Félix Abecassis

The project

During the last school year we had numerous maths courses: theory of distributions, model theory, Lie algebra... For this last course, we were given several projects idea by our teacher, we have chosen to make a 3D representation of the special linear group and of the Julia set.
An online representation of the Julia set can be found here.
You can try some presetted values that yielded nice figures or you can mess around with the different parameters (explained in the following section).

Warning: I recommend using a top notch web browser, if everything is black or you have only a 3D axis, try again with another browser.

This project was my first experiment with Javascript and WebGL using ThreeJS
Using JS was not a requirement for the project, but we thought it would be a good idea to have an online demo !

Explanations

Julia sets are fractals defined by the recurrence relation , with being a constant.
If, for a given , converges, then is in the Julia set.

Julia sets can be represented in 2D (as shown on wikipedia) with ) or in 4D with .
is the set of quaternions, quaternions are hypercomplex numbers of the following form:

With , , being roots of unity, i.e. .

To obtain a representation in a 4D space, each point of this space is represented by a quaternion.
Given a fixed value of , we can compute the corresponding Julia set by testing the convergence of for all points of this 4D space.
In our online demo, the 4 coefficients of the quaternion constant can be modified through the sliders , , and .

We cannot visualize 4D objects, so we need a 3D representation of the Julia set. We have to take a slice of the Julia set, in other words we compute the intersection of the set with an hyperplane.
What this means for our demo is that we assign a constant value to the same coefficient for each quaternion (for example the coefficient associated with ).
The three remaining coefficients will form the three axis of our 3D representation (, et ).
In our online demo we can choose which coefficient should be fixed and modify its value with the slider Hyperplane.

Once we have fixed a coefficient for all quaternions, we iterate through all possible values of , and .
For instance if we have chosen that , the initial value of the sequence is therefore:

Point is in the Julia set (and hence displayed as a white dot) if, after a given number of iterations, the sequence is below a threshold. In our demo we compute 10 iterations.