Fix $\alpha\in\mathbb R$ and $N\in \mathbb N$, consider the set $S(\alpha,N)$ of $\{k\alpha\},k=1,\dots,N$, where $\{x\}$ denotes the fractional part of $x$. Let $a_1,\ldots a_N$ be the elements of $S(\alpha, N)$ arranged in increasing order, and consider the gaps sizes $$ |a_{i+1}-a_i|, i=1,\dots,N-1. $$ The well-known Three Gap Theorem tells us that the number of distinct gap sizes is at most three for any $\alpha$ and $N$.

There are many proofs and generalisations of this theorem, but I haven't gained any intuition regarding the statement. In particular, is there any heuristic/intuitive reason as to *why* the theorem should be true?