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.[1]


Start with two identical coins touching each other on a table, with their "head" sides displayed and parallel. Keeping coin A stationary, rotate coin B around A, keeping a point of contact with no slippage. As coin B reaches the opposite side, the two heads will again be parallel; B has made one revolution. Continuing to move B brings it back to the starting position and completes a second revolution. Paradoxically, coin B has rolled a distance equal to twice its circumference.

As coin B rotates, each point on its perimeter describes (moves through) a cardioid curve.


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 twice as it revolves around the center of the other (still) coin.
Sliding over a circle, without rotation


  1. ^ Weisstein, Eric W. "Coin Paradox". MathWorld.

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