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An Introduction to the fascinating patterns of Visual Math

 Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals Multi Dimensional Forms Mandelbrot Set   |  Visual Mandelbrot Guide Fractal Types & Categories PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns

Visual Guide To Patterns In The Mandelbrot Set

Regions, Patterns, Variations, Combinations, Embedded julia Sets, Number Sequences

This guide to patterns in the Mandelbrot Set is created for general fractal enthusiasts and is not a technical essay. I was inspired by other guides online (such as Mu-Ency) which are a bit too technical for non-mathematicians or don't provide visual examples for region names. I'm using KPT Fraxplorer and it does not have an option to save image coordinates. In the future I will provide numerical coordinates of where each image can be found in the complex plane .

 Spirals as we progress toward Seahorse Valley the spirals become more tightly curved (top 3 are Seahorse Valley, 2nd & 3rd row is Double Spiral Valley) Spokes top level spokes increasing in number as we progress toward Seahorse Valley top level spokes take a different shape as we progress the other way, toward Elephant valley / cusp of main cardoid On the Seahorse Valley side, top level spokes become more like bunches of flowers as we zoom into the spokes along the outer borders of the fractal Top level outer-spokes progression in the direction of Elephant Valley, as we zoom into the delicate outer borders of the fractal

 Scepter Valley Scepter Valley lies in the same orientation as Seahorse Valley but in the cleft one bulb closer to the 'needle' or tip of the set. The bulbs here are decorated with the numbered spokes just as in the previous regions, but each spoke cluster has a Scepter on the tip which repeats the main pattern of the bulb it sprouts from. Exploring the interior of these Scepters reveals many interesting pattern combinations.

 The Mandelbrot Needle at first glance it seems almost featureless but there are interesting sections in this sparse & pointy region.

 Filaments the filaments connecting the mini-mandelbrots are always connected by the cardoid cusp and the needle. Following a filament into the cusp we discover little pearls on the filaments, which progress into a cauliflower shape as we move deeper into the cusp. Inside the filament pearls there are many pattern variations and combinations.

 Amoebas, Islands, Misiurewicz Points & Geometric Nodes In any given area of the mandelbrot set it's possible to find a spot where the pattern period-doubles to infinity and forms a mini-mandelbrot. The way the shape responds to zooming into various features is somewhat constant around the set. For instance, zoom deeper into a spiral and when you get to the mini-brot there will be an equivalent extension in length of the arms spiraling into the attractor. Zoom into a Paramecium tip and then to the center of the sub-paramecium and you'll have 2 sets of 2 paramecium in the period-doubling into the attractor.

 Elephant Valley double horned spiral in period-doubling cycle, turning into twisting armed octopus.

 Deep in Seahorse Valley

 Scepter Valley Formations

 Outer Spoke Patterns

 Elephant Valley Pattern Vortexes Overlays showing underlying 8-fold geometry of the Mandelbrot Set

 Dense Carpet Patterns

 Very Deep in Elephant Valley Very Deep in Seahorse Valley

 Outer Spoke Vortex Patterns (mostly low numbers)

 More Mandelbrot 'Carpet' space-filling patterns

 An Introduction to the Fascinating Patterns of Chaos & Visual Math Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals Multi Dimensional Forms Mandelbrot Set   |  Visual Mandelbrot Guide Fractal Types & Categories PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns

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