A shrink is two numbers — the folded footprint generator
folded-footprint 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: "axes"
view: "census"
view: "axes", which: [accordion, twist, miura]
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.
- 4 families trace curves through 32 measured patterns, of which 9 draw in equally both ways ×2
- on 5 patterns the two directional shrinks multiply to the folded area, measured separately, to a part in a billion ×2
- one direction of the sheet left untouched on 1 of the 5 patterns, equal shrink in both directions on 2, and neither on 2 ×2
- tapering a Yoshimura's columns leaves all 22 interior vertices satisfying every condition, and tapering its rows breaks Kawasaki at 22 of them ×2
- the folded Yoshimura closes into a ring of 6 sides of 0.166667 each, of radius 0.166667, and the ring's perimeter is the sheet's width to a part in a million million ×2
- 11 of the 25 patterns measured draw in by the same factor both ways, and the rest are uneven by up to 3.96 to one ×1
- a twist draws in equally both ways only when its polygon has four sides — the triangle, the pentagon, the hexagon and the heptagon all come out uneven, by as much as 1.102 to one ×1
- across 5 patterns the two shrinks multiplied and the folded area measured agree to 8.9e-16 ×1
- and the square twist's whole curve lies on the diagonal, drawing in by the same factor both ways at every radius — so two of the families here are confined to a line and the rest are not ×1
- at a zigzag of 0.78 the leaf's folded state needs a box 0.6900 across where the flat sheet's is 0.6800 ×1
- at this zigzag the folded state stands 1.47 per cent taller than the flat sheet, measured on the two extents rather than inferred from the factor ×1
- on every one of them the footprint times the average depth returns the whole sheet, to within the sampling ×1
- over a window of its zigzag angle the leaf corrugation's cross factor falls below one — the folded sheet stands taller than the flat sheet it came from — while its areal factor stays above one throughout, so no paper has been created ×1
- so the cross factor is exactly proportional to the row count, at 0.24639 a row, and falls below one for every sheet of fewer than 5 rows ×1
- the accordion's whole curve lies on the line where one direction of the paper is untouched — its cross factor is exactly one at every fold count, which is a line in the plane and not a point on it ×1
- the average depth of each folded pattern is its areal shrink read backwards, to within 1.0% on the worst of 5 ×1
- the folded state's extent across the pattern is 0.6900 at every row count from 2 to 10 — a constant of the cell rather than of the sheet, so adding rows lengthens the flat sheet and not the folded one ×1
- the leaf corrugation's cross factor falls to 0.98555, so over that window of its angle the folded sheet is taller than the flat one ×1
- the leaf's folded cross extent is the same 0.6900 at row counts from 2 to 10, so its cross factor is a straight line through the origin ×1
- the the accordion swept through 8 settings gives along-factors from 2.000 to 16.000 ×1
- the the leaf corrugation swept through 10 settings gives along-factors from 1.175 to 4.785 ×1
- the the Miura fold swept through 9 settings gives along-factors from 1.060 to 3.471 ×1
- the the square twist swept through 8 settings gives along-factors from 1.136 to 1.923 ×1
- the the Yoshimura swept through 7 settings gives along-factors from 3.270 to 4.318 ×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 shrink is two numbers
How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.
The cylinder the pattern chooses
A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.
The direction that gets longer
A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.
The plane the five points were in
Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.
Every generator · The tessellations field · The patterns a reader can fold