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

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.

patch-boundary

7 essays — The vertex the list does not have · Cutting a patch out of a plane · Most of a patch is edge · The property a patch does not have · The crease the drawing cannot show · A patch on a knife edge · The edge is what makes it hard · 10 claims checked

Two ways of cutting a patch out of a tessellationOn the left, the twist units that fit the sheet whole, with the pleats of the outermost ones run out to the rim; the rings mark where two of those run across one another. On the right, the same tessellation generated over a larger region and clipped to the same square, where every crease ends at the edge of the paper and none crosses another.the same tessellation on the same square, cut out of the plane two waysassembled from whole unitsclipped from the plane12 crossings · panels 1.73 apart0 crossings · panels closemountainvalleyraw edge

pleat-ratio

7 essays — The propagation that never had to work · Closing the loops is not folding · One number where the corners wanted four · Every twist writes an equilibrium · The sheet draws in crooked · A dial can turn the fold over · What refused the mirror was the rim · 60 claims checked

The equation on an edge, at both of its endsEach edge of a tiling carries one equation relating the twist sizes at its two ends. The first pair of columns is the tiling as it is drawn; the second is the same tiling with every vertex moved a little. A regular polygon has one interior angle, so both ends of an edge read the same numbers and every ratio is one.the equation on an edge, at both of its endsas drawnevery vertex moved 12 per centratio − 1round a loopratio − 1round a loopthe square grid000.200.90the triangular grid000.170.75the honeycomb000.351.73the rhombille tiling2.0002.992.14the elongated triangular tiling000.171.27the rhombille's tiles are not regular, its ratios are three and a third, and they still multiply to one

rigid-versus-flat

7 essays — What the vertex does on the way · Folding that gets built · From a shell to a solar array · The only pattern that moves · The vertex is geared · Where an error goes · The condition that is not flat-foldability · 1 claim checked

Two different questionsFlat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, most patterns that satisfy the first fail it, and everything that gets manufactured lives in the smaller set.folds flatfolds rigidlyas panelsin the outer setthe bird basemost traditional modelsanything with a squash foldin the inner setthe Miura foldthe Yoshimura patterneverything ever manufacturedthe smaller disc is inside the larger and is not drawn to any measured scale —neither set has been counted; paper cheats by bending very slightly, and sheet metal does not

senbazuru-cuts

7 essays — The oldest book cuts the paper · Which cranes can stay joined · Nearly every cutting fails at one crane · The border is where the cranes come apart · The prediction held at eight and broke at ten · No odd grid holds a perfect tree · A perfect tree is the joins it leaves out · 45 claims checked

The oldest book cuts the paperThe connected cranes of 1797, as the sheet they are cut from: a grid slit along every internal line except at the lattice points, which are left uncut so the birds stay joined. The arrangement is one sheet and it is emphatically not uncut, and the slitting per crane grows with the size of the piece.3×3 — 9 cranes, 4 corner joinsone sheet, and cut2×2 4 cranes 4 sides of slit3×3 9 cranes 12 sides of slit4×4 16 cranes 24 sides of slit5×5 25 cranes 40 sides of slit6×6 36 cranes 60 sides of slitcranes − joins = 2n − 1the rule the subject is usually stated under is one sheet and no cuts; theoldest surviving origami book does not keep it

stack-depth

7 essays — The sheet has a thickness · The pile, not the panel · Three kinds of pile · A panel is not the unit of depth · The slant that stacks in one depth · The slant belongs to the line · Parallel creases forgive every lean · 50 claims checked

Every technique is drawn for two layersThe deepest pile of paper in each pattern this site prints, and what that pile costs a panel with thickness: the length the outside of the pile falls short of the inside when it is folded. Every published way of building a thick fold is drawn and priced at two layers, which is the depth no real pattern has anywhere except its last fold.deepest pile on the shelf: 60 layersin office copier paper, that is 18.5 mm of paper to find at one creaseThe Yoshimura pattern60 layers · 18.5 mm · 59× the two-layer allowanceThe waterbomb tessellation32 layers · 9.7 mm · 31× the two-layer allowanceThe Miura fold16 layers · 4.7 mm · 15× the two-layer allowanceThe tapered corrugation16 layers · 4.7 mm · 15× the two-layer allowanceThe square twist9 layers · 2.5 mm · 8× the two-layer allowanceThe preliminary base8 layers · 2.2 mm · 7× the two-layer allowanceThe hexagon twist7 layers · 1.9 mm · 6× the two-layer allowanceFold and cut — the triangle7 layers · 1.9 mm · 6× the two-layer allowancetwo layers

tree-to-base

7 essays — A flap costs a circle · The last free parameter · Spelling a tree on a grid · The skeleton changes its mind · The second term · Two packings, one radius · A tree cannot argue · 2 claims checked

From a stick figure to a crease patternThe tree method in three steps. A subject is reduced to a skeleton with measured limbs; each limb becomes a circle; the packing that results dictates where the creases go. Robert Lang's TreeMaker automates the middle step, which is the one that is genuinely hard.the subjecta stick figure with limb lengthsthe packingone circle per limb, no overlapthe basea flap for every circlethe lengths in the skeleton become the radii, and the radii become the flaps

universal-molecule

7 essays — The last free parameter · The molecule that does not exist · The skeleton changes its mind · The corner that splits the shrink · The event is an incircle · Six creases belong to the square · Half the difference is a theorem · 24 claims checked

The last free parameterA region left between the flaps, filled by shrinking its outline inward at a uniform rate. Every corner traces a straight line as it goes, and those traces are the creases. The packing fixed everything else about the design; how far this shrink runs before the outline collapses is the one number still free, and it is computed here rather than chosen.the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here

axiom-completeness

6 essays — A fold needs something to align · Seven, and then twenty-two · Twenty-two is a floor · Each fold needs its own two · Counting operations is not counting power · A crease that does not exist yet · 40 claims checked

Eight combinations, seven of them a foldA fold line has two degrees of freedom, so it is determined by alignments worth two constraints. Enumerating the ways to reach two gives eight combinations and no more; seven determine a fold and are the Huzita–Hatori axioms, and the eighth asks for a fold square to two lines at once, which determines nothing.alignments worth one constraintfold through a pointfold square to a linea point onto a lineworth twoa point onto a pointa line onto a linethrough P + through Paxiom 1through P + square to laxiom 4through P + P onto laxiom 5square to l + square to lno foldsquare to l + P onto laxiom 7P onto l + P onto laxiom 6P onto Qaxiom 2l onto maxiom 38 combinations reach two constraints, and there is no ninthseven of them pin a fold down — the axioms Huzita listed in 1991 and Hatori completed in 2001the eighth is square to two lines at once, which is a condition on the lines rather than a fold

gearing-motion

6 essays — Paper that folds itself · The hardest instant · Only four creases decide a Miura · Two drivers and one freedom · A gearing reflects stiffness squared · The deciding set does not move · 20 claims checked

The amplification along the whole motion, crease by creaseEvery crease of a quadrilateral mesh driven in turn, at every point of the fold, and the largest factor by which an error in the driven crease reaches the rest of the sheet. Every crease reports exactly one somewhere, and none of them is flat.the height is the worst amplification anywhere on the sheetone line per crease; the horizontal axis is the driven crease's own fold angle3.010.32.5fold angle of the driven creasea mesh with no two vertices alike6 of 6 creases are worst near the flat sheet0 steps refused as branch changes

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

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