Predicting symmetry structure of 2D "hybrid" fractals

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Offline greentexas

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« on: July 18, 2018, 04:05:37 AM »
I have noticed (when dealing with Kosmic-Stardust's fractals but this is probably true about all fractals) that they share these properties:

The points of the hybrid that are entirely real start and end at the same points of the real axis as the entirely real part of the 'real' fractal, while the points of the hybrid that are entirely imaginary start and end at the same points of the imaginary axis as the entirely imaginary part of the 'imaginary' fractal.

This can be proved because on either axis, the other axis is always zero, so the original axis takes over. For example:
A Burning Ship/Buffalo hybrid will contain the same points on the imaginary axis as the Buffalo since all of the z_reals (which are Burning Ship-related) will be zero.

The symmetry on the real axis is determined by whether the 'imaginary' fractal has symmetry on the real axis, while the symmetry on the imaginary axis is determined by whether the 'real' fractal has symmetry on the imaginary axis.

Sound confusing? Here's an example. I will predict the structure of the Cubic BS Partial Imaginary this way:

The real fractal is the Mandelbrot, while the imaginary one is the Burning Ship.

What the real axis looks like: Shares it with the Cubic Mandelbrot, so it runs from -0.38 to 0.38 or wherever.
What the imaginary axis looks like: Shares it with the Cubic BS. Since it has an imaginary needle, this fractal has a needle on the imaginary part.

Real symmetry: None, since the Cubic BS does not have symmetry around the real axis.
Imaginary symmetry: Since the Cubic Mandelbrot has symmetry on the imaginary axis, the Cubic BS Partial Imag also has symmetry on the imaginary axis.