Quaternion Julia Fractals

What exactly is a quaternion Julia set? Well, it’s beautiful.

These shapes are animated projections of three dimensional slices of four dimensional objects known as quaternion Julia sets. The definition of a Julia set can get a bit complicated, but it can be thought of as an object that carves up four-dimensional space into two categories – belonging to the set, and not belonging to the set. How exactly the shape is carved depends on some very deep mathematics.

Now the big question – how do we look at a four dimensional object if we’re just mere three dimensional humans? Well, first let’s try to describe how we can look at a three dimensional object using only two dimensions.

When I think of two dimensions, I think of a flat sheet like a piece of cardboard. How could we use this flat sheet, or a lot of flat sheets, to make up a three dimensional object? Well, if we were very clever like Yuk King Tan, we could cut a huge number of cardboard sheets carefully and stack them up on top of each other. From far away it would look like a three dimensional object.

But if we look closely.

Very closely.

We can see that this is made up entirely of two dimensional objects cut into specific shapes, each shape cut perfectly to reflect the three dimensional object at a certain height. This is just like how an MRI machine takes “slices” of a three dimensional object (a human!) as it slowly moves upwards. The image below shows the 2D slices of the 3D skull starting just below the eyes.

If we could only see two dimensions, we could flip through each one of these images in turn to get an idea of just what a three dimensional brain looks like. This is what doctors do – all of our current display technology, fancy HDTVs included, currently only display two dimensions. So they take many two dimensional slices and then compare and visualize them in relation to each other, in order to get some idea of what our three dimensional body is actually like.

So we can do the same thing with these four dimensional Julia sets. We can take many three dimensional slices, animate them, and then compare and relate these slices to each other in order to create some idea in our brain of just what this four dimensional structure is.