Sheet shape — where it appears
Named by 29 essays across 5 fields — each of them below, with the objects they name alongside it.
The square is a choice
Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.
The corner is worth four times the middle
A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.
The fifth thing that is not true
Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.
A hole is cheap paper
A flap claims every point of paper within its own length of its tip, but only the paper that is actually there — so an edge is half price and a corner a quarter. A hole in the middle of the sheet is more edge, and holding the area fixed, a square with a hole in it is cheaper paper than a solid square at every flap length measured.
A hole is an edge
A folder's first fold has to be specified by aligning things that are already there, and what is already there is the sheet's outline. Cut a square hole in the middle and the outline doubles: one round of alignments reaches nine references on a plain square and two hundred and twelve on a holed one — more than the plain square reaches in two rounds.
Spending the cheap paper
A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.
Most of a patch is edge
Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry.
The paper a pattern asks for
A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.
A notch is not a hole
Remove the same rectangle of paper from the middle of a square and from its edge, and the two sheets are not worth the same. The hole is cheaper paper at every flap length measured — 4.71% cheaper than a plain square of equal area against the notch's 2.58% — because what a cut is worth is rim with paper on both sides of it, and a notch spends one of its four sides on an edge the sheet already had.
An axiom may name no fold
The seven operations are stated about points and lines in a plane, and a plane has no edges. On a square, three of them always name exactly one fold and always land it on the paper; placing a line on a line names two, and 13.3% of the folds it specifies are creases the sheet never reaches; and placing a point on a line through a second point names two folds, one, or — 23.8% of the time — none at all.
The edge was there first
A folder's first fold has nothing to align to but the sheet's own outline, and it shows in where the marks land: four of the five references the first fold adds are on the paper's edge. By the second fold the edge holds forty-eight of five hundred and fifty-six new ones. The rim is where references are cheap and it fills up, because an edge is a line a fold can cross twice while two folds inside the paper cross once each.
The axiom that names two folds
Bring one line onto another and the operation is satisfied by either of two folds, always exactly perpendicular to one another, creasing the paper in completely different places. Over four thousand random line pairs on a square, both folds land on the sheet two thousand nine hundred and sixty-five times, and the statement of the axiom does not say which one is meant. The fifth axiom is worse: its two answers are at any angle at all, from half a degree apart to square.
How far from the nearest reference
One fold puts nine reference points on a square sheet and two folds put five hundred and sixty-five. That is sixty-three times as many points, and it brings the worst-covered spot on the paper from a third of a sheet away to a twelfth — four times closer. A count of references is not a measure of what a fold buys, because a set of points can be arbitrarily crowded and still leave most of the sheet out of reach.
The edge is what makes it hard
Grids, crumples, leaves, corrugations and fold-and-cut patterns all give up a consistent lettering at one step per panel with no wrong guess anywhere. The one family that does not is a tessellation clipped to a square, and what separates it from the others is not disorder, not size and not irregularity. It is having a rim.
The edge was not what made it hard
Five families of pattern searched at one step per panel and a tessellation patch did not, and the property left standing after four alternatives were killed was having a rim. Measured under a fixed letter order the patches cost between a half and two-thirds of a step per panel, at every tiling and every size — below the line rather than above it, and the rim is why.
A sheet with two edges
Design in this subject starts from a square, and the square's four edges are where every flap ends and every construction begins. A cylinder has two edges instead of four, and what a designer loses turns out to be measurable in the same letters a search counts — and what they gain is that the sheet is what gets manufactured.
The corner premium, with no corners
A square's corners are its most valuable paper: a flap placed there is claimed by a quarter-disc rather than a whole one, so the corner goes four times as far as the middle. A closed sheet has no corners at all, and the accounting that ranks sheet shapes by their corners has nothing left to rank.
A base needs an edge to point at
Every flap of a uniaxial base ends in a point, and every point is made of paper that came from the sheet's boundary. A sheet with no boundary has nowhere for a point to come from, and a sheet with half a boundary can only have points at the half it has left.
The sixth thing that is not true
Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.
A reference on a sheet with no corner
Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.
The proportion a band asks for
√2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.
The field has no edge
Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.
The square is in the answer
The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.
A construction assumes its sheet
Haga's fold gives exactly two thirds on a square. Run the same alignment on an A-series sheet held tall and it gives exactly two sevenths, with the crease meeting the vertical edges at seven sixteenths and eleven sixteenths — every one of them a clean fraction, none of them what the recipe promised. Turn the same rectangle through a right angle and the crease leaves the paper instead, which is the loud failure rather than the quiet one.
Every even polygon beats every odd one
Crossed with what each tool can build, the census of the largest regular polygon a sheet holds gives the same verdict on every proportion from a square to three to one: the best polygon is one a compass already builds, and the best polygon only a fold can build places fourth at best. The ranking itself stops moving at a proportion of 1.1284, where the hexagon overtakes the octagon. On every longer sheet only the short side holds a polygon, each polygon's share is a fixed constant divided by the length, and the constant — its area over the square of its least width — comes down to the circle's π⁄4 for even polygons and climbs up to it for odd ones. So every even polygon beats every odd one.
The axiom that reaches furthest wastes most
The field of origami numbers is defined on an unbounded plane and a folder has a square. Counted axiom set by axiom set on the same sheet, the share of crossings that land off the paper rises with every axiom added: nothing at all from the first two, fifty-seven per cent from the four linear ones at a single round, and seventy-six per cent from the conic axiom at a single round — more, in one round, than the linear four lose in two. The instrument that reaches furthest into the field delivers the smallest share of what it specifies.
The crossing is as hard as the polygon
Lengthen a square sheet and the largest hexagon it holds turns, pressed against all four edges, until it overtakes the polygons held by the short side alone. Every one of those overtakings happens at a proportion with a closed form, √3⁄2 + ½√(8K⁄3√3 − 1), and the number that comes out is exactly as hard to mark as the polygon being overtaken is to build. The octagon's 1.1284 is a compass number of degree eight. The heptagon's 1.0696 has degree twenty-four and needs a fold. The hendecagon's 1.0853 has degree forty and needs two folds at once.
The sheet a polygon fits exactly
A regular polygon with 4k + 2 sides has flat edges along one axis and corners along the other, so there is one sheet, 1⁄cos(π⁄n) long, that it touches on all four edges at once. On that sheet it is beaten only by the multiples of four with fewer sides, and so it ranks exactly (n − 2)⁄4. That puts the fourteen-gon, which only a fold builds, third rather than fourth; the twenty-two-gon, which needs two folds at once, fifth rather than ninth; and the forty-six-gon, beyond two folds, eleventh. Seven sheets from a square to three to one had missed all three.
Turning is uphill all the way
A regular polygon of 4k + 2 sides on a sheet a little longer than a square cannot lie flat: it turns, pressed against all four edges, until the sheet is exactly its own. Its share on the way has a closed form, and the closed form's slope is proportional to h² − 1 for every such polygon — flat on the square, rising all the way to the own sheet, and falling after it. So the own sheet is exactly the peak, the gain from the square to it is the average of one and the sheet's length, and the rank the census measured for polygons of this kind, (n − 2)⁄4, is now a theorem.
Named alongside it
The objects these essays reach for when they reach for this one.
Reference pointBoundaryCircle packingConstructibilityEfficiencyGluingOptimalityBoundary vertexCrease patternDesignInscribed polygonPaper proportion