Origami & Folding
what one sheet can do
Concept

The Resch pattern — where it appears

A tessellation of triangles and twists named for Ron Resch. Its pleat conditions close round every loop of a regular tiling automatically, which is why the construction works and why an irregular tiling can refuse it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge

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.

tessellation · Twists polygonsectors at the twist vertexKawasakitiles the plane3-gon60.0 · 60.0 · 120.0 · 120.0180.0° = 180.0°yes — 6 round a point4-gon90.0 · 90.0 · 90.0 · 90.0180.0° = 180.0°yes — 4 round a point5-gon108.0 · 108.0 · 72.0 · 72.0180.0° = 180.0°no6-gon120.0 · 120.0 · 60.0 · 60.0180.0° = 180.0°yes — 3 round a point7-gon128.6 · 128.6 · 51.4 · 51.4180.0° = 180.0°noevery one of these twists satisfies the local theorems and folds flat on its ownthe interior angle has to divide 360° for the twists to meet, which only 3, 4 and 6 dobeyond 7 sides the assignment search runs out — 21 free creases, and the enumerator refuses above 22

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.

tessellation · Twists degree-4 vertex, 60/90/120/90°4 of 164 creases · 25.0% surviveone degree-6 vertex8 of 646 creases · 12.5% survivethe preliminary base112 of 2568 creases · 43.8% surviveand these are only the local tests — a pattern can pass every vertexand still collide once the layers stack, which is the hard part

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

history · Rediscovery

Named alongside it

The objects these essays reach for when they reach for this one.

Kawasaki's theoremPeriodicityAssignmentThe big-little-big lemmaCompletenessDegree-fourFound patternsIndependent discoveryMaekawa's theoremPleatRotational symmetryTwist tessellation

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