is; drawing a square of it and reading the square is reading the pattern.

What decides it: The square of a two-period cell shows twenty-five panels and forty crease pieces, and the pattern it is a square of has sixteen panels and thirty-two creases. Nine panels and eight creases in that drawing are artefacts of where the square was drawn.

Tested in A sheet with no edge, at the figure it turns on.

The bottom layer of a folded crease pattern is a property of the pattern and its assignment.

What decides it: On patches of one, four and nine periods of the same periodic lettering, the panels with nothing below them number one, two and three on the square tessellation, and every one of them touches the paper's edge. The pattern the patches are cut from has none.

Tested in The bottom layer is at the rim, at the figure it turns on.

A crease pattern that fills its own sheet costs one search step per panel, and that is a property of filling the sheet.

What decides it: The Yoshimura fills its own sheet at every proportion. At a row height of √3 halves of a column its vertices admit thirty labellings and it costs fifty-seven steps on sixty-five panels; at 1.7320509 they admit eight and it costs nineteen.

Tested in One step per panel is a table size, at the figure it turns on.

The Yoshimura, whose vertices are degree six, breaks the one-step-per-panel equality that the degree-four families obey.

What decides it: Measured as a family at six sizes, the Yoshimura costs 16, 32, 47, 57, 82 and 108 steps on 21, 36, 55, 65, 90 and 119 panels — linear, with no decision withdrawn anywhere. It sits at 0.9 of the line rather than off it, and at a different row height at 0.29.

Tested in Six creases and the same straight line, at the figure it turns on.

A backtracking search may discard a branch as soon as it looks unpromising, since the answer will be found somewhere else.

What decides it: Discarding every partial lettering with a loop in its quotient relations makes the search on a glued square cell exhaust with nothing found, in thirty-five steps — a proof of something the same search with a sound prune settles with a witness in nine.

Tested in Pruning on proofs alone, at the figure it turns on.

An exhausted search is the most reliable result available, because it has looked everywhere.

What decides it: The same search exhausts a glued square cell in 3, 35 and 3,455 steps at one, four and nine periods and does not finish at sixteen — and every one of those closed trees is a proof of something the same search with a corrected condition disproves by exhibiting a witness.

Tested in The cost of proving something false, at the figure it turns on.

How much smaller a twist tessellation gets when it is folded depends on which tiling it is built on.

What decides it: The folded lattice of the square, triangular, hexagonal, elongated and rhombille twist tessellations at one turn and fill is the flat lattice scaled by 0.410373441 and turned by 36.62 degrees, identical to eight figures on all five.

Tested in Folding it flat is one similarity, at the figure it turns on.

The number of creases per unit area of a tessellation can be read off a patch of it.

What decides it: A one-period square patch reports a hundred and four creases per unit area, a two-period one eighty-seven and a three-period one eighty-one, against the pattern's sixty-nine. The crease length per unit area is 12.6751 on all three and on the pattern.

Tested in A count is not a length, at the figure it turns on.

Every flat folded state can be described by listing its panels from the bottom one to the top one.

What decides it: A periodic twist tessellation folds flat and has no panel with nothing below it, because every loop in its layer relations travels a cell rather than closing. Its patches have one to three such panels and all of them are at the paper's edge.

Tested in An order with no least element, at the figure it turns on.

A condition that is correct where it was stated stays correct wherever it is applied.

What decides it: Acyclicity of the forced layer order is necessary and sufficient for a bounded sheet and neither for a periodic one, and the difference is invisible in every output either version produces.

Tested in A test imported without its hypothesis, at the figure it turns on.

A sheet of paper folds flat only if a closed path drawn on it crosses an even number of creases.

What decides it: A Möbius band with three creases across it has a two-colouring and a flat folded state; one with four has neither. The even rule is the rule for a sheet with two sides, and the sentence naming that hypothesis is missing wherever the rule is quoted.

Tested in The seam carries a sign, at the figure it turns on.

Every crease in a flat-foldable pattern is either a mountain or a valley, and which one it is is a property of the crease.

What decides it: On a sheet with one side, transporting a crease's letter once round the sheet returns it reversed, so no assignment of letters to creases is consistent. Maekawa's condition survives because the difference of the two counts is unchanged by the swap, and the letters it is about do not.

Tested in The band that needs an odd number, at the figure it turns on.

A pattern folds flat when the reflections composed round every closed path come back to the identity.

What decides it: On a cylinder the composition round the band has to be a translation by the band's length, and on a Möbius band it has to be that translation with a flip. Requiring the identity on either sheet refuses every drawing, including the ones that fold.

Tested in Closure is not the identity, at the figure it turns on.

A rectangle of tessellation glued into a torus differs from the cut rectangle by an amount, and that amount is a property of the pattern.

What decides it: The amount is a sum of two independent amounts, one per pair of edges, and on an anisotropic drawing the two are different sizes — ten letters and two on the elongated tiling's cell. A single number attributes to the pattern what belongs to a direction.

Tested in The rim adds up, at the figure it turns on.

Removing a sheet's boundary removes free choices from the search, so the search gets easier.

What decides it: The total node count falls and the cost per panel rises — a two-period Miura cell costs fifteen nodes over fifteen panels cut out and ten nodes over eight panels glued. The choices removed were the ones that could not be wrong.

Tested in One node per panel, with the rim gone, at the figure it turns on.

A tessellation's period is the rectangle its drawing repeats in, and that rectangle is the unit everything about the pattern can be stated in.

What decides it: The equilateral Yoshimura's drawing repeats every column and its folded state repeats every third, because folding turns the lattice through 240° per column. A gluing at one column produces a sheet with no flat folded state at all.

Tested in The turn a column costs, at the figure it turns on.

A tessellation's unit cell is the unit that quantities about it should be quoted per.

What decides it: The equilateral Yoshimura's folded state repeats every three drawn columns, so any per-cell quantity about the folded object is being quoted per third of a unit. Measured at one to seven columns; the fold turns by 240° per column and closes at three and six.

Tested in The period nobody measured, at the figure it turns on.

A cut relaxes a sheet, so the more cutting there is the freer the paper gets.

What decides it: A closed cut in the middle of a sheet adds a loop that cannot be shrunk, and with it a parity condition on the creases crossing it. Cutting a hole in a sheet that folded can produce one that does not, on the same creases.

Tested in A cut is surgery, at the figure it turns on.

A test that has been correct on every pattern it has been run on is correct.

What decides it: The collection's acyclicity test exhausts on glued cells at three, thirty-five and 3,455 nodes and does not finish at 200,000 for sixteen cells, and the letterings it proves impossible pass every vertex condition and force no loop on ordinary patches up to 1,512 creases.

Tested in The cost of asking the wrong sheet, at the figure it turns on.

Map folding is a question about a rectangular grid, so any grid can be asked it.

What decides it: A grid cell with an odd number of squares in each direction, glued into a torus, has no two-colouring and therefore no folded state at all. Half the sizes have nothing to count.

Tested in A map with no edges, at the figure it turns on.

A metamaterial's behaviour is measured on a unit cell, because the material repeats and one cell stands for all of them.

What decides it: A cell cut out of a patch has a rim, and the rim supplies free choices that no interior cell has: eight of the forty free letters on a two-period square twist cell. A cell that stands for the interior is a glued one.

