How many ways a map folds
map-foldings 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
view: "strip"
view: "dimension"
shapes: [1,4, 1,5, 1,6, 1,7, 2,2, 2,3]
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.
- all 7 map shapes are small enough to enumerate exhaustively, so the counts are complete rather than sampled ×4
- each pair is a strip of an even number of squares beside the two-row rectangle with the same 4 squares ×4
- at every one of the 3 sizes drawn, the strip folds more ways than the two-row rectangle with the same number of squares ×2
- every map drawn folds more ways than the product of its two strip counts, by between 2.0 and 38 times ×1
- the product is exact when one of the directions has nothing in it, which is the one-row case the enumerator has to reproduce ×1
- the ratio between successive counts moves from 2.67 to 3.26 over the lengths counted, so no geometric formula fits the sequence ×1
- the share of stacking orders that are foldings falls from 66.7 per cent at 1 × 4 to 0.3770 per cent at 3 × 3 ×1
- the strip's folding count rises at every one of the 9 lengths counted here ×1
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.
A map with no edges
Counting the ways a rectangular map folds is the oldest open problem in the subject, and every version of it assumes the map has an edge. Join the map's opposite edges and the question changes shape: half the sizes have no folded state at all, and the ones that do have no bottom layer to count from.
More than one way to lie flat
A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.
The count counts labels
One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.
The gadgets that make it hard
Flat-foldability is NP-hard, and the proof is a construction rather than an obstruction: a machine for turning any satisfiability problem into a sheet of paper that folds exactly when the problem has an answer.
The map counted from the layers
The classical map-folding counts are computed from a rule that never places a panel: work out which edge of the folded square each fold wraps around, and refuse the orderings that interleave two folds at one edge. Place the panels instead and order them by the general non-crossing rules, and the same numbers come out — 2, 6, 16, 50, 144, 8, 60, 320, 1368 — on nine sizes, by machinery that shares no line of code with the first.
The map that is not a rectangle
Take one square out of a three-by-three map and the number of ways it folds does not go down by an eighth. It goes up — to 848 if the square came from a corner, and to 8,016 if it came from the middle. Two maps of eight squares in the same box, differing by nearly a factor of ten, and no function of the box tells them apart.
The oldest open problem
In how many ways can a map be folded? The question needs no notation to state, the answer is a small integer for small maps, and after sixty years there is still no formula — only a list of numbers, each one found by searching every possibility.
The tube a map makes
Join one pair of a map's edges and the result is a tube — a real object, foldable in the hand, and neither the strip's problem nor the torus's. It has one loop that cannot be shrunk instead of two, it keeps its bottom layer because it keeps half its rim, and half its sizes are refused by a parity the flat map does not have.
Two directions that will not separate
A map has rows and columns, and a strip of stamps is a map with one row. The obvious hope is that the two-dimensional count is built from the one-dimensional one — fold the rows, then fold the columns. It is not: a two-by-three map folds 60 ways against a product of 12, and the discrepancy grows from a factor of two to a factor of thirty-eight over the counts anybody has.
Every generator · The flat-folding field · The patterns a reader can fold