Origami & Folding
what one sheet can do
Generator

How much of the sheet the flaps claim

A generator in the designing a base library, called 11 times across 7 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

packing-efficiency is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

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

counts: [2, 3, 4, 5, 6, 7, 8, 9]

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

counts: [4, 5, 6, 7, 8, 9, 10, 12]

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.4 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 = 0.166710 discs68.8%r = 0.148012 discs73.4%r = 0.1396hexagonal 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

counts: [2, 3, 4, 5, 6, 7, 8, 9], restarts: 120, steps: 6000

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

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

From a packing to a crease pattern

The circles say where the flaps are. They do not say where to fold, and the step in between is a construction rather than a search — two families of crease, both determined by the packing, neither of them visible in the picture of the discs.

How much paper is wasted

The efficiency of a design is the fraction of the sheet its flaps can claim, and for almost every number of flaps nobody knows the best possible value. The bars in these figures are the best a search could find, which is not the same thing.

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

The shapes the optimum has

Requiring a circle packing to be its own mirror image halves the number of coordinates a search has to find, so the same effort covers a much smaller space. Whether that helps depends on something the search cannot know in advance — whether the best packing was symmetric — and measured flap count by flap count the answer alternates without a pattern anybody could use.

The square is a choice

Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.

When symmetry costs

Design software and designers both reach for symmetry, and for a good reason: it makes the search enormously easier. It is a heuristic and not a theorem, and how much it gives away can be measured — including the case where the optimum is symmetric about an axis nobody imposed.

Every generator · The designing a base field · The patterns a reader can fold