Mandelbrot set with variable exponent

Uses the transformation z[new] = z^x + a, iterated across a on each frame and with increasing x (from 1 to 3) across time. Rendered in FreeBASIC.

Category:

Science & Technology

Tags:

• mandelbrot
• fractal
• freebasic

[danielcdawson]

Experimentation suggests you may have used something like θ = atan(b/a) for z = a + bi. But that is only valid for a > 0; otherwise, you have to correct the result. Some libraries provide a function called "atan2" that gives a correct angle in all cases. I don't know if FreeBASIC has anything like that, but if you're interested, there's an article on Wikipedia that explains it.

[danielcdawson]

Hmm. Interesting. This is not the same as how it looks in most software; I think it's not correct. For one thing, an image with positive integer exponent x should have (x-1)-fold mirror symmetry centered on the complex plane origin.

Do you happen to know how the exponentiation is being calculated in your program? The standard way, I believe, is using de Moivre's formula et al. to calculate z^x as r^x cis xθ, where z = r cis θ, cis θ := cos θ + i sin θ.