Tested in A metamaterial with no edge, at the figure it turns on.

A folded state has a bottom layer, so an enumeration of stackings can always start from it.

What decides it: The minimal panels of a forced order are all at the rim, on every patch measured. Take half the rim away and the candidates halve; take all of it and there is no least element at all, and the enumeration has nowhere to start.

Tested in A bottom layer on half a rim, at the figure it turns on.

A designer's sheet is a shape — a square, a rectangle, a hexagon — and choosing it is choosing an outline.

What decides it: A cylinder is not an outline. It is a rectangle with two of its edges declared to be one edge, it has half the boundary of the rectangle it came from, and every count that depends on the boundary moves.

Tested in A sheet with two edges, at the figure it turns on.

A sheet's corners are valuable, so a shape with more corners is a better sheet to design on.

What decides it: The premium comes from a flap's disc being clipped by the boundary, and how much is clipped depends on the interior angle rather than on there being a corner. A smooth boundary clips half of every disc and has no corners at all; a closed sheet clips nothing and has no premium.

Tested in The corner premium, with no corners, at the figure it turns on.

The uniaxial method is a method for folding, so it applies to any sheet a pattern can be drawn on.

What decides it: Every flap's tip is boundary. A sheet with no boundary produces no flaps at all, and a cylinder produces them only at its two end circles — so the method's output on a closed sheet is empty rather than restricted.

Tested in A base needs an edge to point at, at the figure it turns on.

Box pleating always works, which is why designers use it: every crease is on a grid line and the pattern folds by construction.

What decides it: A box-pleated grid rolled into a tube folds only when an even number of creases run along the tube. Odd column counts have no flat folded state at all, measured at one, three and five columns.

Tested in A grid glued, at the figure it turns on.

Checking that paper does not pass through paper is a matter of comparing every pair of panels.

What decides it: On a glued sheet the pairs are pairs of panel classes rather than of drawn pieces, and the two differ: a class can meet itself round the loop, and two pieces of one class cannot collide with each other at all.

Tested in Two panels that are one panel, at the figure it turns on.

A rigid-foldable pattern stays rigid-foldable when its sheet is closed into a tube, since rigid-foldability is decided vertex by vertex.

What decides it: A closed sheet adds a loop closure: the composed motion round the loop has to be the identity at every stage of the fold, not merely at the flat state. That is a constraint on the whole mechanism and no vertex count sees it.

Tested in A mechanism that closes on itself, at the figure it turns on.

The mathematics of folded structures is done on flat crease patterns, and a manufactured tube is that pattern rolled up.

What decides it: A tube is a different sheet: fewer free letters, fewer panels, a loop that cannot be shrunk, and a parity condition refusing half the crease counts. None of those is a property of the flat rectangle it was made from.

Tested in The tube that gets built, at the figure it turns on.

Thickness accommodation is a local technique: fix each crease and the pattern works.

What decides it: Each technique offsets a hinge from the ideal crease. On a sheet with an edge the offsets accumulate outward and the edge absorbs them. On a closed sheet the accumulation has to return to zero round every loop, which is a global condition on the offsets.

Tested in Thickness round a closed loop, at the figure it turns on.

The seven axioms are a complete list of the ways a single fold can be specified.

What decides it: Each axiom names points and lines on the sheet, and reflecting across a line presupposes that the line has two sides. On a sheet where a line need not separate, several of the seven do not specify a fold at all.

Tested in Which of the seven survive, at the figure it turns on.

Fujimoto's method divides anything into n by halving the error, so it works on any sheet.

What decides it: The iteration measures the leftover against the sheet's far edge and folds it back from there. A closed band has no far edge; the leftover wraps round to where the estimate began, and there is nothing to fold it against.

Tested in Dividing a loop into n, at the figure it turns on.

A paper proportion is a shape a sheet has to be cut to.

What decides it: The √3 a Möbius band needs is a minimum: every strip at least that long folds and no shorter one does. The √2 of the A series is a form: only that proportion has the halving property, and longer or shorter rectangles do not.

Tested in The proportion a band asks for, at the figure it turns on.

A number arrived at by two subjects is the same result found twice.

What decides it: The folded Möbius band and the smooth one are different objects: one is flat except on three lines, the other is curved everywhere. They share a bound and the relation between the problems is a limit that nothing here establishes.

Tested in Found by people not folding paper, at the figure it turns on.

The standard conditions of flat-foldability are stated completely wherever they are quoted.

What decides it: The two-colouring is stated as "the panels take two colours" or "a closed path crosses an even number of creases", and both forms are false on a sheet with one side. The vertex conditions are stated completely and need no hypothesis.

Tested in A theorem with an unstated hypothesis, at the figure it turns on.

A folded surface in a body is a sheet with an edge, like a sheet of paper.

What decides it: A gut, an airway and a blood vessel are tubes — closed in one direction, with a loop that cannot be shrunk, and a parity condition on the creases crossing it. A leaf is a disc. Which of the two a structure is decides which conditions apply.

Tested in Nothing grown has a seam, at the figure it turns on.

Four lineages folding the same pattern have converged on the same design, so the pattern's parameters are what selection settled.

What decides it: The packing a corrugation delivers once each hinge has taken its surface is k(1 - k(pi-2)rho/S), whose maximum sits at k = S/2(pi-2)rho. The optimum fold count is therefore inversely proportional to the hinge radius, and four materials spanning a tenfold range of radius have optima spanning a tenfold range of fold counts. Agreeing on the pattern leaves the number entirely open.

Tested in Four materials, four optima, at the figure it turns on.

A packing ratio quoted for a folded wing is a property of the wing's crease pattern.

What decides it: A corrugation closed to a half-angle theta from shut covers sin(theta) of its open area, so its ratio is one over sin(theta) and is unbounded as the fold closes. The pattern enters only as an exponent - a Miura reads the square - so a quoted ratio of fifteen means 3.8 degrees for a corrugation and 14.9 for a Miura. The number is a report of the angle, and the angle is what the hinge limits.

Tested in The number is the angle, at the figure it turns on.

The degree of the equation a number satisfies says how hard it is to fold.

What decides it: The degree says whether it is reachable at all. What a tower costs is the number of extension steps it takes, which for a degree of two-to-the-a times three-to-the-b is exactly a plus b - so a ninth root, of degree nine, is two steps, and an eighth root, of degree eight, is three. The larger degree is the shorter tower.

Tested in Reachable is not cheap, at the figure it turns on.

What folding adds to the compass is a handful of extra polygons, of which the heptagon is the first.

What decides it: Counted to any bound, the two sets diverge. To forty sides the fold reaches 31 against the compass's 16; to four hundred, 155 against 40; to a thousand, 275 against 52. Both conditions are smoothness conditions on Euler's totient - a power of two for the compass, nothing above three for a fold - so the whole of the gap is the permission to carry a factor of three, and to carry it repeatedly.

Tested in How many polygons a fold reaches, at the figure it turns on.

Two constructions of the same point are equally good, because both are exact.

What decides it: A crossing of two creases at angle phi moves by one over sin phi times whatever either crease moved, so its conditioning is a property of the construction and not of the point. Over two rounds on a square the four linear axioms bottom out at 36.9 degrees - a multiplier of 1.67 - and admitting the conic axiom produces crossings at 6.3, a multiplier of 9.1, with a hundredth of the references below twelve degrees.

