28 Feb
2019
28 Feb
'19
5:06 p.m.
My usual "Arggh!" I should not post that late at night! Let me restate the problem this way: Let the 2^n n-cube vertices be given by C(n) = {0, 1}^n with distances determined in R^n, and let the (n+1) n-simplex vertices be given by a orthonormal set S(n) = {e_j in R^(n+1) | <e_j, e_k> = delta(j,k)}. Then for n >= 2, define f(n) = the smallest number of isometric copies of S(n) that cover C(n). I.e., if ∆_1, ..., ∆_r are subsets of the n-cube C(n) each congruent to S(n) whose union is C(n): C(n) = ∆_1 u ∆_2 u ... u ∆_r then f(n) = the least such r. Questions: What is f(n) and what is it asymptotically as n —> oo ??? —Dan