Origami & Folding
what one sheet can do
Concept

Reflection — where it appears

The map that folds one side of a line onto the other. A flat folding is a composition of them, which is why a panel arrives either the same way up or turned over and never anything between.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

One fold at a time, and there are exactly seven of them

A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.

construction · The axioms the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch

A crease that curves

Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.

material · Curved creases 05101520253000.050.10.150.20.250.3creasesdrift, in panel widthsthe same way each timeat randomeach crease 0.5° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

Error is folded too

A folded position is a composition of reflections, and a reflection in a line that is slightly off turns everything beyond it by twice as much. So an error does not stay where it was made — and whether it grows with the crease count or with its square root depends on whether it is the same error every time.

rigid · Tolerance 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

Named alongside it

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

The Möbius bandGluingOrientabilityClosureFlat-foldabilityParityBoundaryIsometryNecessary conditionSector anglesTwo-colouringCompleteness

All concepts