Origami & Folding
what one sheet can do
Concept

The Möbius band — where it appears

A strip of paper joined end to end with a half twist, so that it has one side and one edge. It is the first sheet in this collection that is not orientable, and it needs an odd number of creases round it where a plain loop of paper needs an even number.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursoff by 4.000 of a widthand turn the paper the right waymountainvalleyraw edge

The seam carries a sign

A loop of paper folds flat when it has an even number of creases round it. A Möbius band folds flat when it has an odd number. The drawing is the same in both cases, the creases are the same creases, and what changed is a factor of minus one contributed by the sheet rather than by anything drawn on it.

flat-folding · Two-colourability 5 creases on a Möbius bandthe panels take two coloursseamthe same seam12345the right edge onto the left, turned over5 creases, 5 panelsinterior vertices: 0two-coloursoff by 4.000 of a widthand turn the paper the right waymountainvalleyraw edge

The band that needs an odd number

A Möbius band is the first sheet in this collection with one side, and the consequence is sharper than a reversed parity. Mountain and valley are defined relative to a side, so on a sheet with no consistent side a crease has no letter — and Maekawa's condition survives the loss while the assignment it is about does not.

flat-folding · Two-colourability the composition, and what it has to equal3 reflections, in order[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )they agree to rounding, so the band foldsand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down

Closure is not the identity

Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.

flat-folding · Flat-foldability the composition, and what it has to equal5 reflections, in order[ -1.000 0 ][ 0 1.000 ]+ ( 2.000, 0 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -2.000, 1.000 )they differ by 4.000 of a width, so it does notand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down

Parity is not enough

A Möbius band needs an odd number of creases round it. Give it three, square across the strip, and it does not fold — nor does five, nor seven, nor any odd number at all. The counting argument is necessary and it is not close to sufficient, and the thing it cannot see is which way the creases point.

flat-folding · Flat-foldability 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

The triangle a strip becomes

A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

flat-folding · Flat-foldability the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

An alternating sum of angles

Kawasaki's condition says the sectors round a vertex alternate to a straight angle. A glued band has no vertices and obeys a condition of exactly the same shape: the crease angles have to alternate to a multiple of a straight angle. Two different quantities, two different sheets, one arithmetic — and in both cases what is being said is that a product of reflections came back the right way.

flat-folding · Sector angles the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

How rare a band that folds is

Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

flat-folding · Genericity one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper

A sheet with two edges

Design in this subject starts from a square, and the square's four edges are where every flap ends and every construction begins. A cylinder has two edges instead of four, and what a designer loses turns out to be measurable in the same letters a search counts — and what they gain is that the sheet is what gets manufactured.

design · Sheet shape 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

The proportion a band asks for

√2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.

construction · Paper proportion

Named alongside it

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

GluingClosureReflectionFlat-foldabilityOrientabilityBoundaryParitySector anglesCountingDesignEfficiencyGenericity

All concepts