The Coin Rotation Paradox

The coin rotation paradox is a counterintuitive result in geometry. When a coin rolls around another coin of the same size, it makes two full rotations, not one as might be expected.

Why Does This Happen?

There are two components to the rotation:

1. Orbital Rotation: The rolling coin rotates once around the fixed coin.
2. Spin Rotation: The rolling coin spins on its own axis as it rolls.

Mathematical Explanation

For a rolling coin of radius r rolling around a fixed coin of radius R:

• Number of rotations = (R + r) / r
• When R = r (same size coins), this gives 2 rotations
• When R = 2r, this gives 3 rotations