Origami & Folding
what one sheet can do
Concept

Found patterns — where it appears

Crease patterns that arise from physics rather than from design — buckling, crumpling, packing in a bud. They satisfy the flat-folding conditions because they were folded rather than drawn.

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

what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

Found before it was designed

Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

history · Rediscovery degree-4 vertex, 60/90/120/90°4 of 164 creases · 25.0% surviveone degree-6 vertex8 of 646 creases · 12.5% survivethe preliminary base112 of 2568 creases · 43.8% surviveand these are only the local tests — a pattern can pass every vertexand still collide once the layers stack, which is the hard part

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

history · Rediscovery

Named alongside it

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

Independent discoveryMaekawa's theoremAssignmentBucklingDegree-fourKawasaki's theoremThe Resch patternThe Yoshimura pattern

All concepts