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