Tested in What buys the reach costs the accuracy, at the figure it turns on.

Two simultaneous folds admit twenty-two operations, by the same enumeration that gives seven for one fold.

What decides it: That enumeration spends 2m constraints on alignments to points and lines already on the paper. An alignment may also name a crease being made in the same instant, which costs one constraint like any other and is unavailable to a single fold. Admitting three such alignments leaves the one-fold count at seven and raises the two-fold count to eighty-six, sixty-four of which use one.

Tested in Twenty-two is a floor, at the figure it turns on.

The paper is what limits how many layers a design can carry.

What decides it: It is one of two ceilings and it is not always the binding one. The stack allows fewer layers than feature over thickness; the grid allows fewer than sheet over the finest crease a folder can place. The two cross at a sheet size equal to the finest crease times feature over thickness, which is 30 mm for copier paper and 167 mm for the thinnest tissue - so on thin paper and small sheets the folder is the limit rather than the substrate.

Tested in Which ceiling is binding, at the figure it turns on.

Which folk stars exist is a matter of which symmetries are pleasing or which folds are easy to describe.

What decides it: A k-pointed star puts 2k thicknesses under the scissors, and the geometry admits any k. Taking a clean single cut as 0.6 mm of stack, copier paper reaches k = 3, washi reaches 6, and the thinnest tissue reaches 16. The three-pointed and five-pointed stars are separated by the paper rather than by the construction, which draws either without difficulty.

Tested in How many wedges the paper allows, at the figure it turns on.

Nothing about the merge can be dated firmly, since every lineage's own dates are contested by centuries.

What decides it: A joint date is the later of two surviving sources, so it inherits the better-attested half of each pair. The worst single claim in this set runs 980 years ahead of its evidence; the year at which all five are simultaneously attested is 1838, one year from where the same five are popularly dated together.

Tested in When two of them are first attested together, at the figure it turns on.

The record shows a general tendency for claims to be dated far ahead of their evidence.

What decides it: It shows two claims doing so enormously and the rest doing so mildly. Removing the five single-witness entries takes the mean overrun from 357 years to 193 and the median from 201.5 to 184.5 - a collapse in one statistic and nothing in the other, which is what a concentrated effect looks like. The two largest overruns in the record, at 980 and 897 years, each rest on exactly one surviving document.

Tested in One lost source and the story changes, at the figure it turns on.

On the way open, a finer corrugation buys clearance and costs nothing, because its hinges have already been paid for in surface.

What decides it: Each hinge is a compliant region that stores energy as it bends, with a stiffness inversely proportional to its arc. The force to hold a corrugation at a span is the slope of that energy along the span, and it comes out as the fold count times one hinge's share. The clearance is the sheet over the fold count times the cosine of the angle. Their product is 75.04 at a half-angle of half a radian for four, eight, sixteen and thirty-two folds alike.

Tested in The fold count sets the spring, at the figure it turns on.

A folded wing can be made to stay both open and folded by giving its creases springs that want different angles.

What decides it: If the creases belong to a corrugation, every crease folds by the same angle, so the springs' energy is a sum of squares of one variable and has one bottom, at the weighted mean of what the springs want — on all 200 random spring settings tried. The same four springs at a degree-four vertex give two or more resting states on 198, 196 and 196 of those settings for three different vertices, one on each branch of the vertex's motion. The second state needs a vertex, not merely disagreeing springs.

Tested in A corrugation has one resting state, at the figure it turns on.

The lattice a DNA origami is designed on is a drawing convention, chosen for convenience, and a square grid is as good as any other.

What decides it: A crossover needs the backbone to face the neighbouring helix. At 10.5 bases a turn a neighbour a third of a turn round is faced exactly at 7, 14, 21, 28, 35 and 42 bases. A neighbour a quarter of a turn round is faced exactly at no whole base — 8b ≡ 21 modulo 84 has no solution — and the nearest bases miss by 4.3°. A design that places a square-lattice crossover every eight bases has drifted 17.1° by thirty-two bases and 137° by two hundred and fifty-six.

Tested in The helix chooses the lattice, at the figure it turns on.

Measuring the distance between two marked points before and after growth reads the interior metric, and so detects the curvature a rim measurement cannot.

What decides it: One distance, or two, or the three distances among three marks, are always the sides of some flat figure, so no set of three marks can refuse flatness on any growth profile. Four marks — a centre and three at one radius a third of a turn apart — must sit √3 times their radius apart on a flat sheet. Uniform growth meets that to 1.5×10⁻⁹; the profile whose total curvature is zero misses by +3.52% at the rim; a spherical cap misses by −1.71%.

Tested in Three marks see nothing, at the figure it turns on.

Restarting a heavy-tailed search at the right cutoff makes it cheap, so the thirty-two-fold saving measured on the rhombille patch is what restarts are worth on that search.

What decides it: The cutoff of a hundred nodes was chosen by reading the runs afterwards. The universal schedule, which knows nothing about the runs, costs 3,222 nodes in expectation on the same 120 runs against 512 for the hindsight cutoff — 6.3 times as much, close to log₂ of that cutoff, 6.64. Of the thirty-two-fold saving over running to completion, the universal schedule keeps about five.

Tested in What the hindsight was worth, at the figure it turns on.

Whether a set of joins holds a slit grid of cranes together is a property of the whole sheet, which a maker can only find out by cutting it.

What decides it: Of the 33,412,811 subsets of a six-by-six grid's twenty-five joins that fail to hold the cranes together, 99.64 per cent leave some crane with none of the joins at its four corners kept — a failure visible at that crane alone. The rest hold every crane by something and still fall apart, and among the subsets that pass the look at each crane the share that hold together falls from 100% at three by three to 54.3% at six by six.

Tested in Nearly every cutting fails at one crane, at the figure it turns on.

Folding cannot help a flat sheet toward a sphere; the only answers are to cut it into gores, to curve its creases, or to stretch it.

What decides it: A tuck folds the circumference a cap does not have under the visible surface instead of cutting it away. Gathering a hemisphere hides 1 − 2⁄π of its rim, 36.3%, and leaves the rim π⁄2 sheets thick on average, exactly; simple tucks of three layers fit side by side up to a cap of 130.6°. The visible pieces between tucks still carry a gore's strain, so a tuck is a seam folded rather than cut — a fourth answer that pays in layers.

Tested in A tuck keeps what a gore cuts, at the figure it turns on.

A gathered sheet approximates a sphere better the more tucks it has, so the number of tucks sets how close to round the result can be.

What decides it: The length a cap needs hidden inside each circle is fixed by the sphere; tucks only decide how it is shared out. Straight tucks hide length along a straight line from where they start, so the error in the hiding is set by how many radii the tucks start from and not by how many tucks there are. From one, two, four, eight and sixteen radii the worst shortfall is 36.9, 12.3, 3.3, 0.8 and 0.2 per cent of the rim's hiding, each doubling dividing it by nearly four.

Tested in A straight tuck is a cone point, at the figure it turns on.

The deepest pile a pattern folds to is the number a thick-panel design has to accommodate, so two patterns with piles of the same depth pose the same problem.

