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.

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a = or enter your own value a=

Created by David Richeson using GeoGebra.