From the cube to the hypercube (Schlegel)
The animation illustrates the prism with whom we get a cube from a square and a hypercube from a cube (in the very first passages of the construction we get a line segment from a point and a square from a line segment).
At every step (apart from the last one) you will be shown the Schlegel diagram of the "new" object, that means a projection of it in a space of smaller dimension, taken from a point of view which is near the centre of a face. So, the square will look like a couple of segments, one inside the other; the cube will be a couple of squares and the hypercube will be represented by a couple of cubes, one inside the other.
You can see this construction in another animation, based on the representation with parallel projection.
See also Beyond the four dimensions.
animazione di Gian Marco Todesco