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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?

David Halitsky
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  • 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 01 '17 at 00:33
  • Sorry for the mistypes I just saw. Working from a phone. – Oscar Lanzi Nov 01 '17 at 00:39
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    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 01 '17 at 07:35
  • 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 01 '17 at 08:29
  • On the other hand, "anon", this link suggests the relationship MAY be there: http://tony5m17h.net/Weyl.html In particular, see this passage: •The128 vertices of the odd 8-dimensional half-HyperCube with graded structure 8 56 56 8 correspond to 128 of the 240 E8 root vectors as follows: ◦8 to the 8 octonion vector space basis elements, with positive sign; ◦8 to the 8 octonion vector space basis elements, with negative sign; ◦56+56 = 112 to the 112 vectors with non-zero components on the octonion real axis; – David Halitsky Nov 01 '17 at 14:31
  • It is important to note that the "6.4" degeneracy we're discussing here (same number of 3-cells and 2-faces in the 8-cube MAY be related to the "6,4" degeneracy which allows E6 to be "folded" into F4. I say this because modern work tends to see E6 as a subgroup of E8. – David Halitsky Nov 18 '17 at 01:25

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