Origami & Folding
what one sheet can do
Field

Tessellations

One vertex repeated until the sheet stops being a sheet and becomes a material.
at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 39.3 sheet-widths of foldingmountainvalleyraw edge

One vertex, repeated

Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.

 nearly flatwidth ×0.87 height ×0.50half closedwidth ×0.58 height ×0.82nearly packedwidth ×0.11 height ×0.99both dimensions shrink together — pulling it open in one direction opens it in the other

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

 found in crushed drink cans, tree bark and deployable boomsthe pattern is a consequence of thin-wall buckling, not of a designmountainvalley

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

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