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

The figure library — page 2

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.

miura-pattern

14 essays — One vertex, repeated · A square that turns · A state no motion reaches · The name is not the date · Nothing meets at three · The only pattern that moves · Where an error goes · The corrugation that curves · One crease decides the sheet · The patterns nobody owns · How much line is on the paper · The paper a pattern asks for · A corrugation agrees with itself · Half the recipe is decoration · 8 claims checked

The Miura foldA grid of identical parallelograms. The assignment is the whole trick: the horizontal creases alternate by row, and the vertical ones change assignment every time they cross a row, so each vertex ends up three of one and one of the other rather than two and two.at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 6.2 sheet-widths of foldingmountainvalleyraw edge
Miura fold — sheet 170×106.83 mm — 22 mountain, 16 valley, 1049.4 mm of crease

deployables

13 essays — Getting thickness round a corner · Folding that gets built · A property you can dial · Panels with somewhere to go · Found before it was designed · From a shell to a solar array · A wing that folds into nothing · Paper through paper · The tube that gets built · What a second deployment costs · The crease count is a reliability budget · Splitting a sheet buys area, not certainty · The pattern cheapest to trust · 19 claims checked

What folding is used forDeployed area against packed area for several engineered folds. The pattern earns its place when something has to be large in use and small in transit, and every one of these is a case where nothing else would fit.Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubemap fold9×the original problempackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable

hole-colouring

13 essays — Even is not enough · A cut that removes no paper · A short reason to say no · Decided before the design · A cut that reaches the edge · A contradiction is even · The seam carries a sign · Parity is not enough · How rare a band that folds is · A cut is surgery · Two holes are two conditions · The cut that changes nothing · A theorem with an unstated hypothesis · 11 claims checked

The same parity, with nowhere to put itThe same creases on a square of paper and on a loop of paper. On the left they meet at one interior vertex, which carries the parity and which every theorem in the subject inspects. On the right the middle has been removed, that vertex is gone, and the parity is still there — in the panels, where nothing local can see it.a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

surface-in-volume

13 essays — Nothing in a body folds on a line · How much surface fits in a body · Four finders, one option · Twice as thick where it is thickest · Nothing grown has a seam · The surface has to be supplied · Two surfaces in one box · A nest pays four a level · Standing up beats lying down by eight · The channel grows with what it feeds · The angle the eight does not know · In a tube the standing members lose · The wedge belongs to one length · 41 claims checked

How much surface fits in a bodyThe surface a corrugation holds inside a fixed box, against the number of folds. With no thickness the answer rises without limit and folding is free. With a thickness the stack eats the depth it is folding into, the curve turns over, and past a certain count the box is full of sheet and holds no surface at all.0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

as-drawn

12 essays — The vertex the list does not have · Two creases that cross · Cutting a patch out of a plane · Two mechanisms at one point · How deep is a crossing · The order the refusals come in · The reader decides the junction · The refusal that reads the list once · The crease the drawing cannot show · The drawing does not say what is glued · A stub is never alone · Two faults, not four · 17 claims checked

The vertices a crease list does not haveEvery crease pattern here, read twice: once as the list of vertices and edges it is built from, and once as the ink on the page. The bar is how many vertices the second reading has to invent, which is how many places two creases cross with nothing recorded there.the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

axiom-set

12 essays — One fold at a time, and there are exactly seven of them · Folding beats the compass, by exactly one degree · Why the list stops at seven · The largest triangle in a square · A fold needs something to align · One crossing, and then another · Cheap where it reaches · What each axiom is worth · The numbers a fold reaches · An axiom may name no fold · The axiom that names two folds · Which of the seven survive · 10 claims checked

The seven ways to specify a foldThe Huzita–Hatori axioms: every fold that can be specified by bringing existing points and lines into coincidence. The list is complete — six were catalogued in 1991, the seventh in 2001, and no eighth exists. The sixth is the one a compass cannot reach.axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

big-little-big

12 essays — The smallest sector decides · Local is not global · The lengths are free · Where the lemma says nothing · How little the conditions decide · Thickness has a sign · The dial that decides nothing · A tie is not a decision · The order decides the count · The vertices nobody checks · Two creases that cross · A region with no lettering · 1 claim checked

The smallest sector decidesTwo assignments of the same four creases. Both satisfy Kawasaki and Maekawa. The left one folds; the right one does not, because the strictly smallest sector has the same assignment on both sides and the paper either side of it has nowhere to go.MVMM40°foldsopposite across the small sectorMMVM40°does not foldthe same on both sidesboth satisfy Kawasaki and Maekawa — the angles and the counts are identical

box-pleating

12 essays — Designing on a grid · The crease has a radius · Publishing the pattern instead of the sequence · What the grid settles · Spelling a tree on a grid · The whole alphabet of a grid · The sheet remembers · The other grid · What a grid costs in circuits · Ninety-nine in a hundred pass · The designer's grid is the dearest thing here · A grid glued · 1 claim checked

