Chapter 4: Systems Based on Numbers

Section 2: Elementary Arithmetic

Multiples of irrational numbers

Instead of powers one can consider successive multiples Mod[h n, 1] of a number h. The pictures below show results obtained as a function of n for various choices of h. (These correspond to positions of a particle bouncing around in an idealized box, as discussed on pages 971 and 1022.)

When h is a rational number, the sequence always repeats. But in all other cases, the sequence does not repeat, and in fact it is known that a uniform distribution of values is obtained. (The average difference of successive values is maximized for h = GoldenRatio, as mentioned on page 891.)

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]