Origami & Folding
what one sheet can do
Concept

Two-colouring — where it appears

An assignment of two colours to a pattern's panels so that any two sharing a crease differ. It is exactly which side of the sheet each panel shows in the folded state, computed without folding anything.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

20 panels, two coloursno crease has the same colour on both sidesall 12 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge

The sheet has two sides

Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.

flat-folding · Two-colourability 00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

design · Colour change a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

Even is not enough

Every vertex theorem in the subject is a statement about one point, and the two-colouring of the panels looks like the exception. It is not — on a square of paper it is a parity at each vertex and nothing more. Cut a hole and the two come apart: a loop of paper with three creases has no interior vertices at all, satisfies every theorem there is, and cannot be folded flat.

flat-folding · Two-colourability a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

A cut that removes no paper

Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.

material · Kirigami the bar is the share of the footprint showing the side that started face upthe rest of it shows the other side, and neither is chosen by anybodyThe preliminary base0.3%2 of 8 panels in viewThe square twist47.4%7 of 9 panels in viewThe hexagon twist90.8%10 of 13 panels in viewFold and cut — the triangle100.0%1 of 7 panels in view · 2 statesa panel out of view from above is not hidden — it is under the pile, and turning the sheet over shows a different set

Which side arrives

A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

design · Colour change 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 period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

flat-folding · Two-colourability what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

A cut is surgery

Two cuts that look identical on the paper do completely different things to the sheet. A slit run inward from the rim changes nothing at all; a closed cut in the middle removes a disc and leaves a sheet carrying a condition it did not have before. What separates them is not the length of the cut or how much paper it removes.

material · Kirigami two holes, two conditionsone loop is odd — the sheet refuses2 out to the left, 1 to the right, 1 betweenround the left hole: 3 creases, oddround the right hole: 2, evenround both: 3, oddno two-colouring exists0 interior verticesa loop round one hole says nothing about a loop round the other

Two holes are two conditions

One hole in a sheet of paper gives one loop that cannot be shrunk and one parity to satisfy. Two holes give two, and they are independent: an arrangement of creases can satisfy the condition round one hole and fail the condition round the other, and the sheet refuses on the strength of the one it failed.

material · Kirigami 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

The sixth thing that is not true

Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.

material · Idealisation what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

The cut that changes nothing

A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.

material · Kirigami a disc, with a vertexa ring, with noneone interior vertex, 3 creases at itodd degree, so they do notno interior vertices at alland the panels still do notboth refuse: two routes round the sheet leave a panel 1.87 sheet-widths apart

A theorem with an unstated hypothesis

Maekawa's and Kawasaki's conditions are quoted everywhere without saying which sheet they are about, and they do not need to be — they are conditions at a point and every point is the same. The two-colouring is quoted the same way and it is not a condition at a point, and the omission there is not harmless.

history · Attribution the bar is the footprint, split by how deep the other side liesone layer downtwo layers downnot under this point at allThe preliminary base8 panels · 1 stateThe square twist9 panels · 1 stateThe hexagon twist13 panels · 1 stateFold and cut — the triangle7 panels · 2 statesa 2 × 2 Miura patch4 panels · 1 statea 3 × 2 Miura patch6 panels · 3 statesa 3 × 3 Miura patch9 panels · 6 statesa 4 × 3 Miura patch12 panels · 11 statesthe letter fold3 panels · 2 statescounted over every folded state and from both faces, so nothing here is one lucky pile

One sheet down

A colour change has been priced by how deep the pile is — eight layers over every point of the preliminary base, six on a small Miura. Ordered, the piles say something else: wherever the other side of the paper lies under a point, it is the next sheet down on every pattern with one folded state, and never more than two down on any. And relettering the same creases cannot reach it. Of 112 letterings of the preliminary base that fold, every one shows either the printed face or the whole face turned the other colour; the square twist's eight only turn its face round.

design · Colour change

Named alongside it

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

BoundaryParityGluingPanelDesign techniqueKirigamiMaekawa's theoremOrientabilityColour changeCountingFlat-foldabilityLocality

All concepts