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The net of the great god Indra was said to span all of space, finer than silk, and strung with pearls reflecting each other and all others along the infinite reach of the net at once forever.

Thus seemingly was foretold in Buddhist scripture the vision of Felix Klein, and the mathematical constructions used in the fractal type of this post, as paraphrased from the book, ‘Indra’s Pearls.’

Good evening, and welcome to the launch of the first weekday of this project’s posts, my attempt to more fully explore this sometimes cross but rewarding fractal type based on the mathematics of the interaction of spirals and the objects it leads to.

Today I’ve got four images to show, what I think to be the very best from today and last night generated by my current favorite app, Ultra Fractal over a period of several hours.


This on was made using my first parameter file created for this series (I generated ten such files, adjusted as needed), using an extremely well-designed coloring algorithm, ‘Double Cusp’ Kleinian group settinmgs and a rather fortuitous combination of color gradient settings.


This one uses a figure altered threefold with a kaleidoscopic transformation, using a different coloring algorithm and different but still striking gradient settings. I like to randomize and shift gradients around when coloring these, just to see what I’ll get, though the app certainly allow easy non-random color adjustments as well.


This one used my second file created, and used a different coloring algorithm than the previous two and different Kleinian groups (Schottky bubbles) to generate this image. Of today’s pieces, it took the longest to render, but was well-worth it I think.

The scope and potential of this fractal type is amazing, despite the relatively simple starting idea of interacting spirals, or perhaps even because of it. I’ll include further images tomorrow, and may you all fare well the remainder of this week.

Talotaa frang.


    • John Seymour
    • Posted Saturday, 1:59, April 19, 2014 at 1:59
    • Permalink
    • Reply

    Beautiful Apollonian Gaskets, would you be willing to release the parameters for the first image ?

    Sorry, I reposted the comment as there was a typo in my address…


    • Certainly! I’ll post them for you this evening.


    • Here are the parameters I used for the first image, save the gradient colors – those were randomized. Good fractaling!

      Image Tab (Fractal Properties)
      Width: 480 pixels
      Height 480 pixels
      Resolution: 300

      Location Tab (Layer Properties)
      Center (Re): 0
      Center (Im): 0
      Magnification: 1.5
      Rotation Angle: 0
      Stretch (X/Y): 1
      Skew Angle: 0
      Left Top (Re): -1.333333333
      Left Top (Im): 1.333333333
      Right Top (Re): 1.333333333
      Right Top (Im): 1.333333333
      Right Bottom (Re): 1.333333333
      Right Bottom (Im): -1.333333333

      Formula Tab (Layer Properties):
      Formula Type: Indra’s Promise (Public Formula file reb.ufm)
      Drawing Method: Guessing
      Periodicity Checking: Off
      Additional Precision: 10
      Maximum Iterations: 511
      Adjust Automatically: Checked
      Calc as slope: Checked
      Max Iters: 100
      Smallest Circle: 0.001
      Rad multiplier: 1.0
      Max circle size: 0.6
      Min display level: 0
      Group Type: Double Cusp
      Cusp: 1/15
      Cusp view: Alternate
      Height Transfer: linear
      Height Pre-Scale: 0.4
      Height Post-Scale: 0.025
      Fill Type: quartic 4
      Power: 2
      alpha: 1
      beta: 1
      gamma: 1

      Outside Tab (coloring algorithms; Layer Properties)
      Coloring algorithm type: Kleinian Group Raytrace (Public File: reb.ucl)
      Color Density: 1
      Transfer Function: Log
      Gradient Offset: 0
      Repeat Gradient: Checked
      solid background?: Checked
      Apply Mapping: Checked
      Group: Double Cusp
      Cusp: 1/15
      Cusp view: Alternate
      Max Iters: 100
      Smallest Circle: 0.001
      Smallest View Circle: 0.001
      Largest Circle: 0.6
      Floor Settings: ‘Add floor’ Unchecked
      All other settings as-is


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