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 .
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| 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) |
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| Spokes top level spokes increasing in number as we progress toward Seahorse Valley |
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top level spokes take a different shape as we progress the other way, toward Elephant valley / cusp of main cardoid |
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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 |
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Top level outer-spokes progression in the direction of Elephant Valley, as we zoom into the delicate outer borders of the fractal |
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Scepter Valley
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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. |
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| Exploring the interior of these Scepters reveals many interesting pattern combinations. |
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| The Mandelbrot Needle at first glance it seems almost featureless but there are interesting sections in this sparse & pointy region. |
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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. |
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| Inside the filament pearls there are many pattern variations and combinations. |
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| 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. |
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| Elephant Valley double horned spiral in period-doubling cycle, turning into twisting armed octopus. |
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| Scepter Valley Formations |
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Very Deep in Elephant Valley
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Very Deep in Seahorse Valley
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| Outer Spoke Vortex Patterns (mostly low numbers) |
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| More Mandelbrot 'Carpet' space-filling patterns |
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