March 2006

### The maths behind the music

The transformation rule which is the basis for Nick's composition uses the complex function $$f(z) = 0.5(-z + \sqrt{12 - 3 z^2}).$$ Iterating this function starting with a given value $z_0$ gives a sequence of complex numbers: $$z_0, z_1 = f(z_0), z_2 = f(z_1), z_3 = f(z_2), ... .$$
Any complex number *z = x+ iy* can also be thought of as the pair *(x,y)* of real numbers, so the formula does indeed transform one pair into another pair, as described above (see *Plus* article Curious quaternions for an introduction to complex numbers).