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| Keywords: Juliasetsdkpictlightpot.jpg en wikipedia 2010 May 7 lighting-effect Transferred from http //en wikipedia org en wikipedia; transferred to Commons by User Thenub314 using http //tools wikimedia de/~magnus/commonshelper php CommonsHelper <br/> Original text I Gertbuschmann talk created this work entirely by myself 09 42 21 April 2010 UTC <br/> 2010 04 21 original upload date Gert Buschmann Original uploader was Gertbuschmann at http //en wikipedia org en wikipedia Released into the public domain by the author en wikipedia Gertbuschmann Colouring the Fatou domains Our method of colouring is based on the real iteration number which is connected with the potential function <math>\phi z </math> of the Fatou domain In the three cases the potential function is given by <math>\phi z \lim_ k\to\infty 1/ z_ kr - z \alpha k </math> non-super-attraction <math>\phi z \lim_ k\to\infty \log 1/ z_ kr - z /\alpha k </math> super-attraction <math>\phi z \lim_ k\to\infty \log z_k /d k </math> d ‰¥ 2 and z ˆž The real iteration number depends on the choice of a very small number <math>\epsilon</math> for iteration towards a finite cycle and a very large number N e g 10<sup>100</sup> for iteration towards ˆž and the sequence generated by z is set to stop when either <math> z_k - z < \epsilon</math> for one of the points z or <math> z_k > N</math> 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 thumb right Nice play of coloursIf we calculate <math>\phi z </math> for the k that stops the iteration and replace <math> z_k - z </math> or <math> z_k </math> by <math>\epsilon</math> 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 <math>\log \epsilon/ z_k - z /\log \alpha </math> non-super-attraction <math>\log \log z_k - z /\log \epsilon /\log \alpha </math> super-attraction <math>\log \log z_k /\log N /\log d </math> d ‰¥ 2 and z ˆž 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 <math>f z </math> 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 page en wikipedia Juliasetsdkpictlightpot jpg 2010-04-21 09 42 Gertbuschmann 800×600× 433930 bytes <nowiki> lighting-effect I ~~~ created this work entirely by myself ~~~~~ Gert Buschmann </nowiki> Julia sets | ||||
