Symmetries of the Three-Gap Theorem

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition unit circle into gaps at most three distinct lengths. We prove a result about symmetries in order with which sizes appear on circle.