Origami & Folding
what one sheet can do
Concept

Two-colourability — where it appears

The property that a pattern's panels can be coloured with two colours so that panels sharing a crease differ. On a disc of paper it is equivalent to every interior vertex having even degree, and on a sheet with a hole it can fail.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases

Nothing meets at three

Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.

flat-folding · Vertex degree 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 what the observer is givenambiguities separated, of 71the outline alonewhat a silhouette carries0the outline and the colourwhat a photograph of duo paper carries, from both sides0the complete layer orderwhat taking the model apart carries69the middle row is what anybody can actually see, and it is the top row

Which side is showing

Two earlier rungs asked what a folded object records about the pattern that made it, first from its outline and then from its complete layer order. Neither observation is one anybody can make. A photograph of duo paper carries the outline, the thickness and the colour showing at every point — and over the whole census the colour separates nothing at all.

flat-folding · Inverse problem patternfootprint with both colours over itlayers thereThe preliminary base8 panels, 4 one way up and 4 the other100%8.0The Miura fold24 panels, 12 one way up and 12 the other100%9.2The square twist9 panels, 5 one way up and 4 the other65%4.1The hexagon twist13 panels, 7 one way up and 6 the other72%4.1The Yoshimura pattern65 panels, 32 one way up and 33 the other100%60.0Fold and cut — the triangle7 panels, 4 one way up and 3 the other4%6.4The tapered corrugation28 panels, 14 one way up and 14 the other100%8.4The waterbomb tessellation52 panels, 26 one way up and 26 the other100%31.1

Decided before the design

A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.

design · Colour change counting rules and counting objects are different measurementsrepeating rulesnine binary choices, one per crease of the repeating unit512pass on a small patchevery vertex of a two-by-two patch satisfies every condition56pass on a larger oneand on a patch that contains all four kinds of vertex32folded objectscounted by comparing the folded panels, not the letters1

Thirty-two rules, one object

Five hundred and twelve repeating rules for the waterbomb tessellation, fifty-six that pass on a small patch, thirty-two that pass on one containing every kind of vertex. Fold all thirty-two and compare their panels: the same panels, in the same places, with the same areas, every time. The rules are thirty-two labels on one object, and a count of them has counted the labels.

tessellation · Waterbomb the bar is how far apart two routes to one panel end up, before the cut3 creases1.751.75 apart · cut open, 0e+05 creases1.591.59 apart · cut open, 0e+07 creases1.431.43 apart · cut open, 0e+09 creases1.321.32 apart · cut open, 0e+011 creases1.241.24 apart · cut open, 0e+0after one cut from the hole to the rim, every one of them places to rounding — with no crease changed

A cut that reaches the edge

A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.

material · Kirigami the bar is how many circles of that many panels were found726 circles, from 6 panels to 32, over every pattern family measured here4 panels0round one vertex — Maekawa forbids it5 panels0odd — the two-colouring forbids it6 panels21129.1% of the circles measured7 panels0odd — the two-colouring forbids it8 panels21129.1% of the circles measured9 panels0odd — the two-colouring forbids it10 panels8211.3% of the circles measured11 panels0odd — the two-colouring forbids it12 panels9513.1% of the circles measured13 panels0odd — the two-colouring forbids it14 panels304.1% of the circles measured15 panels0odd — the two-colouring forbids it16 panels314.3% of the circles measured17 panels0odd — the two-colouring forbids it18 panels172.3% of the circles measured19 panels0odd — the two-colouring forbids it20 panels141.9% of the circles measured22 panels81.1% of the circles measured24 panels152.1% of the circles measured26 panels71.0% of the circles measured28 panels20.3% of the circles measured30 panels20.3% of the circles measured32 panels10.1% of the circles measuredthe empty rows are not rare cases — they are lengths that cannot occur, and each has its own reason

A contradiction is even

A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.

flat-folding · Two-colourability 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

Named alongside it

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

Face graphFolded stateMaekawa's theoremParityBoundaryCrease assignmentDuo paperLayer orderLayer orderingNecessary conditionTwo-colouringAssignment

All concepts