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Author Topic:  Split-complex/Minkovski space fractal  (Read 171 times)

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Offline gerrit

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Split-complex/Minkovski space fractal
« on: January 14, 2021, 09:18:10 PM »
[...] z^2+c, except we're switching our point of view from the complex plane C to the 2D Minskovski plane or hyperbolic plane or "1+1 spacetime"(1 space coordinate and 1 time coordinate). This forms the natural setting for the so-called split-complex numbers:
x+jy where j^2 =1.
Now for complex numbers z^2 is given by
(x^2-y^2)+2ixy
but for split-complex numbers z^2 is given by
(x^2+y^2)+2jxy.
M-set (parameter space image) looks uninteresting for escaping points, but the interior has some fractal structure  (using orbit trap). Below (in normal reading order) M-set, Julia set with c on a "main road" in the "city block" in the left of the M-set, the Julia set with c on a "house" (or noisy region), almost random noise, looks smooth due to oversampling, finally Julia set with c on a "small alley" more to the left in the city block.

Hope the image attaches, it's 1.3 MB but 4000^2.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/split-complexminkovski-space-fractal/3989/

Offline gerrit

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Re: Split-complex/Minkovski space fractal
« Reply #1 on: January 15, 2021, 12:48:17 AM »
In terms of complex numbers this fractal has the iteration \( z \leftarrow z \bar{z} +(z^2-\bar{z}^2)/2 + c \).

Here's an animation of Julia sets with c going from about 0.25 to -2.0.

https://persianney.com/fractal/v/uf385an.mp4.