I was thinking about the characteristics of squares and cubes and know the following: 1D Line has:                                    2 ends 2D square has:                  04 edges        4 corners 3D cube has:            6 sides 12 edges        8 corners 4D hypercube has:  x volumes ,y sides ,z edges ,t corners What are x,y,z,t and can this be generalized to n dimentions?

I was thinking about the characteristics of squares and cubes and know the following: 1D Line has:                                       2 ends 2D square has:                     04 edges        4 corners 3D cube has:               6 sides 12 edges        8 corners 4D hypercube has:  x volumes ,y sides ,z edges ,t corners What are x,y,z,t and can this be generalized to n dimentions? Yes.   In general we have 2^(n-k) (n choose k) k-faces in an n-cube. For example: when n=4 we have  2^4 * (4 choose 0) = 16 * 1 = 16 vertices  2^3 * (4 choose 1) = 8 * 5 = 40 edges  2^2 * (4 choose 2) = 4 * 6 = 24 squares  2^1 * (4 choose 3) = 2 * 4 = 8 faces Here (n choose k) = n!/((n-k)! k!) are the binomial coefficients appearing as the coefficients of x^k in the expansion of (1+x)^n. Chris Hillman

In article < This e-mail address is being protected from spam bots, you need JavaScript enabled to view it , This e-mail address is being protected from spam bots, you need JavaScript enabled to view it (Jason Welter) writes: | | I was thinking about the characteristics of squares and cubes and know | the following: | | 1D Line has:                                      2 ends | 2D square has:            04 edges        4 corners | 3D cube has:              6 sides 12 edges        8 corners | | 4D hypercube has:  x volumes ,y sides ,z edges ,t corners | | What are x,y,z,t and can this be generalized to n dimensions? Consider an n-dimensional hypercube whose vertices are (+-1,+-1,...,+-1). Then consider the coordinates of the centers of the k-dimensional faces. For fixed k, the set of these centers is precisely the set of n-tuples with k 0's and n-k +-1's.  Counting these is easy.  Therefore:   The number of k dimensional faces of an n-dimensional hypercube is         / n   n-k        |     | 2   .         k / Hence, x=8, y=24, z=32, t=16. -Jim Ferry

I was thinking about the characteristics of squares and cubes 1D Line has:                                       2 ends 2D square has:             4 edges         4 corners 3D cube has:               6 sides 12 edges        8 corners 4D hypercube has:  x volumes ,y sides ,z edges ,t corners What are x,y,z,t and can this be generalized to n dimensions? x=8, y=24, z=32, t=16 and this can be generalized to n-dimensions. For more information see Chapter 6 en_title_d The Fourth Dimension , of my book Clifford Algebras and Spinors , CUP, LMS LNS 239, 1997, ISBN 0-5215-9916-4, http://www.cup.org/_title_s/59/0521599164.html.

I was thinking about the characteristics of squares and cubes and know the following: 1D Line has:                                   2 ends 2D square has:                 04 edges        4 corners 3D cube has:           6 sides 12 edges        8 corners 4D hypercube has:  x volumes ,y sides ,z edges ,t corners What are x,y,z,t and can this be generalized to n dimentions? Yes.   In general we have 2^(n-k) (n choose k) k-faces in an n-cube. For example: when n=4 we have 2^4 * (4 choose 0) = 16 * 1 = 16 vertices 2^3 * (4 choose 1) = 8 * 5 = 40 edges 2^2 * (4 choose 2) = 4 * 6 = 24 squares 2^1 * (4 choose 3) = 2 * 4 = 8 faces Here (n choose k) = n!/((n-k)! k!) are the binomial coefficients appearing as the coefficients of x^k in the expansion of (1+x)^n. Chris Hillman Nope - has only 32 edges

I was thinking about the characteristics of squares and cubes and know the following: 1D Line has:                                    2 ends 2D square has:                  04 edges        4 corners 3D cube has:            6 sides 12 edges        8 corners 4D hypercube has:  x volumes ,y sides ,z edges ,t corners What are x,y,z,t and can this be generalized to n dimentions? Of course. I'm using the following names: - a   corner is a 0-volume - an edge   is a 1-volume - a   side   is a 2-volume - a volume  is a 3-volume     ... and so on. In this case, the number of K-volumes of an N-dimensional hypercube is:           N-K    N V(N,K) = 2     (   )                  K                                         Enrico Talinucci