Origami & Folding
what one sheet can do
Generator

What the symmetry is worth, count by count

A generator in the designing a base library, called 15 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.

symmetric-packing 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

What the symmetry is worth, count by countHow far the symmetric search falls short of the free one, as a fraction of the radius, at equal effort. Above the line the constraint costs something real; below it the constraint has made the search easier than the freedom did.2345678910-0.15-0.1-0.0500.050.10.15discsshortfall of the symmetric search14.6%1.7%0.0%-0.0%5.6%0.5%10.8%10.4%4.7%symmetry: mirror · both searches at 90 restartsneither number is a proved optimum — this compares two searches

sym: "c4", view: "pair", n: 4

4 discs, free and symmetricThe same number of discs packed twice at the same search effort: once with every centre free, once with the arrangement required to be a mirror image of itself. The discs are the flaps a design would be asking for, and the radius is how long they can be.every centre freeradius 0.25000symmetric (c4)radius 0.25000the two searches land within 0.00% of each other here

view: "which-n", sym: "mirror", counts: [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Where the requirement costs and where it paysHow much worse the best arrangement is when it is required to be its own mirror image, flap count by flap count, against a free search given the same number of restarts and the same number of steps. Below the line the constraint is winning.23456789101105101520flapshow much the requirement costs, per cent14.61.70.0-0.05.60.510.810.44.72.8

view: "which-n", sym: "c2", counts: [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Where the requirement costs and where it paysHow much worse the best arrangement is when it is required to be its own mirror image, flap count by flap count, against a free search given the same number of restarts and the same number of steps. Below the line the constraint is winning.2345678910110510152025flapshow much the requirement costs, per cent-0.018.60.0-0.01.52.2-0.20.00.22.7

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 corner is worth four times the middle

A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.

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.

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