A circle made of points, shown here with small white balls, appears to roll around the inside edge of a larger circle, but is that what’s really happening? File under: Cycloids, hypocycloids, and optical illusions. This mechanical gear design, demonstrated by the Visual Education Project and originally developed by Italian polymath Girolamo Cardano (1501-1576), demonstrates how circular motion can result from linear motion.

Copernicus’ Theorem states a surprising result that a point on the circumference of the small circle traces a straight line segment – a diameter of the big circle, to be precise…

Cardano’s work with hypocycloids led him to the Cardan joint or gear mechanism, in which a pair of gears with the smaller being one-half the size of the larger gear is used converting rotational motion to linear motion.

Read more about rolling hypocycloids, not to be confused with the hypotrochoids made with a Spirograph, in this blog post by John Baez.

Read about Copernicus’ Theorem at Wolfram.com and Cut the Knot. Plus, a color path demo with the mathematical proof from Mind Your Decisions:

Here’s another video demonstration: Brusspup’s Crazy Circle Illusion.

Plus, more math videos, including Types of Triangles, Mathematica – A World of Numbers… and Beyond, Professor Kokichi Sugihara creates his mind-blowing illusions with math, How High Can You Count on Your Fingers? and How Many Ways Are There to Prove the Pythagorean Theorem?

Bonus: Spirograph pancakes.

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