Generator

A shrink is two numbers — the folded footprint generator

A generator in the tessellations library, called 49 times across 8 essays. 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.

folded-footprint 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

`view: "modules", show: "table"`

`view: "modules", show: "table", C: 8, R: 8, ks: [1, 2, 4, 8]`

`view: "modules", show: "reliability"`

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.

holding a panel whose frame is turned 0.0 degrees gives the factors of the folded state's widths in that frame ×10
4 families trace curves through 32 measured patterns, of which 9 draw in equally both ways ×2
at a slant of 0.35 the folded Miura's smallest box lies 20.1 degrees off the drawing's vertical, which is the slant itself ×2
cut from a 5-sided sheet, the 5-sided twist's folded state needs a box of the same sides in every frame its panels hold ×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 3-sided twist: on the axis square 2 pairs of factors (1.082/1.168, 1.448/1.000, 1.082/1.168); on its own sheet 1.639 along and 1.515 across in every one of 3 frames ×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
3 of the 6 patterns fold to a state lying along the drawing's axes and 3 fold to one that does not ×1
a five-sided twist holds its folded state in 5 frames ×1
a pattern whose creases all run along the drawing's axes has one frame, so it gives one answer whatever its folded shape ×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 every slant from 0.1 to 1.3 radians the Miura fold folds to a strip lying at its own slant to the drawing's vertical, to half a degree ×1
at every zigzag from 0.2 to 1.1 radians the leaf corrugation folds to a strip lying at its own zigzag to the drawing's vertical, to half a degree ×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
cut along its zigzags into 1, 2, 4, 8 modules, a 8 by 8 Miura at a slant of 0.35 folds each module into a box cos a by sec a plus c sin a — 3.808, 2.436, 1.750, 1.407 long — and every cut removes exactly 8 hinges ×1
cut along its zigzags into 1, 2, 4, 8, 16 modules, a 16 by 4 Miura at a slant of 0.35 folds each module into a box cos a by sec a plus c sin a — 6.551, 3.808, 2.436, 1.750, 1.407 long — and every cut removes exactly 4 hinges ×1
held on each of its panels in turn, 5 of the 10 patterns give one pair of factors whichever panel is held, and 5 give more than one ×1
no pattern gives more pairs of factors than its panels have frames ×1
on 22 pairs of twist and regular sheet, every pair the shared rotations and the half-turn predict to agree does, and none gives more answers than its frame classes: 3 on 3: 1/1, 3 on 4: 3/3, 3 on 5: 3/3, 3 on 6: 1/1, 3 on 10: 3/3, 3 on 12: 1/1, 4 on 3: 2/2, 4 on 4: 1/1, 4 on 5: 2/2, 4 on 6: 2/2, 4 on 10: 2/2, 4 on 12: 1/1, 5 on 3: 5/5, 5 on 4: 5/5, 5 on 5: 1/1, 5 on 6: 5/5, 5 on 10: 1/1, 5 on 12: 5/5, 6 on 3: 1/1, 6 on 4: 3/3, 6 on 6: 1/1, 6 on 12: 1/1 ×1
on every one of them the footprint times the average depth returns the whole sheet, to within the sampling ×1
on its own sheet every frame of every twist gives the same two factors ×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 3-, 4-, 5-, 6-sided twists, each cut from a regular sheet of its own number of sides, give one pair of factors whichever panel is held: 3 frames, one answer; 2 frames, one answer; 5 frames, one answer; 3 frames, one answer ×1
the 4-column Miura's folded outline is not its own mirror image across the strip's axis (off by 0.343), and it is that mirror image turned half a turn ×1
the 4-sided twist: on the axis square 1 pair of factors (1.515/1.515, 1.515/1.515); on its own sheet 2.941 along and 2.941 across in every one of 2 frames ×1
the 5-sided twist on a 5-sided sheet: boxes in its 5 frames 0.441×0.432, 0.441×0.432, 0.441×0.432, 0.441×0.432, 0.441×0.432 ×1
the 5-sided twist on the axis square: boxes in its 5 frames 0.812×0.798, 0.677×0.746, 0.707×0.762, 0.812×0.782, 0.638×0.620 ×1
the 5-sided twist: on the axis square 5 pairs of factors (1.232/1.253, 1.478/1.341, 1.415/1.312, 1.232/1.279, 1.566/1.613); on its own sheet 2.156 along and 2.094 across in every one of 5 frames ×1
the 6-sided twist on a 3-sided sheet: boxes in its 3 frames 0.433×0.495, 0.433×0.495, 0.433×0.495 ×1
the 6-sided twist on the axis square: boxes in its 3 frames 0.777×0.856, 0.777×0.856, 0.706×0.577 ×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 box ratio is least at slant 1.05, while the two readings with no axes in them are least at 0.8 and 0.8 ×1
the box ratio is least at zigzag 1.05, while the two readings with no axes in them are least at 0.8 and 0.8 ×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 folded strip is 0.939 wide at every column count from 2 to 10, to a hundredth ×1
the frame classes the symmetry leaves are the exact count of answers on 22 of 22 pairs ×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 Miura with an even number of columns gives one answer on every panel and the Miura with an odd number gives two ×1
the Miura, 3 columns holds its folded state in 2 frames ×1
the Miura's panels hold its folded state in two frames, 0° and 139.9° ×1
the share of the area expected to open rises with every cut whatever the actuators; the chance that everything opens rises with the cuts when an actuator fails less often than 4 hinges together, and falls when it fails more often ×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
while its cross factor falls at every step, because a longer strip at a slant casts a taller shadow across ×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.

