Origami & Folding
what one sheet can do
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The figure library — page 6

Every picture here is generated from code at build time. This is every generator, ordered by how many essays call it, each rendered at its defaults.

graft-strip

6 essays — Paying in paper · The second term · A graft needs a square line · Six rectangles and one term · A design that keeps its lines clear · The width is charged in grid · 25 claims checked

A strip slid into a finished patternA crease pattern cut along a line, with a strip of paper slid into the gap. Every crease that meets the cut continues across the strip at the same angle; every crease that does not is carried along at the length it had. The shaded band is the whole of what the new feature cost.the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

mesh-folded

6 essays — Solving every face at once · Solved is not built · Which crease to push · The Miura folds two ways · Nothing to average over · The hardest instant · 6 claims checked

A mesh with no two vertices alike, foldedEvery panel put where the fold angles say it is, by a walk that rotates each panel about the crease it shares with its neighbour. Nothing in the walk makes the four panels round a vertex meet; they meet because the mesh folds.fold angle 0.9 radians · vertices close to 3.5e-13

number-tower

6 essays — The numbers a fold reaches · Reachable is not cheap · The field has no edge · Twos and threes run out · Gauss's polygon is the expensive one · The axiom that reaches furthest wastes most · 19 claims checked

Twos and threes, and nothing elseEach number with the degree of the simplest rational equation it satisfies. A fold reaches a number exactly when that degree is a product of twos and threes — a compass supplies the twos and one fold supplies the threes — so a fifth root is out of reach at any number of folds, and saying so needs no new argument once the reachable set is known to be a field.the degree of the equation, and what it is made ofnumberdegreemade ofwhere it comes from½11a fold in half√222^1the diagonal of the squareφ22^1the silver rectangle's cousin∛233^1doubling the cube2 cos(2π/7)33^1the regular heptagon∜242^2a square root of a square root∛2 · √262^1 · 3^1a product of two of them2^(1/5)5not twos and threesa fifth root2 cos(2π/11)5not twos and threesthe regular hendecagonchecked by exhaustion: no number here satisfies a rational equation of lower degree with coefficients up to 6

one-vertex-motion

6 essays — A sheet with one freedom · Panels instead of paper · Paper that folds itself · A property you can dial · The condition that is not flat-foldability · The motion has no letters to choose · 1 claim checked

A vertex is a one-degree-of-freedom mechanismOne four-crease vertex at four points in its motion. The four dihedral angles are locked to one another, so setting any one of them sets the rest — which means the vertex has a single degree of freedom and can be built out of rigid panels and hinges.12% folded42% folded72% folded95% foldedno face bends anywhere in the motion — which is what makes it a mechanism rather than a fold

panel-colouring

6 essays — The sheet has two sides · Bringing the other side to the front · The outline is mostly crease · Decided before the design · Which side arrives · A contradiction is even · 10 claims checked

Two colours, and no choice about themThe panels of a flat-foldable pattern, painted in the two colours the creases force on them. The colouring is computed by crossing creases and counting; the parity of the crease counts at the vertices is computed separately from the edges. They are the same fact twice.12 panels, two coloursno crease has the same colour on both sidesall 6 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

paper-proportion

6 essays — The rectangle that keeps its shape · One member of a family · The sheet decides which points exist · A construction assumes its sheet · A stretch keeps crossings · What the square saves · 22 claims checked

The sheet that is the same shape after it is foldedOne rectangle halved repeatedly across its long side, drawn nested, at two starting proportions. On the left the shape alternates between two rectangles and comes back to itself on every second fold. On the right the proportion is √2 and every nested rectangle is the same shape as the sheet it came from, which is what the A series is for.proportion 1.3two shapes, in turn1.3001.5381.3001.5381.300proportion √2 = 1.4142one shape, throughout1.4141.4141.4141.4141.414halving turns a proportion of r into one of 2/r, and those are the same number only at √2the 1.3 sheet is a different shape after every fold; the √2 sheet is the same shape after all of thema square metre at √2 is 840.9 × 1189.2 mm, which is the 841 × 1189 printed on a sheet of A0

pierpont-primes

6 essays — The heptagon a compass cannot reach · The eleven-sided one nobody can fold · The biggest one that can also be folded · Seven, and then twenty-two · Reachable is not cheap · How many polygons a fold reaches · 12 claims checked