What decides it: The Miura and the tapered corrugation both pile sixteen layers deep. On the Miura that depth covers 11.9% of the folded footprint, in three separate places, and 75.5% is at least half as deep; on the tapered corrugation it covers 0.9%, in four places, and 87.9% is at least half as deep. The Yoshimura's sixty layers cover all of its footprint, and the square twist's nine cover a single patch of 17.4% with nothing else half as deep.

Tested in Three kinds of pile, at the figure it turns on.

Any finished crease pattern can be extended by grafting, since a strip slid in along a cut costs its width times its length and changes nothing else.

What decides it: The price holds only along a line every crossed crease meets square. Every line between the vertex columns and between the vertex rows of the eight printed patterns was tried: the preliminary base, the square twist, the Yoshimura and the waterbomb tessellation admit none; the Miura, the tapered corrugation and the hexagon twist admit lines in one direction only; and the fold-and-cut triangle's four admissible lines all run through blank margin and cross no crease. One diagonal crease in a four-by-four grid removes one admissible line in each direction.

Tested in A graft needs a square line, at the figure it turns on.

A folding vertex with two resting states can be made to hold both firmly by choosing its sector angles well, since the geometry of the vertex decides how hard it is to switch from one state to the other.

What decides it: The only road between the two branches of a degree-four vertex's motion crosses the flat sheet, and on 198 of 198 and 196 of 196 two-state spring settings the flat sheet is that road's highest point. Its energy is the sum of each stiffness times its rest angle squared, which contains no sector angle: the same springs on four vertices give a wall of 26.160 on every one, while the states below it move. And the shallower state sits a median of 4.8 per cent of the wall's height below it.

Tested in The wall is the flat sheet, at the figure it turns on.

A Miura's pile takes only four depths, so a thick Miura can be built from panels of four thicknesses, each panel given the thickness of the depth it sits over.

What decides it: Laid on the depth map, every one of the Miura's twenty-four folded panels lies over at least three of the four depths, and eight of them over all four; every one of the tapered corrugation's twenty-eight lies over all four. No panel of either pattern sits over a single depth. On the Yoshimura, the waterbomb tessellation and the preliminary base every panel does.

Tested in A panel is not the unit of depth, at the figure it turns on.

A dome gathered with rings of straight tucks follows a sphere as closely as the number of rings allows, so the only way to make it rounder is to add rings.

What decides it: With the rings placed so that every stretch between them carries the same worst error, four rings on a hemisphere leave a worst shortfall of 2.0 per cent of what the rim hides against 3.3 per cent evenly spaced, and sixteen leave 0.12 against 0.21. Following the sphere within one per cent then takes six rings rather than eight, and within a tenth of a per cent eighteen rather than twenty-four.

Tested in Crowd the tucks toward the rim, at the figure it turns on.

A structure that opens as its bud opens is being driven open, so what a fold count costs a leaf is the force needed to drive it.

What decides it: With hinges that rest flat, the energy stored is k·c·(π − 2θ)² ⁄ 2, which is largest when the corrugation is shut and zero when it is flat. The corrugation therefore opens on its own, and the slope of that energy along the span is the force required to STOP it. It falls from (2πkc) ⁄ S at the shut state to 4kc ⁄ S at flat and rises nowhere in between.

Tested in How far open is a question about the grip, at the figure it turns on.

A level of folding pays a factor of four for its own thickness, so any arrangement that puts a sheet into a clearance pays about the same and the architecture hardly matters.

What decides it: The four comes from members sharing the clearance: k plies across a clearance c from a sheet of thickness τ hold k(1 − kτ ⁄ c), best at c ⁄ 4τ. Walls standing on the base do not share the height — n of them hold 2cn and compete only for footing, nτ ≤ 1, best at 2c ⁄ τ. The ratio is 8 at every clearance from 0.4 to 4 and every thickness from 0.002 to 0.05.

Tested in Standing up beats lying down by eight, at the figure it turns on.

Allowing more folds at once multiplies what can be constructed, because the catalogue of operations grows so fast.

What decides it: The catalogue runs 7, 105, 3042, 145211, 9782771 at one to five folds. What any one operation admits runs 3, 9, 27, 81, 243, because a fold's two alignments are curves of degree at most two in the plane of lines and two parabolas share the line at infinity — so their product loses one. And the largest irreducible degree m folds settle is 2m + 1: 3, 5, 7, 9, 11.

Tested in Counting operations is not counting power, at the figure it turns on.

Two simultaneous folds are hard because a folder has to achieve several coincidences in the same instant, so the difficulty is one of dexterity.

What decides it: The reference graph splits the 105 two-fold operations into 28 whose folds name nothing simultaneous, 49 whose naming has no cycle and can therefore be performed in an order, and 28 whose folds each name the other. Only the last is a simultaneous system, and what it needs is not a steadier hand but a solution: an alternating solve contracts the error to 8 per cent a pass and reaches eleven decimal places in ten passes.

Tested in A crease that does not exist yet, at the figure it turns on.

A uniaxial base exists for a set of flaps exactly when circles of those radii can be packed into the sheet without overlapping.

What decides it: Disjoint circles require |pᵢ − pⱼ| ≥ m(ℓᵢ + ℓⱼ), which is the pairwise condition with every internal edge of the tree set to zero. A tree with a body of 0.8 supports a scale of 0.2783 under the pairwise condition and 0.3228 under the circles; one with two internal edges supports 0.2112 against 0.2740. Only a tree with no internal edge gives the same answer both ways.

Tested in Every pair, not every circle, at the figure it turns on.

The pairwise condition has a requirement for every pair of flaps, so a design with many flaps is held rigid by an enormous number of constraints.

What decides it: In the best arrangement found, 4 of a star's 10 pairs are at their limit, 3 of 10 on a five-leaf tree with a body, and 8 of 21 on a seven-leaf tree with two. Everything else has slack. The tight pairs together with the sheet's edges pin 5, 4 and 6 leaves respectively, and the rest can be moved without lowering the scale at all.

Tested in What the condition does not decide, at the figure it turns on.

A popular date in this field runs about three hundred and fifty years ahead of its earliest surviving evidence.

What decides it: The statistic is the mean of the eight gaps that are positive. Resampling the fifteen entries with replacement twenty thousand times puts five per cent of the recomputed values below 150 and five per cent above 591, with a spread of 133 about a value of 357. Fourteen of the fifteen leave-one-out values differ from it, two of them by more than seventy years.

Tested in The interval is wider than the number, at the figure it turns on.

The record cannot say anything about the claims that left no surviving source, because there is nothing to count.

What decides it: The number of unseen classes is estimated from the low end of the observed distribution — the classes seen once and twice. Here that is 5 and 5, giving 2.5 missing claims against 15 observed. What refuses the question is not that it cannot be asked but the answer's interval: 15.4 to 30.6, and re-reading one entry moves the estimate from 17.5 to 19.5 or to 16.3.

Tested in A question the record is too small to answer, at the figure it turns on.

Crowding the tuck starts toward the rim buys accuracy at the cost of depth, because it puts the starts where the gathering is already thickest.

