Representation Function

Robert P. Munafo, 2002 Apr 20.

The following definitions refer to the standard iteration calculation:

*Z8₀ = 0

Z_N+1 = Z_N² + C

In each definition, N refers to the iteration number, and R refers to the radius or complex magnitude of an iterate.

(also called dwell) The value of N when R reaches a maximum. Because R is constrained by the escape radius, this ends up being equal to Max(N), the maximum value of N.

This is the traditional way of representing the Mandelbrot Set.

External-Angle = R(N(Max(R))) = R(Max(N))

The value of R at the last iteration.

Atom-Domain-Period = N(Min(R))

The value of N when R reaches a minimum. The initial value of R is ignored because it is zero.

This function yields an excellent way of locating mu-atoms of a given period. The image produced contains regions of solid color that surround the mu-atoms, because the minimum R value occurs when the number of iterations is equal to the period of the mu-atom. Each color corresponds to a different period value. All island mu-molecules, no matter how small, can be located easily by this method.

Distance-Estimator = ln(z²) * z / dz

where z = Z(Max(N)), dz = iterated deriviative.

DEM-Dwell Hybrid

Plotting distance estimator and escape iterations at the same time, which is accomplished by using a two-dimensional mapping onto color space (see color).

Iterate Tracks or Buddhabrot

Plotting every Iterate value of Z during iteration of each point (like the plotting used with the Inverse-Iteration Method, except forward iterating using the Mandelbrot iteration, not the Julia Set iteration. See Buddhabrot for more about this plotting method.