Origami & Folding
what one sheet can do
Concept

Vertex — where it appears

A point where creases meet in a pattern. An interior vertex is subject to every local condition in the subject and a boundary vertex to none of them, which is a distinction that decides more than it sounds like it should.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley

Two conditions at a point

Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.

flat-folding · Flat-foldability sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every onelongest ÷ shortest 1.00footprint 0.806longest ÷ shortest 3.09footprint 0.911longest ÷ shortest 3.33footprint 0.623longest ÷ shortest 4.00footprint 1.782every one of them folds; their folded footprints differ by a factor of 2.86

The lengths are free

Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.

flat-folding · Sector angles four conditions with nowhere to holddevelopabilityholds, at 0 verticesKawasakiholds, at 0 verticesMaekawaholds, at 0 verticesbig-little-bigholds, at 0 vertices6 of 12 of these bands have no flat folded stateand only the panel colouring can see it

A crease with no vertex to belong to

Crease density is measured as length of line per area of paper, and everything else about a crease is measured at the vertex it runs into. A band of paper has creases that run from one edge to the other and meet nothing, so it has density and no vertices at all — and it still refuses to fold.

material · Crease density

Named alongside it

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

Crease lengthKawasaki's theoremMaekawa's theoremAssignmentThe big-little-big lemmaBoundaryCrease densityDevelopabilityFlat-foldabilityFolded stateGluingIdealisation

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