The triangle twist
twist-unit 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
kind: "square", mm: 150, R: 0.17
kind: "square", R: 0.17
kind: "triangle"
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.
- developability holds at every interior vertex — its sectors close to 360° — 3 checked ×5
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 3 checked ×5
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 3 checked ×5
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 3 checked ×5
- the twist-census 3-gon is put past all four theorems before it is drawn ×5
- 3 of 5 polygons in the census have an interior angle dividing 360°, which is what lets a twist tile the plane ×1
- a twist vertex has exactly four sectors, and Kawasaki holds on them as an identity rather than by design ×1
- no twist here is drawn with a lettering known to have no folded state; 4 of 5 have an ordering of their panels found by enumeration ×1
- the 3 corners of the triangle twist are alike, their sectors of 60, 60, 120, 120° close to 360°, and none of them is strictly smaller than both its neighbours — so nothing here is ruled out by the lemma before the assignment is even chosen ×1
- the 4 corners of the square twist are alike, their sectors of 90, 90, 90, 90° close to 360°, and none of them is strictly smaller than both its neighbours — so nothing here is ruled out by the lemma before the assignment is even chosen ×1
- the 6 corners of the hexagon twist are alike, their sectors of 120, 120, 60, 60° close to 360°, and none of them is strictly smaller than both its neighbours — so nothing here is ruled out by the lemma before the assignment is even chosen ×1
- the census is taken over 5 polygons, each with at least three sides ×1
- the hexagon twist is put past all four theorems before it is drawn ×1
- the square twist is put past all four theorems before it is drawn ×1
- the triangle twist is put past all four theorems before it is drawn ×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 cut is a licence
What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.
A cut is not local
Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.
A sheet with one freedom
A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.
A square that turns
A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.
Any tiling makes a twist
A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.
The base that tiles
The waterbomb base is the first thing most people fold and the last thing they think about. Repeat it across a sheet and it becomes a tessellation with two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state thirty-two times smaller than the paper.
The creases that cannot move
One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.
The patterns nobody owns
This site prints crease patterns at true scale and prints no designer's work, and that has always been stated as a rule applied at the end. Read the printed shelf as a documentary record instead and the rule turns out to be a property of the record: every pattern that carries a date was published as mathematics, every undated one belongs to nobody, and the two silences are one silence.
The ring is the loop
The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.
The symmetry the letters cannot keep
Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.
The tiling the unit could not promise
Every twist on this site carries the same caveat: the unit is verified and the plane is not, because deciding a whole pattern is intractable. There is one thing about a whole pattern that costs a single pass over its crease list, and it says no. The square twist tiling was drawn with a lettering that contains a loop of twenty-eight panels, so the patch on this site had no flat folded state at all — and only seven of forty independent redraws avoid one.
Which polygons twist
Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.
Every generator · The tessellations field · The patterns a reader can fold