Generator

The equation on an edge, at both of its ends

A generator in the tessellations library, called 41 times across 7 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.

pleat-ratio 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: "collapse", show: "census", turn: 0.5`

`view: "collapse", show: "census", psi: -0.35`

`view: "collapse", show: "sweep", dial: "psi", affine: [1, 0.3, 0, 1]`

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.

a circle on the flat sheet of this pattern folds to an ellipse with axes 0.784 and 0.178 ×3
on ten of their fifteen images it is not, and the sheet draws in by up to 3.46 times as much one way as the other; the five that stay similarities are every image of the triangular grid and the two stretches that leave a square grid rectangular ×3
on this image the one-distance construction fails 36 of its vertices and the one-per-side construction passes every one of its 28 ×3
the collapse is an exact linear map of the plane in every one of these 20 patterns — the twists' positions before and after folding are related by one two-by-two matrix, to 7e-14 of the patch's own width ×3
the collapse of these twist tessellations is a similarity on the tilings as drawn and stretches by up to 3.46 to one on their images ×3
at every stop checked the square grid and the elongated triangular tiling fold by the same map, as do the honeycomb and the rhombille, and the triangular grid under [1, 0.3, 0, 1] folds by exactly the map every tiling as drawn folds by ×2
dividing each side of a twist by the edge it faces gives weights that balance at every vertex to 3e-16 and that the two ends of an edge agree on to 3e-15 — which is a positive equilibrium stress on the tiling, written by a construction that was never asked for one ×2
on all 20 decided patches of the census a stacking exists exactly when no panel carries a ring of flaps — with the rim and without it ×2
round a 4-edge loop of the the rhombille tiling the pleat ratios multiply to 2.755852 ×2
round a 4-edge loop of the the square grid the pleat ratios multiply to 1.000000 ×2
the square grid's collapse under [1, 0.3, 0, 1] keeps the sign of its area over the whole range drawn ×2
a linear program needs an objective ×1
a positive balancing set of weights exists on all 20 tilings and images, with smallest weight between 0.845 and 1.000 ×1
a ring of flaps cannot be stacked, since each flap must lie farther from the panel they share than the next; that half is an argument, and the census checks the other half — that nothing else refuses these patches ×1
after the corner conditions and the two directions that only move the drawing, 3 of these 5 tilings have no choice left and the rest have exactly one ×1
an SVD needs a matrix ×1
and a linear map leaves the weights equal on the honeycomb and the rhombille tiling while making them unequal everywhere else, which is a property of the tiling rather than of the map ×1
and each flap reaches across the crease of the next — by 240, 240, 240 per cent of the middle panel's area ×1
and each flap reaches across the crease of the next — by 66, 66 per cent of the middle panel's area ×1
and on the tilings as drawn it is a similarity: the sheet draws in by the same factor in every direction ×1
and the twist construction folds on the tilings as drawn but on none of their images, at any of the turns 0.2, 0.42, 0.7, 1 — even where the loops close exactly ×1
and the weights are the same numbers before and after every linear map, to 3e-15 — a map takes an equilibrium to an equilibrium because it takes every edge vector by the same map, so this test cannot tell a tiling from its images ×1
and what is left is never more than a single choice: on three of these tilings the turn and the pleat angle fix the pattern outright, and on the other two they leave one number, which trades side lengths between the kinds of vertex without moving the ratio of their sizes ×1
as drawn, the pattern puts the same weight on every edge exactly when every sector at every vertex is the same size — and the elongated tiling, whose vertex has right angles beside sixty-degree ones, is the one that is not, at the square root of three ×1
at every one of 12 sizes and angles, every ring of flaps touches the rim or lies one crease from it, on patches up to 12 creases deep ×1
at one vertex of a stretched grid the conditions ask for sides of 0.90, 0.60, 0.90, 0.60 where a single distance can only produce 0.75, 0.75, 0.75, 0.75 ×1
deforming the same tiling by 12 per cent of an edge takes the loop disagreement to between 0.75 and 2.14 ×1
developability holds at every interior vertex — its sectors close to 360° — 36 checked ×1
every constraint row has one entry per variable ×1
every crease joining the 2 flaps to the panel they fold onto carries the same letter, a valley, so all 2 fold onto one face of it ×1
every crease joining the 3 flaps to the panel they fold onto carries the same letter, a mountain, so all 3 fold onto one face of it ×1
every one of the 8 drawn patches is refused — proved, not timed out — and every one stacks once its outer two rings of panels are cut away, 6 of them mirrored ×1
every one of these 20 tilings and images carries positive weights that balance at every vertex, so every one of them can carry a flat twist tessellation ×1
every ring of flaps on this patch has a panel on the rim or one crease from it, on a patch 6 creases deep ×1
every row is the same length ×1
every tiling and every image folds over a window of pleat angles rather than at an isolated value ×1
Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 36 checked ×1
Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 36 checked ×1
on every tiling by regular polygons the two ends of an edge ask for the same twist size exactly, and the loops close to 4e-15 ×1
on the square grid under [1, 0.3, 0, 1] the collapse's area passes through zero at -0.348 and -0.161, and between them the folded twists are a mirror image of their flat arrangement ×1
on the square grid, the triangular grid and the honeycomb every edge's equation is exactly one at both ends under every map tried, so their loops close however irregular the tiles become ×1
on the the honeycomb under [1, 0.3, 0, 1] the one-distance construction fails 80 vertices and the one-per-side construction passes all 80 ×1
on the the rhombille tiling under [1, 0.3, 0, 1] the one-distance construction fails 204 vertices and the one-per-side construction passes all 202 ×1
on the the square grid under [1.5, 0, 0, 1] the one-distance construction fails 36 vertices and the one-per-side construction passes all 28 ×1
on this vertex the general construction asks for sides of noticeably different lengths, which one distance for the whole polygon cannot produce ×1
one distance per edge folds 20 of 20 tilings and images where one per vertex folds 5 ×1
one right-hand side per constraint row ×1
phase one of a simplex terminates at an optimum ×1
round a 3-edge loop of the the triangular grid the pleat ratios multiply to 1.000000 ×1
round a 6-edge loop of the the honeycomb the pleat ratios multiply to 1.000000 ×1
sliding every twist polygon by one vector satisfies every corner condition exactly, so two of the free directions are the drawing moved rather than a pattern changed, and what the last column counts is what is left after they are taken out ×1
the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 36 checked ×1
the collapse draws the sheet in along both principal directions, by 0.670 and 0.065 ×1
the one-distance construction folds on the 5 tilings as drawn and on none of their 15 images, and one distance per edge folds on every one of the 20 ×1
the the twist on a tiling by regular polygons is put past all four theorems before it is drawn ×1
the twist construction on a deformed tiling returns a pattern failing Kawasaki at 36 of its 36 interior vertices, by up to 12.2° ×1
the weights a twist puts on its vertex's edges close to 3e-16, so the construction writes an equilibrium stress without being asked for one ×1
the weights the construction writes spread by as much as 6.00 to one, and are equal on every edge of 10 of the 20 tilings and images measured ×1
the window of pleat angles runs from -0.05 to between 0.45 and 1.15 radians ×1
while the rhombille's loops close as drawn and fail under every map — its equations are not one at both ends, and only its regular shape makes their products agree ×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 dial can turn the fold over

