Quote:
Originally Posted by tdvance
Yes, but a hypercube has 16 corners 
OT but: Actually, there's a formula for number of r-faces in an n-cube--
r is the dimensionality of the face: 0 for corner, 1 for edge, etc., n the dimensionality of the (solid) cube, 0 for a single point, 1 for a line segment, 2 for a square, 3 for a cube, 4 for a 4-d hypercube, etc.
The number of r-faces in an n-cube is 2^(n-r) * (n choose r)
One way to think of an n-cube is the set of all n-coordinate points
(a,b,c,d,...,x) with each coordinate being between 0 and 1 inclusive.
Then, an r-dimensional face of the n cube is selected by choosing n-r of the coordinates (n choose r ways to do so, since n choose r = n choose n-r), and consider all possible ways of making them 0 or 1 (2^(n-r) ways to do so). Each choice gives a face--and the other points, taking values from 0 to 1, define the r-dimensional coordinates on that face. |
You don't even have to stop there. Dick Anderson had a distinguished career studying- the Hilbert cube, an infinite product of the unit interval, and also the infinite product of lines in which it lives.
This can also be handled with the use of the L-infinity norm (the norm of (x1, x2,..., xn) is the max of {|x1|, |x2|,..., |xn|}). Then the cube is the unit ball in this norm and faces constitute the unit sphere . Faces in this context consist of points in which one of the xi is +/- 1. This is one way to quickly see the validity of your expression for the number of faces, and easily extends to r-faces. (You probably know this, but not everyone will.)