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Straight Lines Only

Two dots slide back and forth through the center of a ring. More dots join, sixteen in the end, and suddenly your eyes see a circle rolling around inside the ring. Then the lines appear, and you can check: every single dot only ever moved along its own straight line.

This is the Tusi couple, described by the Persian astronomer Nasir al-Din al-Tusi in the 13th century. If a small circle rolls inside a big circle twice its size, every point on the small circle travels along a straight line, a diameter of the big one. Here each dot swings along its line in simple harmonic motion, and each one is timed slightly later than its neighbor, in step with the angle of its line. Together they always sit on that small rolling circle.

Fun fact: al-Tusi invented the device to turn circular motion into straight motion for his model of the planets, and a very similar construction later appears in Copernicus's work. On the website you can tap to add more dots.