A twist tessellation drawn on a sheared square grid folds to a sheet that draws in 4.4 times as much one way as the other, and the obvious guess is that the number belongs to the tiling and the shear. It does not: the turn and the pleat angle move it from 1.5 to 383. Between pleat angles of −0.348 and −0.161 the collapse's area changes sign, and the twists fold to a mirror image of their own arrangement.

Closing the loops is not folding

The twist construction propagates one equation along every edge of a tiling, and it can only work where the equations agree round every loop. Asked which irregular tilings pass, a linear map gives a clean answer: the square grid, the triangular grid and the honeycomb pass under every shear and stretch tried, because each edge has a half-turn symmetry that makes its equation exactly one at both ends, and a half-turn survives any linear map. The rhombille passes only as drawn. But passing is not folding. On every one of those images — including the ones whose loops close exactly — the construction produces a pattern that fails the angle condition at every turn tried. The loops were a necessary condition all along, and the construction needs something the tilings' images do not give it.

Every twist writes an equilibrium

Divide each side of a twist polygon by the length of the edge it faces. The polygon closing says those numbers, weighted onto the edges, balance at the vertex; the pleat matching says the two ends of an edge agree on the number. Together they are a positive equilibrium stress — the thing a tiling has when it is the plan of a spider web — and the construction has been writing one at every vertex without being asked for it.

One number where the corners wanted four

The twist construction gives a vertex a single side distance, and every account of these patterns does the same — it is what rotate-and-shrink means. The conditions never asked for it. Written out, the corner condition is one linear equation per pleat crease in the distances taken one per edge, so a degree-four vertex carries four unknowns against two independent equations. Given them back, the twelve sheared and stretched tilings that refused to fold all fold.

The propagation that never had to work

The twist construction carries one equation per edge of its tiling and propagates the twist sizes outward from a seed. On every tiling anybody has drawn a twist on, every one of those equations is satisfied trivially — both ends of an edge read the same two numbers, because a regular polygon has one interior angle. The construction has been running and doing nothing, and the one tiling where it did something is the one whose tiles are not regular.

The sheet draws in crooked

Every twist tessellation measured here has collapsed by a similarity: the folded sheet is the flat one scaled and turned, the same way in every direction. The patterns that exist on sheared and stretched tilings do not. Ten of the fifteen images fold by a map with two different principal factors, up to five and a third to one — and the prediction that said which ten, made from the weights the pattern writes on its edges, is wrong in both directions.

What refused the mirror was the rim

A twist tessellation on a sheared square grid folds, for a window of pleat angles, to a mirror image of its own arrangement, and whether its layers can be stacked was open. Asked of the drawn patches, the answer is no at every pleat angle, mirrored or not. Every refusal is a ring of flaps — flaps folded onto one face of a panel, each lying across the crease of the next — and on this grid every ring sits at the rim. Cut two creases in and every patch stacks, mirrored ones included.

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