Origami & Folding
what one sheet can do
Concept

Equilibrium stress — where it appears

A positive number on every edge of a planar graph such that at each vertex the edge vectors weighted by those numbers sum to zero. It is the nineteenth-century graphic-statics object, and a twist tessellation writes one without being asked: each side of a twist divided by the edge it faces is a weight, and the polygon closing is the balance.

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

Also named here as spider web — the same set of essays touches all of them, so they are one junction rather than several.

each side of the twist, divided by the edge it facesone vertex of the the square grid under [1.3, 0.4, -0.2, 0.9]1.0520.3871.0520.387the weightsthe same edges, scaled by them, end to endthe 4 weighted edges close to 3e-16 of their own total length, and the two ends of every edge agree to 3e-15

Every twist writes an equilibrium

Divide each side of a twist polygon by the length of the edge it faces. The polygon closing says those numbers, weighted onto the edges, balance at the vertex; the pleat matching says the two ends of an edge agree on the number. Together they are a positive equilibrium stress — the thing a tiling has when it is the plan of a spider web — and the construction has been writing one at every vertex without being asked for it.

tessellation · Resch how much more the folded sheet draws in one way than the otherthe ratio of the two principal factors of the collapse, fitted to the twists' positions before and after foldingas drawnshearedstretchedgeneralthe square grid1.00004.41391.00003.5417the triangular grid1.00001.00001.00001.0000the honeycomb1.00002.81795.38555.0597the rhombille tiling1.00002.81795.38555.0597the elongated triangular tiling1.00004.41391.00003.5417one is a similarity — the folded sheet is the flat one scaled and turned, with no direction preferred

The sheet draws in crooked

Every twist tessellation measured here has collapsed by a similarity: the folded sheet is the flat one scaled and turned, the same way in every direction. The patterns that exist on sheared and stretched tilings do not. Ten of the fifteen images fold by a map with two different principal factors, up to five and a third to one — and the prediction that said which ten, made from the weights the pattern writes on its edges, is wrong in both directions.

tessellation · Resch

Named alongside it

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

Spider webTessellationTilingTwistAnisotropyPleatShrinkageSide distance

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