Let a be any real number and f:[0,1)→[0,1) be the function given by f(x)=x+a (mod 1). We can think of [0,1) as a circle of circumference 1 (glue 0 and 1 together). In this case we may interpret f as a rigid rotation by a.

Theorem. If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit.

Three Gap Theorem/Steinhaus Conjecture. The finite orbit segment {0,f (0),f ²(0),f ³(0),...,f ⁿ(0)} divides the circle into arcs of one, two, or three lengths. If there are three lengths, then one is the sum of the other two.

Created by David Richeson using GeoGebra.