Since each quaternion can be described using four numbers, I unfolded these four dimensional quaternion Julia sets into three dimensional space, and animated the final coefficient.

But once I did that I noticed some radial symmetry along the y-z plane – it looks like something that’s been made on a lathe. This means that we can “index” all these shapes in a more sensible manner by collapsing things along this axis of symmetry. While previously we could index all of our shapes with four coefficients a, b, c, and d.

We can now index them with four coefficients a, r, theta, and d after this transformation. But there’s a nice side effect now that our coordinate system reflects our symmetry – if we vary theta, the appearance of the Julia set doesn’t change, the object just appears to rotate about the a axis.

So really we can index all possible shapes using only three coefficients – a, r, and d. This is awesome – it means we can use this symmetry to collapse a dimension and completely illustrate a discrete approximation of this four dimensional set in three dimensional space. The following images (click for 1080p full resolution images) illustrate the full set of these possible shapes – a is the horizontal axis, r is the vertical axis, and values iterate by 0.25. The grey sphere in the first image is the origin, and the images start at a d value of 0 and iterate upward by 0.25. We find that there exists additional symmetry with our d parameter – namely that d = -d, so we only need to illustrate the absolute value to see all shapes.

d = 0.00

d = 0.25

d = 0.50

d = 0.75

d = 1.00

When d = 1.25 there are only a few bits of unconnected dust loops visible. This analysis only covers a single “slice” – namely the plane normal to (0,0,0,1). I’d be very interested to see if there are any other symmetries…

There is a four dimensional Julia set for every four dimensional quaternion. We can label each quaternion using four numbers.

I decided to “unfold” the first two values of the quaternion onto a plane and animate the last two values. The camera is centered at (0,0) and Julia sets are placed at intervals of 0.1 off to infinity for both axes.

You can start to see the larger structure present more clearly. Perhaps a three dimensional unfolding next?

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.

Here’s some of the best free high-definition content I’ve found on the web, if you know of any more please let me know!

Gravitas

Gravitas is a project by John Dubinski of the Department of Astronomy & Astrophysics at the University of Toronto. He works on visualization of galaxy dynamics, and his goal is to “use supercomputer simulations of realistic model galaxies to illustrate these slow and majestic dynamical processes on an accessible timescale and so breathe life into the iconic images of galaxies created by the world’s great telescopes”. He succeeds brilliantly, and has produced a set of captivating animations, some in HD.

Download Future Sky (Quicktime 720p) and Spiral Metamorphosis (Quicktime 1080p).

Fractal Zooms

Eric Bigas has a great website with several fractal animations, including a few in HD.

Cherry Blossom Hexagons is a zoom into a Barnsley fractal, available in 720p XviD or 720p H.264.

19th Hole Terraces is a zoom into a Mandlebrot set, available in 720p XviD or 720p H.264.

Copperplate Chevrons is available in 720p XviD.

Hubble Space Telescope

The European Homepage for the NASA/ESA Hubble Space Telescope has tons of great HD content. Dr. Joe Liske at the European Organization for Astronomical Research in the Southern Hemisphere hosts a video podcast which you can subscribe to in 720p or full HD 1080p. They also have a HD video archive of broadcast quality footage, like this flythrough of the Hubble Ultra Deep Field.

For other HD space videos, NASA has a HD video archive with a section dedicated to Hubble. The Jet Propulsion Laboratory has an HD archive of their own, click “HD” at the bottom to browse.