What the symmetry is worth, count by count
symmetric-packing is one function. Everything below was drawn by it when the site was
last published, at settings 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.
With nothing specified
view: "yardstick", restarts: 120
view: "track", restarts: 120
view: "tightened", restarts: 120
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- symmetric-packing: the annealed packing of 10 discs does not beat the published optimum, which it never consults ×11
- at 20 discs the free search reaches 0.10462 and the symmetric one 0.10899 ×6
- over 9 disc counts the symmetric search falls as much as 14.6% short and as much as 0.02% ahead ×3
- requiring a c2 arrangement matched or beat a free search of the same effort at 7 of 11 flap counts, and cost more than one per cent at 2 ×3
- at 120 restarts the free search falls further behind the best packings known as the discs multiply — 0.13 per cent short at 10, 6.07 at 20 — and no symmetric search ever gains on it by more than its shortfall, since none may beat the optimum ×1
- at 13, 18 discs a symmetric search recovers nine tenths or more of what the free search missed ×1
- at 20 discs and 30 restarts, the search held to a half-turn leads the free one by 8.39 per cent at 3,000 steps a restart and trails it from 24,000 on — by 1.71 per cent at 48,000, where the free search is 0.15 per cent short of the best packing known ×1
- at equal steps a mirror or half-turn search leads the free one by as much as 4.43 per cent; given the free search the same moves for each coordinate, no symmetric search leads by more than 0.56 per cent, at 18 discs, and the free search leads at 8 of 11 counts ×1
- requiring a mirror arrangement matched or beat a free search of the same effort at 1 of 10 flap counts, and cost more than one per cent at 7 ×1
- the best of the three symmetric searches is ahead of the free one at 8 of 11 counts, by as much as 4.18 per cent at 20 discs ×1
- tightened to the arrangement it had found, the free search is within 0.2 per cent of the best packing known at 8 of 11 counts and at least as good as every symmetric search's tightened packing at 9 ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
Equal effort was counted in steps
A circle-packing search required to be symmetric was found to beat a free search of the same effort, and the reason given was that a smaller space is easier to search. Past eleven discs the test gets sharp: the free search falls behind the best packings known, 0.13 per cent at ten discs and 6.07 at twenty, and a symmetric search leads it by as much as 4.18. But tighten the free search's packings to the arrangement they were heading for and most of that shortfall disappears — at twenty discs, all of it. Give the free search twice the steps and the lead reverses. The effort was equal in steps and unequal in what a step does: a symmetric search moves half as many coordinates, so each is moved twice as often.
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.
All the figures · The designing a base field · The patterns a reader can fold