How close the search gets
packing-shortfall 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
counts: [2, 3, 4, 5, 6, 7, 8, 9]
counts: [2, 3, 4, 5, 6, 7, 8, 9], restarts: 120, steps: 6000
counts: [2, 3, 4, 5, 6]
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.
- packing-shortfall: the annealed packing of 2 discs does not beat the published optimum, which it never consults ×8
- every disc count charted has a published optimum to be measured against, so no shortfall is reported against a guess ×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.
Getting close instead of getting it right
When the best answer is out of reach the question stops being what it is and becomes how much is lost. For packing discs into a square the loss is measurable: a seeded search in this repository comes within a fifth of a percent of the best radius anybody has proved, and proves nothing.
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.
All the figures · The what it costs to know field · The patterns a reader can fold