### 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 \( \alpha . \)