Which primes each tool reachesFor each small prime p, the factorisation of p − 1 and whether it clears the bar each tool sets. A compass needs p − 1 to be a power of two; a fold needs it to have no prime factor above three. Eleven is the first prime a fold cannot reach, and it is the first place folding runs out.pp − 1, factoredcompassfoldingthe regular p-gon32both tools52 · 2both tools72 · 3folding only112 · 5neither132 · 2 · 3folding only172 · 2 · 2 · 2both tools192 · 3 · 3folding only232 · 11neither292 · 2 · 7neither312 · 3 · 5neither372 · 2 · 3 · 3folding only11 is the first prime out of a fold's reach — 11 − 1 = 2 · 5the factor of five is the obstruction, and no arrangement of folds produces onea compass needs a power of two; a fold needs nothing above three

shrink-census

6 essays — Found before it was designed · The paper is all still there · What a corrugation costs · A shrink is two numbers · Where two twists share a pleat · Four ways to draw a pattern · 20 claims checked

What each corrugation costsHow much smaller each pattern folds and how many layers deep it gets doing it, both measured off the folded state. The last column is the two multiplied together against the sheet they came from, and it is one everywhere, because the paper has nowhere else to be.patternhow much smaller it foldscreasingper sheet-widthpreliminary8.0 layers, 8 at the deepest8.0×4.81.65×footprint × depth = 1.004 of the sheetmiura8.1 layers, 16 at the deepest8.1×6.21.31×footprint × depth = 1.001 of the sheetyoshimura32.0 layers, 36 at the deepest32.0×11.82.71×footprint × depth = 1.000 of the sheetwaterbomb31.6 layers, 32 at the deepest31.8×14.32.22×footprint × depth = 0.992 of the sheettwist3.0 layers, 9 at the deepest3.0×4.70.64×footprint × depth = 0.995 of the sheetthe shrinkage is the pattern's, not the paper's — nothing here knows what the sheet is made of

waterbomb-pattern

6 essays — Where the paper stops · The base that tiles · A unit that folds is not a tessellation · Thirty-two rules, one object · The rule that breaks the count · Where a rule can close a loop · 7 claims checked

The waterbomb, tiledThe waterbomb base repeated across a sheet: every cell carries both its diagonals, every row of the grid is creased, and the columns are not. That last omission is what makes the corner vertices degree six rather than degree eight, and it is the pattern's whole character.two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge
Waterbomb tessellation — sheet 160×160 mm — 40 mountain, 36 valley, 2290.19 mm of crease

wing-packing

6 essays — A leaf packs by corrugating · A wing that folds into nothing · The same corrugation in four places · The organism is not the model · The census returns one · The number is the angle · 7 claims checked

What each geometry packs toThe packed footprint of four folding geometries as a fraction of the area each covers when open, computed from the pattern rather than measured from a specimen. The spread is the point: a wing that has to disappear under a case is choosing among these, and they are not close.geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal

curved-developable

5 essays — The sculptors got there first · What a flat sheet can become · Where curved creases meet · A curve has no panels · Where the rulings run out · 2 claims checked

A curved crease, and the rulings it forcesConcentric arcs on a flat sheet, and the surface they produce. Each band becomes a cone, the bands alternate which way they open, and the map from the flat sheet to the surface preserves every distance exactly — which is checked here rather than assumed. The straight lines on the surface are the cones' generators, and on the flat sheet they are radii.the patternwhat the sheet doesconcentric arcs with their rulings drawn as radii; the metric matches the flat sheet to 5e-7so nothing here is stretching — every point of the surface is where folding alone can put itthe flat-folding theorems say nothing about any of this: they are about straight creases meeting at a point

dof-census

5 essays — No motor in the fold · Which crease to push · Four finders, one option · The census returns one · A corrugation has one resting state · 3 claims checked

How many things have to pullThe degrees of freedom of four folding geometries and the number of drivers each therefore needs. A wing that opens without a muscle at every crease is not a wing with clever muscles; it is a pattern whose state is determined by one number.geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all