A cut along the zigzag needs a tether

A Miura's folded stack walks along itself by the sine of its slant for every column, so a sheet cut along its zigzags into modules folds into boxes a fraction as long and stacks far deeper for the hinges. But the pieces cannot stay joined by a crease: a zigzag crossed by row creases is a line of four-crease vertices, and joined there the modules are the one Miura again. The join has to be a tether and every module its own actuator — and a cut, which removes a zigzag's worth of hinges, pays for itself in complete success only when an actuator fails less often than those hinges together.

A sheet that turns with the twist

Holding a different panel of a folded twist can give a different pair of shrink factors: on a square sheet the three- and six-sided twists gave two pairs each and the five-sided twist five. Cut each twist from a regular sheet of its own number of sides and every panel gives the same pair. The reason is a count of rotations. A K-sided twist's panels hold its folded state in frames that are multiples of 180⁄K or 360⁄K degrees; a sheet that shares g of the twist's rotations makes the widths agree in every frame those rotations reach, with the half-turn a width cannot see thrown in. On twenty-two pairs of twist and regular sheet, that count gives the exact number of answers every time — a hexagon twist on a triangle gives one, a pentagon twist on a square five.

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.

Every panel holds a frame

A flat-folded pattern can be put down on any of its panels, and on some patterns the pair of shrink factors changes with the choice. Holding a panel does not move the paper so much as turn the frame it is measured in, by twice the alternating sum of the crease directions crossed to reach that panel — a number Kawasaki's condition makes independent of the route. So the frames are read off the crease pattern before anything is folded. A pattern with one frame gives one answer whatever its folded shape; the five-sided twist has five frames and five answers; and the four-column Miura agrees on every panel although its folded strip is not its own mirror image, because a box cannot tell a shape from its half-turn.

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 folded strip lies at its own slant

The two directional factors read a folded state against the axes the pattern was drawn on, with its first panel held where it was drawn. A flat-folded Miura is not lying along those axes: it is a strip of fixed width laid at exactly its own slant to them, and the leaf corrugation's strip lies at exactly its zigzag. So the Miura's cross factor changes with the number of columns — 1.748 on four, 0.948 on ten, at one slant — and on the leaf, on every Miura with an odd number of columns and on the twists of three and six sides, it changes with which panel is held. Read without axes, both families compact least at 0.80 radians, not at the 1.05 the box on the axes reports.

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.

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