The reason is that in connecting the two cubes we've created 6 other cubes, which connect to faces of the 2 cubes, just as taking two squares and connecting them with four edges we create four new squares which connect to the four edges of our original squares. Everything by analogy!

Finally we'll count the number of square faces in the hypercube. Each corner should now have 4 edges coming out of it, any 2 of which will define a square face. Then every corner has six square faces coming out of it (in 4-D space these squares do not intersect except along edges and at the corner). There are 16 corners. But counting 6*16=96 we've counted each square 4 times, since we've counted each square at each of its corners. Then there must be 96/4 = 24 square faces in a hypercube. Let's see, the pattern from a 2-D square, in numbers of squares, in dimensions 2, 3 and 4 goes 1,6,24. The pattern of third highest dimensional "face," starting with a square, which has four 0-dimensional parts, goes, 4, 12, 24.

(see Sloane's integer sequences A001788 and A046092.)

A solid shape in any dimensional space, made up of flat pieces of lower dimensional spaces, i.e., the higher dimensional analogs of polygons and polyhedrons, are called polytopes.

It turns out that in 4 dimensions there are 6 distinct regular polytopes, but that for dimensions n>4 there are just 3. This result required the hard work of a number of mathematicians, so don't expect to figure out how it works. But here's an exercise I leave you with, you should know how to do by now.

Describe the analog of the triangle and the tetrahedron for 4-space. Describe means list the number of corners, edges, faces, and tetrahedrons, reasoning by analogy or other computation.