Seeing in 4D
If you were a creature who lived in a 2-dimensional universe, what would you make of a 3-dimensional object, like a cube? The visualization above gives us an idea. We could hold a light above the object and project a shadow onto our 2-dimensional world and try and glean some information that way.
Using the same idea, we can try and understand a 4-dimensional object, such as a tesseract, also known as a hypercube, by projecting it onto a shadow that fits in our 3-dimensional space. We've seen this technique before when we looked at the Hopf fibration.
This is exactly what is done in the visualization below. The difference in this case, is that we no longer see the light source, nor the 4-dimensional hypercube itself. The only thing that we perceive is its 3-dimensional shadow.
In order to better understand the object, it is very valuable to be able to examine the shadow while rotating the object in all manner of ways. But first, we need to understand what a rotation even is in 4 dimensions.
In 3 dimensions, we think about rotations as being done around an axis. In 4 dimensions, it is better to think about subplanes that are rotated. This view applies to both 3 and 4 dimensions. In 3 dimensions, a rotation around the \(z\)-axis can be thought of as a rotation of the \(xy\) plane.
One major difference regarding rotations in 4 dimensions, is that you can simultaneously rotate two sets of mutually orthogonal planes. If only one plane is rotated, it is called a single rotation, which is very similar to a 3-dimensional rotation. On the other hand, it is possible for rotations to occur simultaneously on both orthogonal planes. This kind of rotation is called a double rotation.
In general the rotation angles for the mutually orthogonal planes do not need to be the same. If they are, the rotation is called an isoclinic rotation.
The tesseract has 8 subcubes which correspond to the inner cube, the outer cube, and the six truncated pyramids at each of the 6 faces. Analogously with the 2D shadow, the inner cube is the projection of the sub-cube with a smaller \(w\) coordinate. The isoclinic and simple rotations move each of the 8 subcubes onto another subcube for each quarter rotation.
The simple rotation in the \(xy\)-plane, when projected into 3 dimensions, is recognizable as a rotation in the \(xy\)-plane. The simple rotation in the \(zw\)-plane behaves quite unusually by cycling the subcubes along the \(z\)-axis. The inner cube moves onto the top cube, which moves onto the outer cube, which moves onto the bottom cube, which moves back onto the inner cube. The isoclinic \(xy\)/\(zw\) double rotation is a simultaneous combination of these two rotations.
Right isoclinic rotations distinguish themselves from their left counterparts by the fact that one of the two orthogonal plane pairs rotates in the opposite direction. If you play around with the viz you'll see what I mean.