Multibrot Set

A Multibrot set uses the iteration of the Mandelbrot set with another power: the map is z \to z^d + c, and c is in the set when the iteration does not escape to infinity. Change d to go from the Mandelbrot set at d = 2 to a set with more parts at d = 8. At a whole number d, the set has a rotational symmetry of order d - 1: d = 3 gives two parts and d = 5 gives four.

The power is a real number, so the function M of the document's prelude uses the polar form on real numbers. With s = a^2 + b^2, it calculates r = s^{d/2} and w = d\arctan(b, a), and then a \to r\cos(w) + p and b \to r\sin(w) + q. Between two whole numbers, the angle w has a jump where z crosses the negative real axis, so the picture has straight cuts that are not in the sets of whole powers.

M gives the escape count of a point divided by 128, and 1 for a point that does not escape in 128 steps. To make the count continuous, the test for the escape is |z|^2 > 4096. This does not change the set, because a point with |z| > 2 always escapes for d \ge 2. The count is then n + 1 - \ln(\ln(|z|^2) / \ln 4096) / \ln d. The loop stops when the point escapes, so only a point that does not escape does all 128 steps.