Origami & Folding
what one sheet can do
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The figure library — page 10

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.

held-and-free

3 essays — The flap nobody holds · The shapes the optimum has · Two packings, one radius · 4 claims checked

One flap the packing does not placeThe tightest arrangement of these discs, with the contacts drawn. A disc is held when the directions of its contacts surround it; a disc whose contacts all lie to one side can be moved, and the ring round it is everywhere its centre could go without anything overlapping.7 flaps, packed as tight as they will goradius0.174457630loose flaps1room to move0.1127the inner ring is where that flap's centre may sit; the algorithm reports one point of it and stops

kawasaki-residual

3 essays — Almost every pattern fails · A no costs more than a yes · A near miss is nearly as rare · 4 claims checked

How much coincidence a flat-foldable pattern isSectors drawn at random at one vertex, two vertices and three, and the fraction that come within a given tolerance of Kawasaki's condition. Each vertex is one equation, so each vertex costs another factor of the tolerance, and the lines are the powers.-3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding

mesh-collision

3 essays — Closing is not building · A collision is an order · Refused at one lettering · 3 claims checked

Closing and being solid are two conditionsQuadrilateral meshes solved so that every closure holds to within a millionth of a radian, each followed through its motion and asked whether any two panels that share no crease pass through one another. Most are solid the whole way. One is not solid anywhere, and the equations that were solved cannot tell it from the others.5 of 6 solved meshes are solid at every angle sampledthe bar is the deepest interpenetration found anywhere in the motion, in panel widthsmesh 11closes to 9e-14solid at every anglemesh 17closes to 1e-121.18 — panels 3:0 and 3:2, 2 steps apartmesh 19closes to 6e-14solid at every anglemesh 23closes to 5e-14solid at every anglemesh 27closes to 2e-12solid at every anglemesh 71closes to 4e-12solid at every angle

mesh-tolerance

3 essays — Solved is not built · A tolerance is a direction · The allowance is spent at the end · 2 claims checked

What each mesh's reason for folding survivesEvery length of each mesh made wrong by the stated amount, and what is left of the closure. The solved mesh folds because an equation holds; the Miura folds because one crease family runs straight through every vertex, which a badly cut sheet still does.the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

metamaterial-response

3 essays — A material made of creases · Bought with holes · A metamaterial with no edge · 1 claim checked

The property belongs to the patternPoisson's ratio against fold state, measured off the solved motion for three Miura panels and for a plain accordion of the same paper. The accordion holds at zero for its whole travel, so this is not something folding does in general; and the Miura's value changes by an order of magnitude as it closes, which is not something a material can do.00.20.40.60.80123how far the sheet is closed−ν, so every curve shown is a negative ratioMiura, slant 0.25Miura, slant 0.42Miura, slant 0.6accordionexactly zerothe same paper,three behaviours,chosen by the creasepattern alone

move-graph

3 essays — Walking between two foldings · The creases that cannot move · The pieces without the list · 5 claims checked

The letterings that fold, and what joins themEvery mountain-and-valley lettering a single vertex folds flat in, joined wherever one small change turns one into another. Whether the picture is one piece or several is the question: a change of any two creases joins all of them, and a change of two neighbouring creases does not.8 letterings fold · 2 pieces under two neighbouring creasesflip two creases that are next to one another, which is the change a folder makes by handMMVVVVMMMMVVMVVVMVMVMMMVVMVVVMVMMMVMVVVVMMVVMMMMsectors 43° · 110° · 121° · 57° · 16° · 13°the two pieces are each other turned overevery crease at once: changes piece

multifold-reach

3 essays — Two creases at once · Counting operations is not counting power · A crease that does not exist yet · 2 claims checked

Which polygons need more than one fold at a timeFor every regular polygon up to twenty-four sides: the totient of its side count, that number's prime factors, and which tool reaches it. A compass needs the factors to be twos, a single fold allows threes as well, and two folds at once allow fives.nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

offset-panels

3 essays — Panels with somewhere to go · From a shell to a solar array · Error is folded too · 3 claims checked

Thickness that does not move the hingeThe same fold given real thickness two ways, in cross-section. Growing each panel symmetrically about the ideal surface makes the two sides of every crease fight for the same space. Offsetting each panel entirely to one side leaves the hinge axes exactly where the zero-thickness pattern put them, so the mechanism moves along the ideal path — and pays for it by no longer being a surface.grown symmetricallyoffset to one sidepanels overlap over 4% of their areawhich is the jam every thick-panel design meetspanels do not overlap at alland the hinge axes have not movedwhich is not free: fold it tighter and this offset runs out toosheet 0.16 panel-lengths thick, folded to 100° — the overlap is measured from the geometry

packing-shortfall

3 essays — Getting close instead of getting it right · When symmetry costs · The square is a choice · 2 claims checked

How close the search getsFor each number of discs where the optimum has been proved, the radius a seeded annealing search in this repository found and the radius somebody proved is best. The bar is the shortfall as a fraction of the optimum. The figure refuses to draw if the search ever exceeds a published value, which would mean one of the two is wrong.discsfoundproved bestshort by20.292880.292890.00%30.254310.254330.01%40.250000.25000matched50.207050.207110.03%60.187580.187680.05%70.174360.174460.06%80.170220.170540.19%90.166670.16667matchedworst shortfall 0.19% of the radius, at 8 discsthe search never consults the published values, so the comparison measures the searchbeyond nine discs there is nothing to compare against, because nothing has been proved

patch-conditions

3 essays — Where the paper stops · A unit that folds is not a tessellation · Thirty-two rules, one object · 5 claims checked

Conditions arrive with the interiorInterior vertices per crease as a patch of one tessellation grows. Every interior vertex is four conditions and every boundary vertex is none, so a small patch is not a small version of the pattern — it is a much less constrained one.12345600.10.20.30.4patch, in cells acrossinterior vertices per creasewaterbomb1 interior vertices at 1 across, 61 at 6the ratio has a ceiling and approaches it from below, so a big patch is the honest test

same-outline

3 essays — The shadow does not name the pattern · The order does not name it either · Which side is showing · 3 claims checked

Different patterns, one folded objectCrease patterns that a photograph of the folded strip cannot tell apart. The creases are in different places and the folded profiles — the outline, and the number of layers over every point of it — are identical.4 patterns, one folded profile1/122/124/127/1225311/123/126/128/1225312/124/125/127/1225312/125/127/128/122531foldedthe layer counts under each band are the same in every row, and so are the widths

sat-gadget

3 essays — The gadgets that make it hard · Hardness is about the worst one · A short reason to say no · 3 claims checked

A wire made of paperA strip of four creases assigned V M M V. Every local condition holds, the shape is fixed, and there are exactly two ways to stack it — so the strip carries one bit, and the bit lives in the layer order rather than in the paper. This is the piece the hardness proof is built out of, and it is the piece that can be checked here.state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions