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Colouring the Fatou domains

Our method of colouring is based on the real iteration number, which is connected with the potential function ${\displaystyle \phi (z)}$ of the Fatou domain. In the three cases the potential function is given by:

${\displaystyle \phi (z)=\lim _{k\to \infty }1/(|z_{kr}-z{*}|\alpha ^{k})}$ (non-super-attraction)

${\displaystyle \phi (z)=\lim _{k\to \infty }\log(1/|z_{kr}-z{*}|)/\alpha ^{k}}$ (super-attraction)

${\displaystyle \phi (z)=\lim _{k\to \infty }\log |z_{k}|/d^{k}}$ (d ≥ 2 and z* = ∞)

The real iteration number depends on the choice of a very small number ${\displaystyle \epsilon }$ (for iteration towards a finite cycle) and a very large number N (e.g. 10¹⁰⁰, for iteration towards ∞), and the sequence generated by z is set to stop when either ${\displaystyle |z_{k}-z{*}|<\epsilon }$ for one of the points z* or ${\displaystyle |z_{k}|>N}$, or when a chosen maximum number M of iterations is reached (which means that we have hit the Julia set, although this is not very probable).

If the cycle is not a fixed point, we must divide the iteration number k by the order r of the cycle, and take the integral part of this number.

If we calculate ${\displaystyle \phi (z)}$ for the k that stops the iteration, and replace ${\displaystyle |z_{k}-z{*}|}$ or ${\displaystyle |z_{k}|}$ by ${\displaystyle \epsilon }$ or N, respectively, we must replace the iteration number k by a real number, and this is the real iteration number. It is found by subtracting from k a number in the interval [0, 1[, and in the three cases this is given by:

${\displaystyle \log(\epsilon /|z_{k}-z{*}|)/\log(\alpha )}$ (non-super-attraction)

${\displaystyle \log(\log |z_{k}-z{*}|/\log(\epsilon ))/\log(\alpha )}$ (super-attraction)

${\displaystyle \log(\log |z_{k}|/\log(N))/\log(d)}$ (d ≥ 2 and z* = ∞)

In order to do the colouring, one needs a selection of cyclic colour scales: either pictures or scales constructed mathematically or manually by choosing some colours and connecting them in a continuous way. If the scales contain H colours (e.g. 600), we number the colours from 0 to H − 1. Then the real iteration number is multiplied by a number determining the density of the colours in the picture. The integral part of this product modulo H corresponds to the color. The density is in reality the most important factor in the colouring and if its judicious choice can result in a nice play of colours. However, some fractal motives seem to be impossible to colour satisfactorily and in these cases we have to leave the picture in black-and-white or in a moderate grey tone.

Lighting-effect

We can make the colouring more attractive for some motives by using lighting-effect. We imagine that we plot the potential function or the distance function over the plane with the fractal and that we enlight the generated hilly landscape from a given direction (determined by two angles) and look at it vertically downwards. For each point we perform the calculations of the real iteration number for two points more, very close to this, one in the x-direction and the other in the y-direction. The three values of the real iteration number form a little triangle in the space, and we form the scalar product of the normal (unit) vector to the triangle by the unit vector in the direction of the light. After multiplying the scalar product by a number determining the effect of the light, we add this number to the real iteration number (multiplied by the density number).

Instead of the real iteration number, we can also use the corresponding real number constructed from the distance function. The real iteration number usually gives the best result. Using the distance function is equivalent to forming a fractal landscape and looking at it vertically downwards.

The effect is usually best when ${\displaystyle f(z)}$ is a polynomial and when the cycle is super-attracting, because singularities of the potential function or the distance function give bulges, which can spoil the colouring.

Original upload log

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• 2010-04-21 09:42 Gertbuschmann 800×600× (433930 bytes) {{Information |Description = lighting-effect |Source = I (~~~) created this work entirely by myself. |Date = ~~~~~ |Author = Gert Buschmann |other_versions = }}