Toronto Math Forum

MAT475-2017S => Presentations and discussions => Topic started by: Laura Campbell on March 14, 2017, 03:18:47 PM

Title: Julia Set
Post 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
Title: Julia Set
Post by: Thierry Serafin Nadeau on March 15, 2017, 06:07:38 PM
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.
Title: Re: Julia Set
Post by: Leonora Boci on March 27, 2017, 11:19:20 PM
Julia set with the parameter µ taken from the center of the circle on top of the cardioid.
Title: Re: Julia Set
Post by: kojak on April 05, 2017, 12:56:26 AM
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.