Box pleatingDesigning on a grid, with every crease running along a grid line or at forty-five degrees to it. It gives up the efficiency of a free circle packing and gains something worth more for complex work — the creases meet where they are supposed to, and the errors do not accumulate.16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

tiling-twist

12 essays — Any tiling makes a twist · Where two twists share a pleat · Fenced at both ends · The propagation that never had to work · The dial that decides nothing · Four ways to draw a pattern · The tiling the unit could not promise · Cutting a patch out of a plane · Letters that agree get rarer · The ring is the loop · Where a sector crosses sixty · The dial and the tiling that is not alike · 15 claims checked

The the square grid's twist tessellationThe crease pattern the offset construction produces, with its assignment found by propagating the two vertex conditions rather than drawn on. Every interior vertex has four creases and passes developability, Kawasaki, Maekawa and the big-little-big lemma.what the construction produced9 twists, 36 interior verticesturned 24.1° from the tiling's edgespleats 0.118 to 0.118 wide2.20× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge
square twist tessellation — sheet 165×165 mm — 41 mountain, 43 valley, 2180.74 mm of crease

twist-unit

12 essays — A sheet with one freedom · A square that turns · Which polygons twist · The base that tiles · Any tiling makes a twist · The patterns nobody owns · The creases that cannot move · A cut is a licence · The symmetry the letters cannot keep · A cut is not local · The tiling the unit could not promise · The ring is the loop · 15 claims checked

The triangle twistA triangle twist: a triangle with a pleat running out from each of its 3 corners, drawn at 150 mm and carrying 5 mountain and 4 valley creases — 563 mm of folding on a sheet 150 mm across. As the sheet closes the triangle rotates, which is what gives the family its name. The sector angles are fixed by Kawasaki and the assignment is chosen for having a folded state rather than for reading well — and the unit is verified, while the tessellation it belongs to is not.3 corners, all alikesectors 60°, 60°, 120°, 120°two equal pairs, so no sectoris strictly the smallestthe assignment64 of 512 fold5 mountain, 4 valleythe ring takes two lettersthe panels can be orderedwhat is checked3 interior verticesand not the tilinga twist of radius 0.17 sheet-widths9 creases, 3.75 sheet-widths of foldingmountainvalleyraw edge
triangle twist — sheet 150×150 mm — 5 mountain, 4 valley, 562.66 mm of crease

crease-density

11 essays — How much line is on the paper · Fourth of eight, and still not chosen for it · The paper a pattern asks for · Where the length sits · The shortest crease is not a crease · A crease with no vertex to belong to · A length needs a scale · The density a paper allows · A paper limits spacing, not density · A vertex creases the paper twice · The pattern cheapest to trust · 25 claims checked

How far a hand travels to fold each printed patternThe total length of crease in every pattern this site prints at true scale, in millimetres at the size it is printed. It runs from 258 mm to 6,679 mm, and it does not rank the patterns the same way counting their creases does.the bar is millimetres of crease at the printed sizea pattern with more creases is not always a pattern with more folding in itThe preliminary base724 mm8 creases · printed at 150 mmThe Miura fold1,049 mm38 creases · printed at 170 mmThe square twist704 mm12 creases · printed at 150 mmThe hexagon twist916 mm18 creases · printed at 150 mmThe Yoshimura pattern2,380 mm86 creases · printed at 170 mmFold and cut — the triangle258 mm6 creases · printed at 150 mmThe tapered corrugation1,057 mm45 creases · printed at 160 mmThe waterbomb tessellation2,290 mm76 creases · printed at 160 mmsix point seven metres of crease on a sheet seventeen centimetres across

folded-layers

11 essays — The paper is all still there · Bringing the other side to the front · Error is folded too · The base that tiles · The cylinder the pattern chooses · The creases a sheet gives itself · The creases that cannot move · The pile, not the panel · The rim lies over less · The property a patch does not have · Where the length sits · 16 claims checked

The pattern, and where its panels landEvery panel of the pattern drawn at the place folding puts it, at the same scale as the pattern itself. The outlines are left in so the layers can be counted; which panel lies above which is a separate question this construction does not answer.the patternthe panels, foldedsheet 12.000footprint 1.966 · 6.11 layers on average · 12 at the deepest1.966 × 6.11 = 12.007, which is the sheet