Where is the cube?

In a former blog I showed a relation between the tetrahedron and the cube. If you try to build it from marbles, you will probably have no success. Rather you get something like this:

Stella Octangula

This is not a cube. The reason it is not, would I explain like this:

If the side of the tetrahedron is a, then the cube is ½*a*sqrt(2), which is an irrational number.

diamond.

Still there must be a rational relation, since the tetrahedron is called "cubic". The cube looks like this:
The unit cube. Scary! I should say that this cube is among the last things you discover, when looking at the model. Probably predilection to ortogonal coordinate system brought the term "cubic" to a system that seems to be something else. When you look up "chrystallography", you will most likely get to know, that it is an experimental science. It can also be handled geometrically. The spatial capacity of human mind is often limited, even when concerning professors'. The cure is building models of what you can't imagine. Hand and mind together brings the most out of it. Certainly you know that a coin on the table could be closely surrounded by six equally sized coins. But did you know that a marble in space will have 12, and this can be accomplished in more than one way? Assuming there is a fourth dimension, modelling is not very easy. So this is my problem to you: How many is it in 4D-space? Due to crisis in my economy, I can not give you a prize this time, but I will publish your name and photo.