Origami & Folding
what one sheet can do
Generator

One flap the packing does not place

A generator in the designing a base library, called 8 times across 3 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.

held-and-free 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

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

n: 7

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

n: 9

Every flap heldThe 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.9 flaps, packed as tight as they will goradius0.166666667loose flapsnoneevery disc is wedged, and the arrangement names every flap's position

n: 8, view: "packing", ns: [5, 6, 7, 8, 9, 11, 13, 14]

Every flap heldThe 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.8 flaps, packed as tight as they will goradius0.170540689loose flapsnoneevery disc is wedged, and the arrangement names every flap's position

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.

The flap nobody holds

An optimal packing is presented as an answer: here are the circles, here is where they go. For some numbers of flaps that is not what it is. The best arrangement of seven discs in a square leaves one of them free to wander over an eleventh of the sheet without changing the answer at all — and the algorithm reports one point of that region and stops.

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.

Two packings, one radius

A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.

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