Toronto Math Forum
MAT4752017F => Presentations and discussions => Topic started by: Laura Campbell on March 14, 2017, 03:18:47 PM

Link to a photo of a Julia set: https://commons.wikimedia.org/wiki/File:Julia_set_(Rev_formula_03).jpg

Turns out that there are 4D analogues to the 2D Julia sets which use quaternions ($i^2 = j^2 = k^2 = ijk = 1$) instead of regular complex numbers. Seeing as they're 4D they can only be visualised as 3D slices of the whole set, which end up looking quite a bit different than the regular Julia sets depending on the chosen slice.
Here's a link to a video which passes through multiple 3D slices of a quaternion Julia set: https://www.youtube.com/watch?v=VkmqT6MQoDE (https://www.youtube.com/watch?v=VkmqT6MQoDE)
And bellow is an image of such a slice.

Julia set with the parameter ยต taken from the center of the circle on top of the cardioid.

Julia Sets:
zn+1 = c sin(zn) zn+1 = c exp(zn)
zn+1 = c i cos(zn) zn+1 = c zn (1  zn)
A property of the the Julia Set is that if the domain of c is real numbers the the Julia Set it mirrored about the Real axis. If c is a complex number with an imaginary component then then the symmetry is rotational at 180 degrees.