# The Japanese Theorem for Nonconvex Polygons - Irrational Rotations of the Circle

Author(s):
David Richeson

### Irrational Rotations of the Circle

We now illustrate the previous theorem with a concrete example. Recall the following well-known theorem (see [p.53,R], for instance).

Theorem. If $\alpha$ is a irrational number, then the sequence $\left( \left( \cos(2 k \pi \alpha), \sin(2 k \pi \alpha) \right) \right)_{k=0}^{\infty}$ is a dense subset of the unit circle.

This theorem says that if we take the point $(1,0)$ and repeatedly rotate it by an angle $2 \pi \alpha$ about the origin, then the orbit will be dense in the circle. Note that if $\alpha = p/q$ is rational and is expressed in lowest terms with $q > 0 ,$ then the orbit consists of $q$ points.

Use the applet below to see the sequence of total inradii for various irrational values of $\alpha .$

rotation = $2\pi\cdot$ or enter your own value

David Richeson, "The Japanese Theorem for Nonconvex Polygons - Irrational Rotations of the Circle," Convergence (December 2013)