Origami & Folding
what one sheet can do
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The figure library — page 13

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.

curved-rigid

1 essay — A curve has no panels · 4 claims checked

A curved crease, cut into panelsOne curved crease and its straight-line approximations at 4, 8, 16 segments, with the sheet either side drawn as the flat panels each approximation demands. The panels grow narrower and more numerous and the picture improves; the total turning collected at the joints is identical in all of them, which is why no finite number of flat panels is a curved crease.the creaseno panels at all128.3° of turning4 segments8 panels128.3° of turning8 segments16 panels128.3° of turning16 segments32 panels128.3° of turningone crease, cut into flat pieceseach panel keeps its width across the crease and loses length along it as the count risesa finer approximation is a better picture and the same total kink, spread over more joints

error-field

1 essay — Where an error goes · 4 claims checked

Flat-foldability does not noticeHow far the panels come apart against how far one row was moved. The mesh is exactly as flat-foldable at every point of this graph as it was at the start — the vertex conditions are satisfied to within rounding throughout — and the sheet stops folding rigidly the moment the displacement is anything but zero.6.6e-42.0e-34.7e-31.1e-2Kawasaki: 5e-14°, flatthe gaphow far the row was moved, as a fraction of its own stephow far two panels leave a shared cornerone quantity is a condition at a point and the other is a distance between panels, and only the second can see the mistake

field-ledger

1 essay — The organism is not the model · 2 claims checked

What the field computes, and what it therefore does not knowThe rule this field runs under, made into a table. Everything on the left is run by the build and would stop it if it failed. Nothing on the right is established anywhere on this site, and no figure in the field should be read as though it were.computed herenot established herethe curvature a growth field forcesthe growth solverthat any leaf grows that waythat a corrugation with tapered columns folds flatthe pattern librarythat a leaf's creases are where these arethe packed fraction of four folding geometrieswing-packingthe packed fraction of any wingthat one freedom needs one driverdof-censushow an insect actually deploys a wingthe surface a fold count fits in a boxsurface-in-volumethe surface area of any organwhich lattice shapes admit a single routethe routing modulethat any published design used this routeevery row's left side is a computation in this repository; every right side is a claim it does not make

grain-census

1 essay — The fifth thing that is not true · 8 claims checked

Which way up to lay the paperEach printed pattern, with the share of its crease length lying within a few degrees of the sheet's grain at its best placement and at its worst. Nothing in the subject's theorems can choose between the two, because none of them mentions a direction.patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

grid-packing

1 essay — What the grid settles · 2 claims checked

What the grid costs, and what it settlesFor each number of flaps: the largest equal circles a free packing can reach, and the largest reachable with every centre on a lattice of two, four and six divisions. The lattice figures are enumerated rather than searched, so each is the optimum of its own problem; the free figure beside them is the best a search has found and may not be the best there is.flapsfree search4×4 grid8×8 grid30.25430.2500 −1.7%0.2500 −1.7%40.25000.2500 −0.0%0.2500 −0.0%50.20710.1768 −14.6%0.1768 −14.6%60.18760.1250 −33.4%0.1398 −25.5%the 6-flap case on the finest lattice here is one of 3.25e+8 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

hinge-travel

1 essay — Thickness has a sign · 4 claims checked

The travel is a step, and the step is at the surfaceHow far a fold in a panel of real depth closes, in each direction, against where its hinge axis sits between the panel's two faces. The two curves are mirror images and they do not overlap anywhere: whichever face the hinge is on, the fold has a direction.how far the fold closes, against where the hinge sitslower faceupper facemid-surface180°0°closing upwardclosing downwardtapering the panels buys most of it back: 166° of the 180°, less twice the taper

hole-packing

1 essay — Spending the cheap paper · 3 claims checked

A holed sheet against a solid one of the same areaThe longest flap a seeded search could pack onto a square with a hole in it, against the same search on a solid square holding the same amount of paper. The bar is the difference as a percentage; it is a comparison of two searches at one budget and not of two optima.the bar is how much longer a flap the holed sheet carriesboth sheets hold the same area of paper, so this is about arrangement2 flaps4.20%0.7213 against 0.69233 flaps2.00%0.5165 against 0.50634 flaps2.09%0.4983 against 0.48815 flaps-0.27%0.3164 against 0.31736 flaps1.07%0.2907 against 0.28767 flaps1.30%0.2540 against 0.25078 flaps0.73%0.2386 against 0.2369a bar to the left is the search doing worse on the holed sheet, which is a search result rather than a finding

order-type

1 essay — The order decides the count · 5 claims checked

Same arrangement, same letteringsVertices drawn at random and sorted by which of their sectors are strictly smallest. Inside each group every vertex admits exactly the same letterings — the same list, not merely the same number of them — and a count that never sees a vertex reproduces it.smallest sectors atverticesletteringspredictedfoldpositions 1, 443888positions 2, 525888positions 0, 325888positions 0, 417888positions 1, 314888positions 2, 413888positions 3, 512888positions 1, 511888the fourth column is computed by walking the cycle with no vertex present

overlay-union

1 essay — The sheet remembers · 16 claims checked

Two models, and the sheet that has carried bothThe two patterns and their union. The union's creases are exactly the creases of the two patterns; what it has that neither of them had is the vertices where one pattern's creases cross the other's, marked in magenta where they fail.two models, and the sheet that has carried both4 crossings, of which 0 cannot fold flatThe preliminary baseThe square twistthe sheet after bothno crease has moved and none has been added; what is new is where they cross

pattern-symmetry

1 essay — The symmetry the letters cannot keep · 6 claims checked

Which of a pattern's symmetries its letterings keepEvery symmetry that carries a crease pattern to itself, and how many of the mountain-and-valley letterings that fold are carried to themselves by it. A symmetry no lettering keeps is a symmetry the drawing has and no folded object made from it does.The preliminary base: 8 symmetries, 112 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turnnone of 112 — this symmetry cannot be foldeda half turnnone of 112 — this symmetry cannot be foldedthree quarters of a turnnone of 112 — this symmetry cannot be foldeda mirror across the sheet12 of 112a mirror up the sheet12 of 112a mirror in one diagonal12 of 112a mirror in the other diagonal12 of 112

photo-observer

1 essay — Which side is showing · 5 claims checked

Three observersEvery crease pattern of 4 creases on a grid of 12ths that folds flat, grouped by the outline it folds to. Of the groups holding more than one genuinely different pattern, how many each observer can separate.what the observer is givenambiguities separated, of 71the outline alonewhat a silhouette carries0the outline and the colourwhat a photograph of duo paper carries, from both sides0the complete layer orderwhat taking the model apart carries69the middle row is what anybody can actually see, and it is the top row

pleat-hole

1 essay — The gap between two curves · 4 claims checked

The hole a curved pleat leavesConcentric circular creases at a fixed spacing, drawn inward until the paper between two of them can no longer exist. The shaded disc is what is left: its radius is set by the spacing and the ruling angle, and by nothing about the sheet.17%7 circles fit; the middle 17 per cent cannot carry anyspacing 0.06, rulings at 0.9 radians to the crease