Origami & Folding
what one sheet can do
Generator

What the grid costs, and what it settles

A generator in the designing a base library, called 6 times across 1 essay. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

grid-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

What the grid costs, and what it settlesFor each number of flaps: the largest equal circles a free packing can reach, and the largest reachable with every centre on a lattice of two, four and six divisions. The lattice figures are enumerated rather than searched, so each is the optimum of its own problem; the free figure beside them is the best a search has found and may not be the best there is.flapsfree search4×4 grid8×8 grid30.25430.2500 −1.7%0.2500 −1.7%40.25000.2500 −0.0%0.2500 −0.0%50.20710.1768 −14.6%0.1768 −14.6%60.18760.1250 −33.4%0.1398 −25.5%the 6-flap case on the finest lattice here is one of 3.25e+8 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

counts: [3, 4, 5, 6], grids: [4, 8], n: 6

What the grid costs, and what it settlesFor each number of flaps: the largest equal circles a free packing can reach, and the largest reachable with every centre on a lattice of two, four and six divisions. The lattice figures are enumerated rather than searched, so each is the optimum of its own problem; the free figure beside them is the best a search has found and may not be the best there is.flapsfree search4×4 grid8×8 grid30.25430.2500 −1.7%0.2500 −1.7%40.25000.2500 −0.0%0.2500 −0.0%50.20710.1768 −14.6%0.1768 −14.6%60.18760.1250 −33.4%0.1398 −25.5%the 6-flap case on the finest lattice here is one of 3.25e+8 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

counts: [3, 4, 5], grids: [4, 8], n: 5

What the grid costs, and what it settlesFor each number of flaps: the largest equal circles a free packing can reach, and the largest reachable with every centre on a lattice of two, four and six divisions. The lattice figures are enumerated rather than searched, so each is the optimum of its own problem; the free figure beside them is the best a search has found and may not be the best there is.flapsfree search4×4 grid8×8 grid30.25430.2500 −1.7%0.2500 −1.7%40.25000.2500 −0.0%0.2500 −0.0%50.20710.1768 −14.6%0.1768 −14.6%the 5-flap case on the finest lattice here is one of 2.56e+7 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

counts: [3, 4, 5, 6], grids: [4, 6], n: 5

What the grid costs, and what it settlesFor each number of flaps: the largest equal circles a free packing can reach, and the largest reachable with every centre on a lattice of two, four and six divisions. The lattice figures are enumerated rather than searched, so each is the optimum of its own problem; the free figure beside them is the best a search has found and may not be the best there is.flapsfree search4×4 grid6×6 grid30.25430.2500 −1.7%0.1667 −34.5%40.25000.2500 −0.0%0.1667 −33.3%50.20710.1768 −14.6%0.1667 −19.5%60.18760.1250 −33.4%0.1667 −11.1%the 5-flap case on the finest lattice here is one of 1.91e+6 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

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.

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.

What the grid settles

Box pleating is usually defended as a trade: give up efficiency, buy creases that land where they should. There is a second thing it buys and nobody quotes it — on a lattice the best possible packing is a finite question with an answer, while off the lattice nobody knows the best packing of six circles in a square and probably never will.

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