Machine Kinematics
solved, then drawn
Concept

Developability — where it appears

The property that the sector angles at a vertex sum to a full turn, so that the surface can be flattened onto a plane without stretching. It distinguishes nothing among crease patterns, because a vertex drawn on a flat sheet has sectors summing to a full turn by construction, and it is worth stating for exactly that reason.

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

What a pattern that cannot fold leaves behind. The best a least-squares solve can do with the vertex closures, against the fold it is asked for. The Miura pattern closes at every angle, at the arithmetic's own floor — the line along the bottom is 10⁻¹⁵ and below. The same grid with its vertices moved by a tenth of a panel does not close at any angle at all: its residual starts at 5.9e-6 at the smallest fold and grows with it, and no seed and no number of iterations moves it. Both patterns have the same panels, the same creases, the same graph and the same developable vertices, and every one of those vertices folds perfectly well on its own.

Each one moves, and together they do not

Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.

networks · Network A vertex is a spherical linkage, and the sectors are its link lengths. The four creases of one folded vertex, drawn as directions from the vertex itself, with the great-circle arcs between consecutive ones. Those arcs are the sector angles of the flat pattern — 80°, 60°, 100°, 120° — and they are those angles at every fold, to 8.9e-16 radians. That is the whole of the claim in the title: four axes through a point at fixed arcs from each other is a spherical four-bar, the object this site's spatial field built two phases ago, and a crease pattern's vertex is one of them with the arcs printed on the paper. The dihedral angle of the sheet at each crease is a half turn less that crease's fold angle, which here run 68.8°, 14.4°, 68.8°, 14.4°. positioned by solving, not by drawing.

Every vertex is a spherical linkage

Four creases through a point at fixed arcs from one another is a spherical four-bar — the object the spatial field is built on — with its link lengths printed on the paper as sector angles. The arcs hold to four parts in ten thousand million million at every fold, and on a flat-foldable vertex the half-angle tangents keep a ratio constant to nine figures.

networks · Network

Named alongside it

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

Crease patternMobilityNetworkRigid origamiSpherical linkageAssembly branchCompatibilityDihedral angleFold angleRedundant constraintResidual

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