### Hypercubes within Hypercubes

```                    Hypercubes within Hypercubes

An extremely simple expression characterizing an aspect of the structure
of hypercubes of  1/3  of all dimensions, namely,  2, 5, 8, 11, ..., 26,
... .

Let a 0-face be a vertex, a 1-face be an edge, a 2-face be a face,
a 3-face be a cube, a 4-face be a hypercube or tesseract, etc.
Obviously, an f-face is an f-dimensional hypercube.
Let  d  =  3n + 2 .
Let  c(f)  be the number of f-faces of a d-dimensional hypercube.
Now  c(n) = c(n+1) .
E.g., the number of vertices of a square equals the number of edges.

-

Generalizing, let  c(d,f)  be the number of f-faces of a d-dimensional
hypercube ( or d-dimensional hyper-rectangular parallelepiped ).
Let  d_C_f  be the number of combinations of  d  things taken  f  at a
time.

Now,  c(d,f)  =  d_C_f * 2^(d-f) .  Since

d - n
----- = 2                                                     (*)
n + 1

and

2^(d-(n+1))   1
----------- = - ,
2^(d-n)     2

thus

d_C_n * 2^(d-n)  =  d_C_(n+1) * 2^(d-(n+1)) .  Q.E.D.

(*) is equivalent to noting that in every third row of Pascal's
triangle, there are 2 consecutive entries, the second of which is twice
the first.

-

Further examples:

In a hendekeract(11), there are as many cubes as there are tesseracts,
42240.

In an icosihexeract(26), there are as many octeracts(8) as there are
eneneracts(9),  409541017600.

Walter Nissen
2004-03
```