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 6 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 an odd polygon, its half-turn, and the regular polygon of twice the sides between themsolid: the polygon; dashed: turned half a turn; shaded: the shape halfway between, drawn in the sheetthe triangle on a sheet 1 by 1.1547averaged with its half-turn: a regular hexagonthe same two widths, so it fits the same sheetthe pentagon on a sheet 1 by 1.0515averaged with its half-turn: a regular decagonthe same two widths, so it fits the same sheet

An odd polygon fits like its double

A regular polygon with an odd number of sides has a corner opposite every flat, so it looked as if the closed form for the turning polygons of 4k + 2 sides would need redoing with a different offset. It does not. An odd polygon is exactly as wide, in every direction, as the regular polygon with twice its sides — that polygon is the odd one averaged with its half-turn — and a box sees only widths. So the triangle fits wherever a hexagon fits, the heptagon wherever a fourteen-gon does, and on every sheet the odd polygon's share is its double's times 2 − sec²(π/2n). It peaks on its double's own sheet with the same gain; on a square it is tilted π/4n, the triangle's fifteen degrees; and on its own sheet it beats nothing but smaller odd polygons, never once passing the circle.

construction · Optimal constructions the largest regular polygon a triangle of paper holds, for each number of sides, best firstshaded: sides a multiple of 3; plain: the rest; the upright line is the largest circle, 60.5%bars start at 50%3 sides100.0% · compass6 sides66.7% · compass9 sides63.0% · one fold12 sides61.9% · compass15 sides61.4% · compass18 sides61.1% · one fold21 sides60.9% · one fold24 sides60.8% · compass23 sides60.2% · neither22 sides60.1% · two folds at once20 sides60.1% · compass19 sides60.0% · one fold17 sides59.9% · compass16 sides59.9% · compasscircle

A sheet rewards the multiples of its sides

On a rectangle every odd polygon fits exactly as its double does, because a box measures widths and a width cannot see a half-turn. A triangle of paper has no half-turn, and the ranking it gives is a different kind of thing. On a regular sheet of m sides, a regular polygon beats the largest circle exactly when its sides are a multiple of m — it can then face every side of the sheet with one of its own and share the sheet's circle, taking n·tan(π/n) over m·tan(π/m) of it — and every other polygon to forty sides falls short. On a triangle that puts the triangle first, the hexagon second at two thirds, and the nonagon third: the first sheet on which a polygon only a fold can build takes a place on the podium.

construction · Optimal constructions twists cut from sheets of their own shape, each drawn with its creasesdash-dot: mountain; dashed: valley; the outline is the sheettriangle twist3 frames, one answersquare twist2 frames, one answerpentagon twist5 frames, one answerhexagon twist3 frames, one answer

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.

tessellation · Shrinkage

Named alongside it

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

ConstructibilityInscribed polygonOptimalitySheet shapeTotientFolded stateFootprintPeriodicityPleatRegular polygonThe Resch patternShrinkage

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