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Author Topic: All about the Quasi Mandelbrot - Multibrot and variations  (Read 697 times)
Description: All about the Quasi Mandelbrot - Multibrot and other formulas
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Posts: 64

« on: November 08, 2016, 01:49:42 AM »

This is a simple variation of the Mandelbrot set that produces strange results with a non-integer power:

(-z)n + pixel,

where n is any real, imaginary, or complex number.

If n is 2.05, you will see a split towards the needle part of the M-set, but you will also see an outgrowth near the Elephant Valley. The normal 2.05 power Mandelbrot does not have this outgrowth.

If you keep increasing the power, the needle will split into two, and the outgrowth will become the other half of the Cubic Mandelbrot. The cubic Mandelbrot is on its side due to a property of real numbers:

3^3 = 27

(-3)^3 = -27

and partial powers work the same way.

0.2^2.5 ≈ 0.0179

(-0.2)^2.5 ≈ 0.0179i

Even powered Mandelbrots of the form n = 4a + 2, where a is an integer are an exception.

(-0.9)^6 ≈ 0.5314

0.9^6 ≈ 0.5314

The fourth power Mandelbrot, eighth power Mandelbrot, and other multiples of 4, are another exception. The needle on these fractals is on the right, and not on the left.

The Burning Ship has odd behavior when it is given a partial power formula: |(-z)n + pixel| is its formula.

Instead of a "leg" forming under the Burning Ship, a structure arises out of the needle, and the mast of the Burning Ship tilts right and splits. If the power is exactly 21/3, the Celtic is formed. Up until I learned the formula for the Celtic, I assumed this was the formula for the Celtic. Dinkydau's video even has the Celtic fractal oriented the same way.

If the power is 12/3, you will see the Celtic Mandelbar. This is amazing because this gives an intricate relationship (no pun intended) between the Burning Ship and Celtic.

I do not have much to say about other fractals, except for that the Perpendicular Burning Ship gains other strange structures unlike its own higher powers, and that this formula destroys the balloon shape of the power 4 Simonbrot.
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