# Is octonion structure related to the fact that an 8-cube has an equal number of 2-faces and 3-cells?

Background: We know that octonions exist in 8-space, and we know that in 8-space, the 8-dimensional "measure polytope" ("hypercube") just so happens to have the SAME number of 2-dimensional faces (squares) and 3-dimensional cells (cubes). (This number is 1792,) Further, Walter Nissen has stated (and proved) the relevant general property of CERTAIN n-cubes (n=2,5,8,11,...) at this link:

http://upforthecount.com/math/hypercubes.html

Question: So my question is the following. Is this property of the "8-cube" (i.e. the property of having the same number of square faces and cubic cells) related in any way to the structure of octonions and/or the permissible operations on octonions?

• Is the 8-dimensional space of opinions isomorphois with that containing the hypercube? – Oscar Lanzi Oct 31 '17 at 23:19
• Thanks for taking the time to respond, Oscar. Don't know the answer to your question, but I agree it's a good initial question to ask. The 8-cube of course picks out 2**8 points in Euclidean 8-space (the vertices of the 8-cube), so I think you're right to ask how these relate to octonions. If they don't in any way, then the answer to the question is probably "no". – David Halitsky Nov 1 '17 at 0:33
• Sorry for the mistypes I just saw. Working from a phone. – Oscar Lanzi Nov 1 '17 at 0:39
• I see no relationship whatsoever. Furthermore, the link illustrates the fact this phenomenon occurs in every dimension of the form $d=3n+2$, not just $8$. – anon Nov 1 '17 at 7:35
• Thanks for what seems to be a definitive answer, "anon". However, I do want to observed that the possibility of the relationship is NOT logically ruled out by the fact that "dimensional degeneracy" is perfectlty general, i.e. not restricted to the case of 2 and 3 within 8. – David Halitsky Nov 1 '17 at 8:29