Slope Formula Tutorial for UltraFractal - Page 9
Indra's Promise
Kleinian group fractals have been popularized by the book "Indra's Pearls" by David Mumford, Caroline Series and David Wright. The key to fractals of this type is an understanding of Möbius transformations. Möbius transformations form a mathematical group. They are also known as linear fractional transformations, and are represented as:
where z is the complex number being transformed, and a, b, c and d are complex constants. A Möbius transformation can be viewed as a composition of translations, scalings and inversion. Properly chosen Möbius transformations can be iterated, with the limit set of the iterated points defining a fractal.
The Indra's Promise ufm is based upon Indra's Pearls, and has extensive non-slope coloring options. This tutorial will concentrate on the slope options. The following image is an example of non-slope coloring using traditional coloring ucls.
FrescoAtTwilight |
The discussion that follows will deal primarily with use of Indra's Promise were the user has checked the "calc as slope" box. If it is checked, the following parameters will be visible. Full use of some parameters requires a thorough knowledge of Indra's Pearls by Mumford, Series and Wright or of Kleinian Group theory.
(radiuspower - alpha*xpower - beta*ypower)1/power/gamma
(radiuspower + alpha*xpower - beta*ypower)1/power/gamma
(radiuspower + alpha*xpower + beta*ypower)1/power/gamma
(radiuspower - alpha*xpower + beta*ypower)1/power/gamma
The first image is the the full 1/15 Kleinian double cusp. Its fractal nature is clearly visible. 3D Texturizer Enhanced III is used for coloring. Generally, Direct Color Slope will not give satisfactory results with this ucl. The background was done with the non-slope coloring mode of Indra's Promise. Examine the upr to see how the image was created and the gradients were used.
1/15 Cusp |
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One of the best known images from the book Indra's Pearls is called Indra's Net, and is a depiction of the net cast by the Buddhist god Indra. The user should study the upr to see how the image is created. The top layer uses non-slope coloring.
Indra's Net |
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The Min display level parameter can create some interesting effects. When it equals Max iter you will get an approximation to the limit set, the set of circles that is the convergence of all the earlier circles. For small values of Max iter the image can have artistic interest. Indra's Necklace is such an image:
Indra's Necklace |
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