What decides it: A gathered cap is α ⁄ sin α sheets thick on average, which is 1.57 at a hemisphere's rim and 1.02 a fifth of the way out. A simple tuck is three sheets. So the deepest start of the equal-error placement sits in 3.37 sheets and the deepest of the even placement in 3.32 — a difference of 1.5 per cent, against an error the crowded placement reduces by 38.

Tested in Crowding outward costs almost nothing, at the figure it turns on.

Tucks, gores, curved creases and stretch are four different approaches to the sphere, so choosing between them is choosing which is most efficient.

What decides it: Each disposes of the same excess, 1 − sin(s) ⁄ s of the circle at arc s, and each does it by dividing the circle so that the residual inside a piece is within what the material absorbs. The count is ⌈f ⁄ ε⌉ for all of them: 19 on a hemisphere at 2 per cent of stretch, 8 at 5, 4 at 10, 2 at 20. Only the cost of one division differs — a cut, three sheets of pile, or neither.

Tested in Three answers, one count, at the figure it turns on.

Driving one crease of a rigidly folding quadrilateral mesh settles every other one, so an actuator may go on any crease.

What decides it: It holds on a mesh whose vertices all differ: every one of 24 creases leaves exactly one consistent assignment. On a four-by-four Miura driven to 0.6 radians, 4 creases leave one, 8 leave two, 4 leave four and 8 leave eight — and the four that decide it are c:3:2, c:3:3, r:3:2 and r:3:3, all at the same corner.

Tested in Only four creases decide a Miura, at the figure it turns on.

Two actuators on a sheet with one freedom are redundant, so a small disagreement between them is absorbed harmlessly.

What decides it: They are over-determined rather than redundant: two commands for one number. Minimising the stored energy with equal stiffnesses leaves gδ ⁄ (1 + g²) of movement at the first crease and δ ⁄ (1 + g²) of residual at the second, for a gearing g. On one rigid mesh the gearings from a single crease run 0.316 to 1.761, so the residual left standing runs from 91 per cent of the disagreement down to 24.

Tested in Two drivers and one freedom, at the figure it turns on.

A folded deployable that has to work repeatedly is the same structure as one that works once, with a more durable hinge.

What decides it: A crease of radius ρ in a sheet of thickness t strains its outer fibre by t ⁄ 2ρ, and a material taking N cycles takes about C·N^(−b) of strain, so the smallest surviving radius goes as N^b. The packing optimum is S ⁄ 2(π−2)ρ, so it goes as N^(−b): 87.6 folds at one cycle, 27.7 at ten, 8.8 at a hundred, 2.8 at a thousand, with the packing falling from 43.8 to 1.4.

Tested in What a second deployment costs, at the figure it turns on.

A pattern should be folded at the count that packs best, since compaction is what a deployable is sold on.

What decides it: A deployment needing all n hinges succeeds with probability p^n, so what a mission gets is the compaction times that. On a sheet of ten with hinges of radius 0.05 at 0.99 each, compaction peaks at 88 folds and compaction times the chance of it peaks at 54 — and the shift costs under a per cent of compaction while adding tens of points of deployment probability.

Tested in The crease count is a reliability budget, at the figure it turns on.

A staple's binding domain has to be long enough that the stretch it pairs with occurs nowhere else in the scaffold, and that requirement is what sets the length of a staple.

What decides it: Both requirements were computed against the same 7,249-base scaffold. Uniqueness across a design of 454 domains is met at eleven bases. A duplex of eleven base pairs melts at 24 degrees, and a design held at 45 needs seventeen. The uniqueness requirement is real and it is slack by six bases at the length the other one demands, so it decides nothing.

Tested in A domain too short to be unique, at the figure it turns on.

The three simple-fold machines are ordered by strength, so a measurement of what they can produce will order them the same way a measurement of what they can decide does, with the gaps in the same places.

What decides it: The reach of all three was enumerated on four evenly creased strips and five unevenly creased ones. The ordering holds and says almost nothing: the some-layers machine reaches every state of every strip, the one-layer machine reaches exactly four on all nine, and the all-layers machine reaches everything on the even strips to six stamps and nothing at all on the uneven ones. Three machines, and one of them has no dependence on the strip, one has no dependence on the strip, and only the middle one varies.

Tested in Deciding is not making, at the figure it turns on.

Folding reaches every degree that is a product of twos and threes, which is far more than the compass reaches, so folding settles most algebraic equations and the compass settles few.

What decides it: The degrees of each were counted to a million. A fold reaches 142 of them and a compass 20, so the first claim is right and the second is the wrong comparison: 142 of a million is 0.0142 per cent. Both instruments reach a share of the degrees that falls to zero, one as a logarithm squared over n and one as a logarithm over n.

Tested in Twos and threes run out, at the figure it turns on.

Adding an axiom to a folder's repertoire adds fold lines and therefore adds reference points, so a larger axiom set is a strictly better instrument on a sheet in the same way it is in the field.

What decides it: Every crossing of every pair of fold lines was formed on a unit square for four axiom sets and two rounds, and the ones missing the paper were counted rather than dropped. The first two axioms lose nothing. The four linear ones lose 57 per cent at one round and 72 at two. Adding the conic axiom loses 76 per cent at one round, and at two rounds it specifies 16,762 distinct folds and cannot be computed at all. The reference count still rises; the share delivered falls at every step.

Tested in The axiom that reaches furthest wastes most, at the figure it turns on.

Two crossing grafts cost their separate bills plus the rectangle where they cross, so p strips crossing q strips cost their separate bills plus p times q rectangles, and a design with many features has a budget that grows with the product of the counts.

What decides it: Five arrangements of strips were slid into a grid and every area summed over the panels the creases enclose. The p·q rectangles are real and the excess over the separate bills is exactly the product of the two families' total widths — one term at every arrangement, including three strips by two with unequal widths. The budget grows with the two totals, not with the two counts.

Tested in Six rectangles and one term, at the figure it turns on.

A graft costs exactly the paper slid in for it and nothing else, so a designer may choose the strip's width freely and pay only for the area.

What decides it: The greatest common divisor of the vertex coordinates was computed before and after grafting, for seven widths on an eight-by-eight grid. Widths of one, two and three spacings leave the grid exactly as it was. A width of 0.10 against a spacing of 0.125 divides the grid by five; a width of 0.13 divides it by twenty-five. The area bill is exactly as promised in every case, and the design that comes out of five of the seven is not on a grid anybody would draw on.

Tested in The width is charged in grid, at the figure it turns on.

The substrate's constraint on a design is its thickness: thinner paper allows more layers, and the sheet can always be made larger if a design needs more of it.

What decides it: The mass a hand mould carries is its own weight plus a newly formed sheet holding about ten times its mass in water, so the largest sheet it can be lifted and shaken with is bounded in area. Over mould weights from four to twenty kilograms a square metre and lifts from five to twenty kilograms, the largest square sheet runs from 495 to 2,122 millimetres — and a sixty-four-layer model finished at a hand's width wants 1,200 of them.

Tested in A sheet is as large as two arms, at the figure it turns on.

The substrate bounds a design's layer count, so a folder who wants more layers needs better paper and a folder who wants a larger model needs a larger sheet, and the two demands are independent.

