Imagine a coin rolling around a pentagon, the coin will rotate in regular angular increments along each edge with an additional fifth of a rotation at each vertex. Now imagine increasing the number of edges. When the edges shorten and approach the vertices the Archimedean principle of exhaustion holds even when infinitely short edge and vertex are one. There is therefore an extra - you could say hidden - rotation. Of sorts.

## circle round a circle

Imagine a coin rolling around a pentagon, the coin will rotate in regular angular increments along each edge with an additional fifth of a rotation at each vertex. Now imagine increasing the number of edges. When the edges shorten and approach the vertices the Archimedean principle of exhaustion holds even when infinitely short edge and vertex are one. There is therefore an extra - you could say hidden - rotation. Of sorts.