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 at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

grid-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 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 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.

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.

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