# 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," Loci (December 2013)