Tau (τ) is a mathematical constant equal to 2π. Tau is the ratio of a circle’s circumference and its radius, approximately 6.28. Actually, tau is not in popular in mathematics, the use of π is far more common. (π is the ratio of the diameter to the circumference). Here we use tau as the exponent (instead of 2) in the common Mandelbrot formula. Such use has no significant meaning other than general curiosity, but it does give us good excuse to inspect a fractal with a fractional exponent.
There has long being an argument that circumference to radius relationship (tau) is more important than π. Michael Hartl popularised the idea of using tau with his “The Tau Manifesto”, and while it is full of good arguments, I’m not sure that it will be adopted. The biggest problem is that pi is almost exclusively reserved for the constant, but tau is used in other contexts, especially physics. For example, tau is ‘proper time’ in relativity, the primary invariant of relativity. So, for now physicists are just happy to write 2π because it works fine.
Fractals with non-integer exponents have branch-cuts. Because we iterate over the function so many times, we start to find branch cuts everywhere the whole way down. Interestingly, it still seems to retain some of the properties of the original Mandelbrot, and we find lots of interesting mini-Mandelbrots.
Fractal: Mandelbrot with decimal exponent (Power=6.283185307179586)
Music: SoloAcoustic5 by https://audionautix.com (Creative Commons)
Other Fractional Power Mandelbrot Videos
Increasing the power of the Mandelbrot Set - An animation showing the exponent slowly increasing from 2 to 100.
This video's keyframes was made with Ultra Fractal 6. 52 keyframes were rendered at 4k. I merged the Key frames using Adobe After Effects, with a customised script. These fractals get very slow to render after we get beyond double precision range. If own a copy of Ultra Fractal you can explore the parameters I used below.
Ultra Fractal 6 Parameters
(Copy and paste text below)
title="Fractional - tau" width=1920 height=1080 layers=1 frames=52
caption="Background" opacity=100 precision="[email protected]#[email protected]#8160"
[email protected]#8160" magn="[email protected]#[email protected]#8160E"
maxiter="[email protected]#[email protected]#8160" filename="Standard.ufm"
entry="Mandelbrot" p_start=0/0 p_power=6.283185307179586/0
density=0.3 transfer=linear filename="Standard.ucl" entry="Smooth"
smooth=yes rotation=-177 index=0 color=0 index=68 color=6470344
index=138 color=6026495 index=161 color=2368512 index=186
color=229887 index=232 color=597041 index=-127 color=4144127
index=317 color=1966084 index=-58 color=16252927 index=376
color=11513775 index=-3 color=789516
smooth=no index=0 opacity=255
Working in Ultra Fractal:
Working in After Effects:
Some selected images from the video. All 52 original keyframes and the original mp4 video in 4k can be downloaded if you are a Patreon member.