What decides it: The three bounds were drawn as one region. The crease floor fixes the paper at about 38 microns and its stack at 80 layers. The sheet ceiling gives (largest sheet over finished size) squared, which at a 1,200 mm sheet is 80 layers at 134 mm and falls as the square after that: 36 at 200 mm, 16 at 300, 4 at 600. Above the corner the paper is not the binding constraint at all and better paper buys nothing.

Tested in Eighty layers and the sheet decides the rest, at the figure it turns on.

A crease pattern's density is limited by the paper: a pattern fine enough will have creases closer together than the paper can carry, which is what stops a tessellation from being made finer.

What decides it: A crease occupies a band a few sheet thicknesses across, so a sheet of thickness t carries at most 1/(6t) metres of crease a square metre — 1,667 on copier paper, 4,167 on washi. The densest pattern on the printed shelf is the waterbomb at 89. Solving the Miura family's linear growth for the ceiling puts it at 139 cells across on copier paper, which is cells 1.22 mm wide — finer than a hand can place, so the limit arrives somewhere else first.

Tested in The density a paper allows, at the figure it turns on.

The best regular polygon only a fold can build ranks fourth at best on any rectangular sheet, and a polygon needing two simultaneous folds ranks ninth.

What decides it: Both ranks were read on seven sheets. On a sheet 1⁄cos(π⁄14) = 1.0257 long the fourteen-gon touches all four edges and ranks third among every polygon to forty-eight sides; on 1⁄cos(π⁄22) = 1.0103 the twenty-two-gon ranks fifth. Every 4k + 2-gon from six to forty-six ranks exactly (n − 2)⁄4 on its own sheet.

Tested in The sheet a polygon fits exactly, at the figure it turns on.

Creases that meet at a vertex may come as close as they like, so a paper's limit on fineness is set by creases that do not meet, and the finest Miura a 170 mm sheet of copier paper carries is 262 cells a side.

What decides it: Near a vertex the bands of two creases at sector angle θ overlap for w⁄sin θ along each. On the Miura family every straight crease is 2.13 band widths of overlap at its two ends per cell of its own length, and it becomes overlap from end to end at 135 cells a side on copier paper; the waterbomb's half-diagonals do so at 82 cells, against a spacing bound of 141.

Tested in A vertex creases the paper twice, at the figure it turns on.

An edge that costs nothing to lengthen can absorb length for nothing, so a design with a free edge should spend all its slack there.

What decides it: The bird's tail is free as drawn. Moving a tenth of a unit to it from the head raises the scale 3.8 per cent, and re-pricing then puts the tail at 0.044 and the legs, which were not touched, at 0.079 against 0.031 before. The free edge was free for the length of one step, because the length added to it brought it into a pair at its limit.

Tested in A price holds until the arrangement moves, at the figure it turns on.

Rounding a subject's limbs to the nearest grid unit is the neutral choice, costing the design only the shape error it introduces.

What decides it: Every rounding of the bird's five edge groups up or down was solved on grids of four, six and eight units. On eight units the nearest rounding gives a model of size 0.2631 — smaller than the unrounded bird's 0.2651 — and rounding the head and body down and the rest up gives 0.2782, the largest of all eight roundings, at a worst-limb error of 20 per cent against 15.

Tested in Rounding in the cheap direction, at the figure it turns on.

The layer order a crease pattern file leaves out is worth about log₂ of the number of orderings of its panels — fifteen bits for the preliminary base, thirty-three for the hexagon twist — so every file is missing that much of its folded object.

What decides it: Listing every legal stacking of the printed patterns small enough to list, the preliminary base, the square twist and the hexagon twist each have exactly one folded state, so their layer order is fixed by the crease pattern and the missing information is zero bits. The fold-and-cut triangle has two states, one bit. The Miura patches are the exception: the four-by-three has eleven states, 3.46 bits.

Tested in The field is empty where it would say nothing, at the figure it turns on.

A deployable is most reliable with a single degree of freedom and a single actuator, because every extra freedom is an extra actuator that can fail.

What decides it: True of the chance that the whole area opens: three hundred hinges at 0.999 with actuators at 0.99 open fully 73.3 per cent of the time as one module and 67.0 per cent as ten. False of every partial requirement: at least nine tenths of the area opens 73.3 per cent of the time as one module and 94.4 per cent as ten, and the expected share rises from 73.3 to 96.1 per cent.

Tested in Splitting a sheet buys area, not certainty, at the figure it turns on.

A pattern that converts crease length into compaction efficiently is also cheap to qualify, so ranking deployable patterns by crease length per layer ranks them by the testing they need.

What decides it: Testing grows with the hinge count, not the crease length. Ranked by hinges per layer of compaction the printed patterns reorder: the preliminary base moves from third to first and the square twist from seventh to fourth. And on three families refined from two cells a side to eight, the Miura's hinges per layer rise from 1.32 to 5.69 while the waterbomb's settle at 2.47 and the Yoshimura's at 1.47.

Tested in The pattern cheapest to trust, at the figure it turns on.

A structure packed long enough in its container comes to hold itself, so time spent packed is force the container no longer has to supply and the structure is simply better off.

What decides it: With the remembered turn relaxing toward the held turn, the container's torque falls as e^(−T⁄τ) and the torque needed to open the corrugation afterwards rises by the same amount, so the two sum to C(φₕ − φₒ) at every holding time to 10⁻¹². Held three relaxation times, the container supplies 4 per cent and the opening needs 96, and the barrier to the mirror state is 8.41 times its creased value.

Tested in Holding a fold moves the force, at the figure it turns on.

A tiling on which the twist construction's pleat equations close round every loop is a tiling the construction can fold, so consistency of the loops is the condition that decides which irregular tilings carry a twist tessellation.

What decides it: Under a shear, a stretch and a general linear map the square grid, triangular grid and honeycomb close every loop exactly, with every edge's equation equal to one. At turns of 0.2, 0.42, 0.7 and 1.0 radians the construction on those images fails the angle condition at its worst vertex by between 0.08 and 0.33 radians, and folds on none of them, while folding on all five tilings as drawn.

Tested in Closing the loops is not folding, at the figure it turns on.

A twist tessellation's polygon is fixed by the tiling once the turn is chosen: the angles come from the tiling's own sectors and nothing is left but how large the polygon is, so a tiling on which the construction fails carries no twist tessellation of this kind.

What decides it: The corner condition constrains the sums of side distances along each edge, not the distances one at a time, so a vertex of degree $K$ carries $K$ unknowns. Solved that way at a turn of 0.42 radians, all twenty of the five tilings and their three linear images fold every vertex condition, where one distance a vertex folds five. On the stretched square grid the sides the conditions ask for measure 0.904 and 0.603 against the 0.753 a single distance can produce.

Tested in One number where the corners wanted four, at the figure it turns on.

Whether a tiling carries a twist tessellation is settled by whether the construction's pleat equations are consistent round every loop, so a tiling that fails the loop test carries no twist tessellation.

What decides it: The loop test changes under a linear map — it passes the rhombille as drawn and fails every linear image of it — while the condition a twist tessellation actually needs does not, because a linear map carries a balancing set of edge weights to the same numbers. Measured on five tilings and three maps, the balancing weights are identical before and after every map, and the weights the pattern writes balance at every vertex to 3.1 × 10⁻¹⁶ and agree at the two ends of every edge to 1.4 × 10⁻¹⁴.

