[math-fun] What is the ratio, Kenneth?

27 Feb 2019

      Let C(n) denote the 2^n vertices of the n-cube [-1, 1]^n in R^n:

   C(n)  =  {-1, 1}^n

Let S(n) denote the n+1 vertices of the n-simplex in R^(n+1):

   S(n)  =  {e_j  |  1 <= j <= n}

where 
                                { 0,  i ≠ j
   <e_j, e_k>  =  delta(i, j) = {
                                { 1,  i = j 

(so all interpoint distances = sqrt(2)).

Question:
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For n >= 2, what is the smallest number f(n) of isometric copies of S(n) needed to cover C(n) ???

E.g., f(2) = 2; f(3) = 2, f(4) = ???

Question:
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What is a nice function asymptotic to f(n) as n —> oo ???

—Dan

[math-fun] What is the ratio, Kenneth?

Dan Asimov