Tusi Couple - gears that move in straight lines (desk toy, parametric)

A 20-tooth gear rolls inside a 40-tooth ring. Every point on the small gear's pitch circle then moves back and forth in a straight line, never a curve. Nasir al-Din al-Tusi described this in 1247, and it's still surprising to watch. Two pins sit on that circle and run in a crossed-slot lid, one along each slot. Push either pin straight along its slot and the gear rolls round the ring. A ring mark at the centre of the gear goes round in a circle while the pins slide in lines. There are three par…

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