Chapter 4: Systems Based on Numbers

Section 6: Mathematical Functions

Three sine functions

All zeros of the function Sin[a x] + Sin[b x] lie on the real axis. But for Sin[a x] + Sin[b x] + Sin[c x], there are usually zeros off the real axis (even say for a = 1, b = 3/2, c = 5/3), as shown in the pictures below.

If a, b and c are rational, Sin[a x] + Sin[b x] + Sin[c x] is periodic with period 2 π /GCD[a, b, c], and there are a limited number of different spacings between zeros. But in a case like Sin[x] + Sin[2 x] + Sin[3 x] there is a continuous distribution of spacings between zeros, as shown on a logarithmic scale below. (For 0 < x < 106 there are a total of 448,494 zeros, with maximum spacing 4.6 and minimum spacing 0.013.)

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