Branch Busters: Work arounds for those pesky branch lines

[kjknohw]

Non-integer powers are a pain. For every number there are two square roots because both 1 and -1 square to 1. Trying to make the square-root continuos on the complex plane is like trying to paint one side of a Mobius strip red and the other green. What if you make a tree by taking both "branches" at once each iteration and color by the one that takes the longest to escape? You get this:
http://www.righto.com/2011/12/new-multi-branch-algorithm-to-render.html
I took this formula and adapted it for all rationals, it is included in the pack below. The calculation time isn't bad on the periphery because all but a select few choices fly off quickly and can be pruned. No pretty pics for you, most regions are unexplorable because the number of branches explodes exponentially, unless you come up with an optimization  .

But what about arbitrary powers? Any exponents can be rewritten in terms of logarithms (a^b = exp(b*log(a))). You can avoid the log(z) branch line by forcing the imaginary part to be positive. This introduces distortion (even for real, integer powers), however it gets rid of branches for ANY complex power. Although it looks ugly at first, zoom an and you can get interesting shapes:


More images:
http://www.developmentserver.com/kevin/2014_mandelPower/special_log/special%20log.html

Here is the formula pack (includes Julias):
http://www.developmentserver.com/kevin/2014_mandelPower/branch%20busters.ufm
To download use the "save link as" option instead of directly clicking.

[laser blaster]

Wow, nice work! The first idea is very clever, and the second one is especially gorgeous. It looks like it contains a staggering amount of variety.

[kjknohw]

The first idea is not mine, and the second is obvious so other people must have tried it. However, it looks ugly at first glance so you tend to avoid it. But, like the burning ship, it is (mostly) explorable. When you are exploring, distortion is easier to avoid/mitigate than branch lines.