Rotating 4D Spacetime Hypercube

The movie depicts a rotating spacetime hypercube, projected from 4 dimensions on to the 2 dimensions of the screen in the same way as the rotating 4D hypercube. The prototypical example of a hypercube is the 4-dimensional geometrical object with corners at (t, x, y, z) = (±1, ±1, ±1, ±1).

A spacetime hypercube differs from an ordinary 4-dimensional spatial hypercube in that one of the dimensions, the time dimension, is, essentially, imaginary. That is, the time dimension behaves in many ways as if it were i (the square root of minus one) times a spatial dimension. How and why this comes about will becomes apparent in subsequent pages, leading up to The Spacetime Wheel.

What is a spacetime rotation? It is a Lorentz transformation, the combination of an ordinary rotation in space with a Lorentz boost by some velocity in some spatial direction. Just as a spatial rotation transforms two spatial directions amongst each other, leaving one spatial direction (the rotation axis) fixed, so also a Lorentz boost transforms time and one spatial direction, the direction of the velocity, as illustrated for example on the Centre of the Lightcone page.

In the movie, the direction of the Lorentz boost has been chosen to be the same as the direction of the spatial rotation axis. The diagram at left shows the direction of this boost/rotation axis, together with the direction of the time axis.

Updated 3 Apr 1998