Coin rotation paradox

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The coin rotation illusion. Although to some the moving coin appears to have made a complete revolution, its point in contact with the stationary coin at the beginning (Washington's "nose") is not in contact at the end. It has only made a half revolution.
The path of a single point on the edge of the moving coin is a cardioid.

The coin rotation paradox is the counter-intuitive observation that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes two full rotations after going all the way around the stationary coin.


The problem begins with two identical coins. One is rotated around the other without slipping so that it ends up on the opposite side of the other coin from where it began. It has made a full rotation meaning it has rolled a distance equal to its circumference.

This can be visualised by placing two coins that are touching each other at one point flat on a table. Let both of them have the heads side up and be parallel to each other. Now keeping one coin stationary, rotate the other coin such that there is always one point of contact. Rotate until it reaches the opposite side. The coin has rolled a distance equal to its circumference.


The rolling coin actually participates in two separate motions not unlike the moon relative to the earth (except that the moon completes only one rotation about every 28 days):

  • The moon rotates once while revolving around an elliptical pathway relative to true north.
  • The moving coin rotates once as it revolves around the center of the other (still) coin.
Sliding over a circle, without rotation

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