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.

• Apply Mapping - This is a Boolean parameter that must be checked if the user wants to apply transforms to the image.
• Oversize by (%) - This is visible only if Apply Mapping is checked. The formula uses an array containing a virtual viewing screen. This parameter increases the size or the virtual viewing screen.
• Max Iters - This is the maximum number of Möbius iterations.
• Smallest Circle -  This is the "bailout" for Möbius inversions. Iteration continues until the circle size is less than the smallest circle.
• Rad multiplier - Alters the radius of all circles by this amount.
• Max circle size - The largest circle to be displayed.
• Min display level - The minimum iteration level to display. If Min display level equals Max Iters, this will display an approximation of the limit set.
• Group - The Kleinian group to be calculated
• Double Cusp - Calculated a Kleinian double cusp group. The particular double cusp to be calculated is listed below:
• 0/1
• 1/15
• 1/10
• 2/19
• 1/9
• 7/43
• 3/10
• 2/5
• 1/2
• 21/34
• 1/1
• Grandma's Special - This parameter utilizes a special formula developed in the book Indra's Pearls, and requires a thorough knowledge of the use of trace parameters. The use of the explore mode in UltraFractal is not recommended.
• Grandma's Special #2 - This parameter utilizes a special formula developed in the book Indra's Pearls, and requires a thorough knowledge of the use of trace parameters. The use of the explore mode in UltraFractal is not recommended.
• Masket - This parameter utilizes a special formula developed in the book Indra's Pearls, and requires a thorough knowledge of the use of trace parameters. The use of the explore mode in UltraFractal is not recommended.
• Nearby Group - This is a Kleinian group near the 1/10 cusp, but is not a cusp itself.
• Schottky - The user can chose from several options
• Kissing - All the base circles are tangent. The explore mode can be used on the numeric Kissing parameters.
• Disjoint - None of the base parameters touch.
• Apollonian 1 - This is a mapping of the standard Apollonian Gasket. All circles are centered on the limit set. This option is very slow unless Max Iters is decreased to a small value.
• Apollonian 2 - This is a Masket type mapping of the standard Apollonian Gasket. All circles are centered on the limit set. This option is very slow unless Max Iters is decreased to a small value. This option is very slow unless Max Iters is decreased to a small value.
• Apollonian 3 - This is the standard Apollonian Gasket, except all circles are centered on the limit set.
• Indra's Net - This option can be used to create the Indra's Net image in Indra's Pearls.
• Slice - This is a special formula developed by the author. The groups are a mapping of the double cusps. The particular double cusp to be calculated is listed below:
• 0/1
• 1/15
• 1/10
• 2/19
• 1/9
• 7/43
• 3/10
• 2/5
• 1/2
• 21/34
• 1/1
• Cusp View - This parameter is visible for some group selections. For Grandma's Special, Grandma's Special #2 and Masket groups use only the Standard option unless you have a thorough knowledge of Kleinien groups.
• Height Transfer - Slopes are calculated on the "stacked" orbits to determine the lighting effects. Height transfer applies a function to the height before the slope is calculated. It is useful for special effects.
• Height Pre-Scale - This is the ratio between height and distance. Larger values will exaggerate the lighting of highs and lows in the texture. This needs to be adjusted for most formulas.
• Height Post-Scale - This is similar to Height Pre-Scale but is applied after the transfer function. It is useful for special effect
• Fill Type -  This determines the lighting shape for the iterated circles.
• Quartic1 - This is the ellipsoid lighting:
`(radiuspower - alpha*xpower - beta*ypower)1/power/gamma`
• Quartic2 - Hyperboloid lighting #1:
`(radiuspower + alpha*xpower - beta*ypower)1/power/gamma`
• Quartic3 - Hyperboloid lighting #2:
`(radiuspower + alpha*xpower + beta*ypower)1/power/gamma`
• Quartic4 - Hyperboloid lighting #3:
`(radiuspower - alpha*xpower + beta*ypower)1/power/gamma`
• Power - The power in the Quartic formulas.
• alpha - See the Quartic formulas.
• beta - See the Quartic formulas.
• gamma - See the Quartic formulas.
• global fBm weight - The two global fBm parameters are unique in that they perturb the structure of the gasket shapes and not the coloring of the shapes. They put texture on the shape surface so that light reflecting off the surface shows the texture. This parameter determines the intensity of the texture.
• global fBm scale - Determines the scale of the texture.

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
```Cusp1_15 {
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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
```IndrasNet {
::xTEv9jn2tbVTPutNQ07Gw/HE0leaXrPsssbBPkFJHMQaQR3UgezgmiSmdpI1SSvr1+rvzQSp
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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
```IndrasNecklace {
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