Origami & Folding
what one sheet can do
How this site is made

The figure library — page 7

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.

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

evidence-strength

5 essays — Nothing here is as old as it sounds · A record is not a proof · Two traditions and a merge · The cure was named first · When two of them are first attested together · 4 claims checked

What each claim rests onThe record grouped by what kind of thing its oldest source is, strongest first, with one dot per independent surviving source. The claims resting on a single source are marked, and they are a third of the table — which is the honest statement of how much of this field's dating is one document away from being unsupported.15 claims · 5 resting on one sourceartefacta surviving folded object, or a picture of one made at the timePaper is made in Europe1056The pajarita is folded in Spain1793manuscripta hand-written document that survivesFolded paper is used ceremonially in Japan1600Paper reaches Japan720one sourceprinteda printed book or paper with a publication datePaper folding is taught as geometry1838The conditions at a flat-foldable vertex1979The Miura fold1970The diamond pattern in a crushed cylinder1951The dashed-and-dotted diagram notation1954Any straight-line drawing, from one straight cut1998Paper is folded for amusement in Japan1680one sourceThe thousand cranes1797one sourceOne fold solves a cubic1936one sourcesecondarysomebody later reporting it, with no surviving primary sourcePaper is made in China105A five-pointed star from one straight cut1873one sourcea source is dated; it is not thereby rightthis ranks what a source can bear, not what it says

fujimoto

5 essays — Folding a strip into thirds · The kindergarten was a geometry class · One crossing, and then another · Exact is not accurate · Dividing a loop into n · 5 claims checked

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

n-division

5 essays — One crossing, and then another · Cheap where it reaches · The numbers a fold reaches · Exact is not accurate · The grid a division makes · 5 claims checked

One crossing, and then another: 1/5 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/25 folds to reach 1/5, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops

sector-ties

5 essays — Where the lemma says nothing · The dial that decides nothing · A tie is not a decision · The order decides the count · Walking between two foldings · 3 claims checked

The count halves the moment the tie is brokenHow many mountain-and-valley assignments a degree-four vertex admits, over a family in which the two smallest sectors stay equal, and then at a vertex a tenth of a degree away from that family. The tied family holds twice as many throughout, and the fall is a step rather than a slope.8, with the tie4, without it0the two smallest sectors, kept equalfoldable assignments of one interior vertexthe dashed line is a vertex 0.1° off the family: the lemma wakes up and takes half of them

thickness-technique

5 essays — Getting thickness round a corner · Panels with somewhere to go · Paper through paper · Nowhere to put the error · Thickness round a closed loop · 4 claims checked

Thickness: chamferA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does