TITLE: Intuition for the Universal Chord Theorem
QUESTION [21 upvotes]: So the Universal Chord theorem is a statement and proof that; 
The numbers of the form $r = \displaystyle \frac{1}{n} \ \ n \ge 1$ are the only numbers such that for any continuous function $\displaystyle f:[0,1] \to \mathbb{R}$ such that $\displaystyle f(0) = f(1)$, there is some point $\displaystyle c \in [0,1]$ such that $\displaystyle f(c) = f(c+r)$. 
The proof is straightforward to understand. I don't have difficulty with any step in it or anything. As well, I understand the counterexamples for any non rational, and why they fail. But even still it seems absurd that this is how it is. I've been drawing a bunch of graphs here on a page and I just can't see any sort of insight as to why rationals should work but not non rationals. I heard once that there was intuition to be found with topology, but I can't find it anywhere online. Could someone please enlighten me as to why this theorem makes sense intuitively? (without just giving a proof)

REPLY [5 votes]: Intuitively, the problem begs for us to consider the function $g$ where $g(x) = f(x+r)-f(x)$ and see whether it is ever 0. If it is never 0, then $g$ is always of the same sign on $[0,1-r]$, and it is natural to hop from $f(0)$ to $f(r)$ to $f(2r)$ and so on, which must be a strictly monotonic sequence, which is impossible if $1$ is an integer multiple of $r$. On the other hand, if $1$ is not an integer multiple of $r$, then we can actually let $f$ be such that $g$ is a constant $c$, which makes the hop points a regularly spaced collinear sequence, by drawing a straight line segment from the last hop point to $(1,f(1))$ and filling in what follows and finally the rest of the graph with parallel line segments.