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Author Topic: I figured out why there are no minibrots in the HPDZ Buffalo!  (Read 407 times)
Description: There are no minibrots in the HPDZ Buffalo.
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Posts: 64

« on: August 29, 2017, 06:02:48 PM »

I remember Kalles Fraktaler wondering if there was a minibrot if he zoomed forever into the HPDZ Buffalo. I have news for you:

I can guarantee you, there are no minibrots, with the exception of certain structures on the needle. I think there's a finite number of those.

Here's why:
The formula for the HPDZ Buffalo is abs(z2 - z + z0.) The highest exponent is two, meaning the fractal technically has a power of two. But the lowest order of zn controls the Minibrots. Here's an example:

If you zoom into the fractal with the formula z = z2 + z6 + z0 (a sixth order fractal according to Kalles Fraktaler), you can start moving towards a minibrot at 10^100 zoom level. Because the order is 6, you'd expect the minibrot to be at (10100)1 + 1/(6 - 1) (or e120), but it is at (10100)1 + 1/(2 - 1) (or e200).

This (zoom level)1 + 1/(n - 1) formula formula governs the density of Minibrots in an nth order fractal. If the fractal has two or more exponents with addition or subtraction, n is the number of the lowest exponent. In the example, n = 2, since it is the lowest.

If you zoom into a fractal where n = 1.5, the Minibrot is located at (zoom level)1 + 1/(1.5 - 1), or (zoom level)3.
If you zoom into a fractal where n = 1.2, the Minibrot is located at (zoom level)1 + 1/(1.2 - 1), or (zoom level)6.
If you zoom into a fractal where n = 1.01, the Minibrot is located at (zoom level)1 + 1/(1.01 - 1), or (zoom level)101.

At n = 1 (since 1 is the lowest order of z in the HPDZ Buffalo), the Minibrot is located at (zoom level)1 + 1/(1 - 1), or (zoom level)infinity. Because the power of the zoom level is infinity, you will have to zoom infinitely.

In the same way abs(z2 - z + z0) has n = 1, if you take away the abs, n = 1, since it's the lowest order.
Fractal Senior
Posts: 1616

« Reply #1 on: October 16, 2017, 02:37:05 AM »

Nice discovery. I never even wondered if there was such a relationship for formulas with different exponents.

Kalles Fraktaler
Fractal Senior
Posts: 1458

« Reply #2 on: October 17, 2017, 10:08:48 AM »

But with a seed value <0.5, 0i> the HPDZ Buffalo do have minibrots.
Unfortunately perturbation breaks and cannot render these correctly, otherwise I would have done many such Movies sad

Want to create DEEP Mandelbrot fractals 100 times faster than the commercial programs, for FREE? One hour or one minute? Three months or one day? Try Kalles Fraktaler http://www.chillheimer.de/kallesfraktaler
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