Tested in Every twist writes an equilibrium, at the figure it turns on.

A twist tessellation draws in by the same amount in both directions, which is what distinguishes it from a corrugation and from the Miura.

What decides it: True of the square twist and of no other. Measured at a radius of 0.17, a twist on a triangle draws in 1.082 along and 1.168 across, on a pentagon 1.232 and 1.253, on a hexagon 1.287 and 1.168, and on a heptagon 1.374 and 1.369. The square twist alone is equal, at every radius from 0.06 to 0.33.

Tested in The plane the five points were in, at the figure it turns on.

A curved fold's reach is the distance to where its rulings cross, so a design that stays inside that distance on either side of the crease has surface everywhere it needs it.

What decides it: The reach is a property of one surface, not of the fold. The two surfaces' envelope denominators are exact negatives, so their bounded stretches partition the crease: measured at 241 points on four crease curves and four ruling angles, no point has both sides bounded, and on the circle, the ellipse and the parabola the second surface is bounded at no point at all.

Tested in Only one side can run out, at the figure it turns on.

A comb of walls beats a stack of plies by eight because its members stand perpendicular to the base, so a body that wants the factor has to build its members square.

What decides it: The angle cancels before anything is evaluated: a member leaning at an angle is the clearance over the sine of it long, and occupies the thickness over the same sine of the base, so the reach is twice the clearance over the thickness at every angle. Measured at ten angles from 90° to 1°, the reach is 200.000000000 on all ten from a sheet 0.01 thick in a clearance of 1, while the member length runs from 1.000 to 57.299 and the count from 100 to 1.745.

Tested in The angle the eight does not know, at the figure it turns on.

A comb of standing members holds eight times what layers lying flat hold, so a body that wants surface in a volume should stand its members across the space rather than lay them along it.

What decides it: The eight is a fact about a flat base. Inside a tube, radial fins must clear one another where their tips are, so their count falls as their height rises; their best height is half the radius and they then occupy exactly half the cross-section. Concentric layers occupy all of it. Measured at radii 0.5, 1, 2 and 4 out of a sheet 0.01 thick, the layers hold 2.0400, 2.0200, 2.0100 and 2.0050 times what the fins hold, the excess over two being the half-layer a whole count leaves over.

Tested in In a tube the standing members lose, at the figure it turns on.

Gluing a period cell's edges is what makes the search for a lettering dear, so a sheet cut from the plane stays cheap however large it is.

What decides it: True of three of the four tilings swept and false of the fourth. On the square grid, the honeycomb and the triangular grid the cut cell costs between 0.31 and 0.37 of a node per free letter at every size from one period to five, never backtracking. On the rhombille it costs 0.37 at one period and 64.05 at two — 13,834 nodes against 216 free letters — with no gluing at all.

Tested in Each drawing has its own threshold, at the figure it turns on.

Gluing a period cell's edges makes the search for a lettering three orders of magnitude dearer, so a sheet with no rim is that much harder an object than a sheet with one.

What decides it: The factor is a property of the pair. Under eight branch orders on the square grid's four-period cell, the cut sheet costs 42 to 55 nodes and the torus 69 to 24,636, with one order failing inside its budget. The torus's cheapest order costs 1.6 times the cut sheet's cheapest, against the 26 a single fixed order reports.

Tested in The route, not the sheet, at the figure it turns on.

A drawing can fail in several independent ways, so a checker has to test for each of them — a crease that crosses another, and a crease that stops in the middle of the paper.

What decides it: Across 120 twist patches at three periods and eight turns, 76 carry a crossing and 15 carry a stub, and the second set lies inside the first: no patch has a stub without a crossing. The 66 stubs are 33 pairs — the count is even on every patch, and every distance from the rim occurs an even number of times.

Tested in A stub is never alone, at the figure it turns on.

A crease whose curvature never changes sign has one surface that is never bounded, because the sign of the curvature is what decides which of the two runs out.

What decides it: The sign is carried by the curvature plus the rate the ruling angle turns at, and only a constant angle makes the second zero. On a circular crease with the angle running 1.4 plus a rate times a sine, the outer surface is unbounded along the whole crease at every rate up to 1.00 and bounded over 14% of it at 1.10, 20% at 1.25 and 25% at 1.40.

Tested in An angle that turns faster than the crease, at the figure it turns on.

A regular polygon's best sheet is found by sampling proportions: the peak can only be located to the resolution of the sheets tried, and whether it sits exactly on the sheet the polygon fits is a matter of measurement.

What decides it: For a polygon of 4k + 2 sides the share on a sheet one wide and h long, while it turns, is (n·cos(π⁄n) ⁄ 4 sin(π⁄n))·(sin²(π⁄n) + (h − cos(π⁄n))²) ⁄ h, which agrees with a search over rotations to a part in a billion. Its derivative is proportional to h² − 1 for every n, so the share rises on every sheet from the square to 1⁄cos(π⁄n) and falls after it: the peak is exactly the own sheet.

Tested in Turning is uphill all the way, at the figure it turns on.

A drawn crease pattern can fail in four independent ways — two creases crossing, a crease stopping short, a crease ending on another crease, and a crease too short to see — so a checker needs a test for each.

What decides it: On 120 extension drawings and 120 clipped drawings of every twist patch, no crease ends on the interior of another crease; the reading's junctions are all creases meeting the rim, which every drawing has. All fifteen drawings with a stub also have a crossing. Crossings and fragments are the only two faults that occur on their own, and five clipped drawings carry a fragment and nothing else.

Tested in Two faults, not four, at the figure it turns on.

Radial fins inside a tube lose to concentric layers by a factor of two, and the loss is a property of radial arrangement, of which a hierarchy of shorter fins can recover only a part.

What decides it: The factor of two is the best that fins of one length can do. Fins of m lengths with their tips equally spaced along the radius hold exactly m ⁄ (m + 1) of the ceiling 2πR² ⁄ τ, checked against the fins counted one length at a time and against four thousand random schedules of two, three and four lengths. The schedule of halving heights measured before converges to two thirds; equal spacing passes eighty-eight per cent at eight lengths and reaches the ceiling in the limit.

Tested in The wedge belongs to one length, at the figure it turns on.

A cut sheet's search cost is a point — the same whatever order the letters are taken in — and the wide spread of costs over branch orders is what gluing a sheet's edges does to it.

What decides it: On the cut sheets of four tilings from two to four periods, eight branch orders agree to within a factor of 1.3. The rhombille's cut sheet does not: at two periods its eight orders cost 67 to 13,834 nodes with one out of budget, and at three periods seven of eight run out. The spread is the signature of a sheet past its threshold, glued or cut, and gluing is one way of getting there.

Tested in The cheapest route crosses later, at the figure it turns on.

Origami is an ancient Japanese art, practised for something over a thousand years.

What decides it: The record, laid out as a table with one row per claim. Paper reaches Japan in the seventh century and the earliest surviving reference to folding it for amusement is Saikaku's poem of 1680; the oldest surviving book of recreational folding is of 1797. Eight of the fifteen claims in this site's record are popularly dated earlier than anything that attests them, by a median of 201 years, and the largest single gap is 980. The generator refuses any entry whose popular date precedes its evidence without saying so.

