Re: D=5

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's homework problem.

> Homework:  As a function of D (the number of dimensions
> in the hypercube), from D=1 to D=6, calculate the
> number of d=0 corners, the number of d=1 edges,
> the number of d=2 faces, the number of d=3 hyperfaces,
> et cetera.  Hint:  there's a relationship between rows,
> somewhat reminiscent of Pascal's triangle.  Duplicate
> and connect corresponding parts.....

The hint I like is to consider the binomial expansion of
(2 + x)^D as the generating function for the number N(D,n)
of n-dimensional hypercubes that are part of the boundary
of a D-dimensional hypercube (where 0 <= n <= D).

We have already discussed these things a couple of years
ago (16-18 JUL 2000) in this forum in a thread whose
subject was, appropriately, "Re: Hypercube".

Try seeing:
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=40968
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=42249
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=42340

See also:
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=42670
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=42761
http://lists.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&P=43076

if you are interested in the numbers of lower dimensional
parts on the boundary of a D-dimensional simplex.

David Bowman
georgetowncollege.edu