Title: I figured out why there are no minibrots in the HPDZ Buffalo! Post by: greentexas 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. Title: Re: I figured out why there are no minibrots in the HPDZ Buffalo! Post by: Dinkydau 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.
Title: Re: I figured out why there are no minibrots in the HPDZ Buffalo! Post by: Kalles Fraktaler 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 :( |