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.

rim-cost

16 essays — What the rim was doing · A loop that goes somewhere · The lettering that was proved impossible · A sheet with no edge · The rim is four letters a cell · The bottom layer is at the rim · One step per panel is a table size · Pruning on proofs alone · The cost of proving something false · Where you cut hardly matters · The most decided vertex here · The edge was not what made it hard · The designer's grid is the dearest thing here · A count is not a length · A test imported without its hypothesis · Nothing grown was cut out of anything · 18 claims checked

Cutting a square tessellation out of the plane, and gluing it upSearch cost in nodes, on a logarithmic scale, against how many periods of the tessellation the rectangle holds. The lower line is the rectangle cut out of the plane in the ordinary way; the upper is the same rectangle with its opposite edges joined, so that no crease is divided. Both search the same drawing under the same rule at the same vertices.the same drawing, cut out of the plane and glued upnodes, log scale, against periods across the sheet10100100010⁴10⁵1×12×23×34×4glued upcut outan open mark is a search that ran out of budget rather than a cost

family-glued

15 essays — A grid that will not close · The corrugation that closes on itself · One node per panel, with the rim gone · The turn a column costs · The period nobody measured · Which pair is glued · A map with no edges · The tube a map makes · The seam that is not a symmetry · A metamaterial with no edge · The arc that arrived twice · A grid glued · The symmetry a gluing adds · A mechanism that closes on itself · Nothing grown has a seam · 9 claims checked

The period cell of the Yoshimurathe Yoshimura drawn over the plane, with one period rectangle marked on it and a ring of its neighbours around it. The rectangle's edges are placed to miss every vertex, so identifying opposite edges can neither make nor destroy an interior vertex — there are 2 of them either way. one column wide and two courses high, because the courses alternate their offset.the period cell of the Yoshimuraone period, with its neighbours round it2 interior vertices in the cell12 crease pieces drawnperiod 0.167 × 0.289one column wide and two courses high, because the courses alternate their offsetthe cell is a rectangle of ordinary paper until somebody says its edges are one edge

as-drawn

14 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 · No placement cleans every drawing · A fragment counts creases, not length · 30 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

crease-density

14 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 · A disc as deep as the degree · Where two bands part · The floor at degree four · 47 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

machine-census

14 essays — The patient machine is the weak one · The machine that may choose · Four questions about one sheet · Hardness is about the worst one · Publishing the pattern instead of the sequence · The crease that stops in the middle · Where the machine catches up · Deciding is not making · The easiest strip needs the deepest reach · A shallow machine pays in states, not folds · Fourteen states are one pile · Nine stamps have piles of their own · The machine's misses have no short list · An odd strip can hook its own ends · 54 claims checked

What each machine can reachFor three one-dimensional crease patterns, the number of mountain-and-valley assignments that fold flat at all, and the number each kind of folding machine can actually reach. Every bar is a separate search: the top one over stackings, the next three over sequences of folds, the last over rewritings of the segment lengths.evenly spaced — 4 creasesany flat folding16 of 16some-layers16 of 16all-layers16 of 16one-layer2 of 16crimping only6 of 16one short segment — 4 creasesany flat folding4 of 16some-layers4 of 16all-layers0 of 16one-layer2 of 16crimping only4 of 16uneven — 4 creasesany flat folding8 of 16some-layers8 of 16all-layers0 of 16one-layer2 of 16crimping only0 of 16a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move

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

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

inscribed-polygons

13 essays — The largest triangle in a square · The biggest one that can also be folded · The square is in the answer · Every even polygon beats every odd one · The crossing is as hard as the polygon · The sheet a polygon fits exactly · Turning is uphill all the way · An odd polygon fits like its double · A sheet rewards the multiples of its sides · The rule has a width · The worst angle sets the price · The window has an exact edge · Two sides at sixty are enough · 64 claims checked

More sides is not more paperThe area of the largest regular polygon of each number of sides that fits inside a square sheet. The square and the octagon share the sheet's own symmetry and do well by it; the polygons between and after them do not, and the sequence goes up and down rather than up.sidesshare of the sheet the largest one usestilt346.4%15.00°4100.0%45.00°567.4%9.00°669.6%15.00°772.9%6.43°882.8%22.50°975.1%5.00°1075.3%9.00°1176.3%4.09°1280.4%15.00°the 8-sided polygon is the peak, and every one of the 4 polygons after it does worse

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

growth-curvature

12 essays — A sheet that grows cannot lie flat · Which way the disc curves · The excess does not choose its waves · A crease carries no curvature · What one cut buys · A leaf ends its pattern · The test measures the rim · Three marks see nothing · A cut reads a slope · A circle sees only the average · A cut cannot tell which way a leaf grew · Growth that only turns · 50 claims checked

Growth is a change of metric, and a metric decides a curvatureThe linear growth factor against radius, and the Gauss curvature that metric forces. Nothing about a material enters either panel: given how much each part of the sheet grew, the curvature is determined, and a curvature that is not zero is a sheet that cannot lie in the plane.00.20.40.60.8100.511.5radius on the flat sheetgrowth factor Ωhow much each ring grew00.20.40.60.81-2-112radius on the flat sheetcurvature Kclosed formthe curvature that forcesgrown more at the rim · K(0) = −4a = -2.40