Quote:
Originally Posted by Frog march
well, I wasn't sure what I was asking. 
I suppose that you could define the 8 corners of a 4D cube, with different length sticks(with the ends being the defining points)....It's going to be my entry for the Turner prize.. 
does a cube even have only 8 corners in 4D? |
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.