Tested in Nothing here is as old as it sounds, at the figure it turns on.

Traditional origami is folded from a single uncut square; cutting is a later corruption, or a different art altogether.

What decides it: The oldest surviving book of recreational folding does not keep the rule. The connected cranes of the Hiden Senbazuru Orikata are cut from one sheet along a grid and left joined at the lattice points. The generator computes the arithmetic: an n by n arrangement gives n² birds held at (n−1)² corners and needs 2n(n−1) sides of slit, so the slitting per bird rises with the size of the piece. It refuses any arrangement requiring no cutting at all.

Tested in The oldest book cuts the paper, at the figure it turns on.

The Miura fold and the diamond buckling pattern are inventions of the space age, designed for deployable structures.

What decides it: Yoshimura published the diamond pattern in 1951, in a paper about how a thin cylindrical shell fails under axial load, and Miura described his fold in 1970 in work on the same subject. Neither was about deployment. The generator builds the buckled pattern, verifies it at every interior vertex, and asserts that the assignment the physics produces carries Maekawa's three-to-one split everywhere — which no designer chose.

Tested in Found before it was designed, at the figure it turns on.

Paper folding is one continuous tradition that began in Japan and spread outward.

What decides it: At least three lineages are visible in the record with different purposes, different social locations and different earliest sources: Japanese ceremonial wrapping, attested through codified manuals around 1600; recreational folding, attested from 1680 and published from 1797; and the European kindergarten syllabus of 1838, whose folds came from an existing German-speaking practice. The three met in the 1870s when kindergartens reached Japan. Merging them assigns the oldest date in any lineage to all of them.

Tested in Two traditions and a merge, at the figure it turns on.

A five-pointed star was cut from a folded sheet in a Philadelphia workshop in 1776, and the general one-cut theorem is a formalisation of a technique already understood.

What decides it: Two separate claims and both fail. The earliest source for the workshop story is an account of 1873, ninety-seven years later, given by a descendant — a secondary source with nothing primary behind it, and this site's record marks it as the only such entry. And the wedge method is not the theorem: it is a symmetry construction that gives a regular star for nothing and reaches no shape lacking that symmetry. The generator computes the released outline and asserts its regularity to 1e-12.

Tested in The star that was cut before it was proved, at the figure it turns on.

The Miura fold was invented in 1995 for the Space Flyer Unit's solar array.

What decides it: Miura described the pattern in 1970, in work on pseudo-cylindrical concave polyhedral shells, and the satellite is the date it became visible rather than the date it was known. The record marks the entry and the generator asserts what the pattern does — one degree of freedom, both in-plane dimensions shrinking together at every fold state sampled — none of which needed a satellite to be true.

Tested in From a shell to a solar array, at the figure it turns on.

The Miura pattern appears in leaves, in buckled shells and in spacecraft because engineers copied it from nature.

What decides it: The pattern was published in 1970 as an analysis of shell buckling, and the buckling was described as mathematics rather than as biomimicry. What the three cases share is a set of requirements — one degree of freedom, a monotone opening, two-directional collapse — each of which this repository computes, and the set of patterns satisfying all of them is small.

Tested in The same corrugation in four places, at the figure it turns on.

The square is the natural sheet for folding constructions: the classical recipes are written for it because it is the sheet on which they are cheapest.

What decides it: The fewest creases that mark each of the twenty-three fractions up to twelfths, found by exhaustive search with every fold of the first four axioms, average 2.96 on a square and 2.91 on an A-series sheet. Neither sheet is cheaper everywhere: the square wins on six fractions, the A-series sheet on seven, and they tie on ten. What the square is cheapest for is the recipes that were written on it.

Tested in What the square saves, at the figure it turns on.

Thickness is a manufacturing detail — get the crease pattern right and the hardware follows.

What decides it: Thickness is the one idealisation that changes the kinematics rather than the accuracy. A stack of panels binds where a zero-thickness sheet pivots freely, and the accommodation techniques are not corrections but different mechanisms: the offset-panel construction is measured here against the symmetric arrangement and overlaps strictly less at its worst pair, which is the claim it exists to make.

Tested in The sheet has a thickness.

A crease pattern is a description of what a sheet of paper does.

What decides it: It is a description of what an idealised sheet does, and the idealisation has four parts — zero thickness, no stretch, creases that are lines, perfect memory — every one of which is false about paper. The essay prices each and finds they are not independent: the same complex model fails on two of them at once, which is why competition paper is thin rather than merely strong.

Tested in Four things that are not true.

A figure computed from a model of a biological structure is evidence about that structure.

What decides it: Each row of the ledger names a computation this repository runs and would fail the build on, beside the claim it does not establish. The left column is checkable and the right column is not established anywhere on this site, and no figure in the field should be read as though it were.

Tested in The organism is not the model, at the figure it turns on.

The number of folds that maximises the surface a body holds is set by the thickness of the folded sheet.

What decides it: It is set by the thickness of the sheet plus whatever depth is needed to service each unit of it. Both are charged per fold and in the same units, so the optimum moves from S/2t to S/2(t+delta) and the peak area falls in the same proportion. At a servicing depth three times the sheet's own the best fold count is a quarter of the unserviced one.

Tested in The surface has to be supplied, at the figure it turns on.

A specimen that lies flat at its margin has a flat metric, so flattening separates a sheet that folded from one that grew.

What decides it: The integral of curvature over a grown disc equals minus two pi R times the growth profile's logarithmic slope at the rim, by parts - so it is fixed at the boundary and every detail of the interior cancels out of it. A profile whose rim slope vanishes integrates to nothing while carrying curvature reaching 1.4 in the interior, changing sign at R over root two. A margin measurement reports it as flat.

Tested in The test measures the rim, at the figure it turns on.

The set of origami numbers describes what a folder can mark on a sheet of paper.

What decides it: It describes what an unbounded plane carries. A crossing of two fold lines is a number in the field wherever it falls and is a usable reference only on the paper, and counting both shows the square discarding 1,440 crossings against 565 kept at two rounds of the four linear axioms. The share discarded is a property of the proportion of the sheet, running from 47% to 72% over the four compared.

Tested in The field has no edge, at the figure it turns on.

What the substrate limits about a folded design is how many layers it can carry.

What decides it: It limits the finished size as well, and through a different quantity. The folded footprint times the mean layer count is the area of the sheet, so a model finished at a stated size needs a sheet larger by the square root of its layer count: a sixty-four-layer model at a hand's width wants 1,200 mm of paper, which is larger than an A0 sheet.

Tested in A sheet has a size as well, at the figure it turns on.

A boundary measurement of a grown sheet can only ever report the total curvature, so the distribution of growth across the interior is out of reach of boundary measurements.

What decides it: A circular cut makes a new boundary, and the piece inside it reads −2πr(ln Ω)′(r). Cut readings on the cancelling profile run −0.258, −0.825, −1.100 and −0.677 at radii 0.25, 0.5, R⁄√2 and 0.9, while the whole disc reads nothing. Nine cuts recover its growth profile to within 0.0022 in ln Ω, and enlarging the whole sheet by one and a half or two times changes no reading at all.

Tested in A cut reads a slope, at the figure it turns on.