Concept

Side distance — where it appears

How far a twist polygon's side sits from the vertex it belongs to. The construction that draws these patterns gave a vertex one of them for the whole polygon; the flat-folding conditions constrain only sums along edges, so a vertex of degree four may have four, and that difference is what decides which tilings the construction reaches.

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

One number where the corners wanted four

The twist construction gives a vertex a single side distance, and every account of these patterns does the same — it is what rotate-and-shrink means. The conditions never asked for it. Written out, the corner condition is one linear equation per pleat crease in the distances taken one per edge, so a degree-four vertex carries four unknowns against two independent equations. Given them back, the twelve sheared and stretched tilings that refused to fold all fold.

tessellation · Resch

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

Named alongside it

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

Pleat Tessellation Tiling Twist Construction Equilibrium stress Kawasaki's theorem Spider web

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