Origami & Folding
what one sheet can do
Concept

Rotational symmetry — where it appears

A turn about a point that carries a shape onto itself. For a regular polygon it decides which edges face which directions, and so which rectangles the polygon can touch on all four sides at once.

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 the best rank a tool's polygons reach, on a few sheets and on all of thema proportion between the named sheets is where each tool's best polygon does bestthe polygonon seven sheetson every sheetits own sheetonly one fold builds it14 sides · 4th14 sides · 3rd1.0257only two folds at once build it22 sides · 9th22 sides · 5th1.0103no two folds build it47 sides · 12th46 sides · 11th1.0023ranks among every polygon from three sides to 48; the seven sheets run from a square to three to one

The sheet a polygon fits exactly

A regular polygon with 4k + 2 sides has flat edges along one axis and corners along the other, so there is one sheet, 1⁄cos(π⁄n) long, that it touches on all four edges at once. On that sheet it is beaten only by the multiples of four with fewer sides, and so it ranks exactly (n − 2)⁄4. That puts the fourteen-gon, which only a fold builds, third rather than fourth; the twenty-two-gon, which needs two folds at once, fifth rather than ninth; and the forty-six-gon, beyond two folds, eleventh. Seven sheets from a square to three to one had missed all three.

construction · Optimal constructions 11.051.11.151.20.850.90.9511.05the sheet's length, with its width oneshare, over the share on a square6 sides · +7.7%10 sides · +2.6%14 sides · +1.3%22 sides · +0.5%each curve is one polygon's share over its share on the square; the dot is its own sheet

Turning is uphill all the way

A regular polygon of 4k + 2 sides on a sheet a little longer than a square cannot lie flat: it turns, pressed against all four edges, until the sheet is exactly its own. Its share on the way has a closed form, and the closed form's slope is proportional to h² − 1 for every such polygon — flat on the square, rising all the way to the own sheet, and falling after it. So the own sheet is exactly the peak, the gain from the square to it is the average of one and the sheet's length, and the rank the census measured for polygons of this kind, (n − 2)⁄4, is now a theorem.

construction · Optimal constructions

Named alongside it

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

ConstructibilityInscribed polygonOptimalitySheet shapeTotientPeriodicityPleatThe Resch patternTwist tessellation

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