Origami & Folding
what one sheet can do
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The figure library — page 5

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.

crease-radius

7 essays — The crease has a radius · How many times can it be halved · The paper had to arrive first · What a corrugation costs · The rectangle that keeps its shape · Closer than a crease is wide · How deep is a crossing · 7 claims checked

A crease is not a lineA fold carries the paper round a small radius rather than through a point, and the arc uses more of the sheet than the stack advances by. One crease loses a fraction of a millimetre. A grid with hundreds of them loses that on every line at once, which is why an ambitious tessellation is folded from thin paper and why a large grid comes out short.ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

fold-sequence

7 essays — The fold a machine can make · The patient machine is the weak one · What a dashed line can say · Two directions that will not separate · The half no notation records · Where the machine catches up · Deciding is not making · 3 claims checked

A strip folded by the all-layers machineThe pile of paper after each fold, drawn from the simulator's own states rather than from a description of them. The dashed line marks where the next fold happens. Each layer is one run of the strip that has not yet been folded anywhere along its length.creases at 0.20, 0.40, 0.60, 0.80 — assignment MVMVflat1 layerafter fold 12 layersafter fold 23 layersafter fold 34 layersafter fold 45 layers4 folds, and the finished pile satisfies the assignment — checked against the layer-ordering rules

inscribed-polygons

7 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 · 28 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

maekawa-reason

7 essays — Why the difference is two · A sheet that routes itself · The sheet has two sides · Thickness has a sign · A ring and a line · Two creases that cross · The loop a vertex cannot close · 1 claim checked

Why the difference is twoThe cross-section of a flat-folded vertex is a closed path that turns through exactly one full circle. Every mountain turns it one way by half a turn and every valley the other, so the counts must differ by exactly two — which is Maekawa's theorem, and it is a statement about winding rather than about paper.MVMwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started

mesh-solve

7 essays — Solving every face at once · Solved is not built · The Miura folds two ways · Nothing to average over · Closing is not building · A tolerance is a direction · The motion has no letters to choose · 11 claims checked

Twenty lengths, four conditionsThe same mesh before and after. Nothing about the flat-folding conditions has changed — both are exact at every interior vertex — and only the lengths of the creases are different. The one on the right folds rigidly and the one on the left does not.mismatch 0.066 radiansmismatch 8.5e-14 radiansevery vertex of both is developable and Kawasaki-exact to the last bit a double holds

packing-efficiency

7 essays — How much paper is wasted · From a packing to a crease pattern · The last free parameter · The same vertex, found four times · When symmetry costs · The square is a choice · The shapes the optimum has · 1 claim checked

How much of the sheet the flaps claimThe fraction of a square filled by n equal discs, for the best arrangement a seeded search could find. The dashed line is the density of the hexagonal packing of the whole plane, which is proved and which no packing inside a square reaches, because the boundary wastes a strip. For most n the true optimum is unknown.2 discs53.9%r = 0.29293 discs61.0%r = 0.25434 discs78.5%r = 0.25005 discs67.3%r = 0.20716 discs66.3%r = 0.18767 discs66.9%r = 0.17448 discs72.8%r = 0.17029 discs78.5%r = 0.1667hexagonal density 90.69%every bar is the best a seeded search found, not a proved optimum —which is the honest state of the problem for all but the first few values of n

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

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

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

uniaxial-base

7 essays — Every flap on one axis · Getting close instead of getting it right · Every pair, not every circle · What the condition does not decide · The price of a limb is not its length · A price holds until the arrangement moves · Rounding in the cheap direction · 25 claims checked

Every flap along one lineThe tree method does not produce an arbitrary shape; it produces a base whose flaps all lie along a single axis, with each flap as long as its edge of the skeleton. That restriction is what makes the circle argument work, because the base's shadow square to the axis is the tree itself.the treeleg0.62leg0.62body0.34arm0.50head0.78the basethe axis0.620.620.340.500.78the flaps are the tree's edges, at the tree's lengths, all square to one lineso the base's shadow along the axis is the tree, and nothing else can be designed this waywhich is the restriction the circle argument quietly depends on

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