Yup. Maybe there is an easier way to see what the cosine of the angle between any two vectors from the center to a vertex of a regular n-simplex is, but the way I know is this: __________________________________________________________________________________ The n+1 standard basis vectors in R^(n+1) form the vertices of a regular n-simplex whose center lies at c = (1/(n+1), ..., 1/(n+1)) in R^(n+1). So, two vectors in R^(n+1) forming the angle in question are v = e_1 - c = (1-1/(n+1), -1/(n+1), -1/(n+1),..., -1/(n+1)) and w = e_2 - c = (-1/(n+1), 1-1/(n+1), -1/(n+1),..., -1/(n+1)) and so <v, w> / (||v|| ||w||) = (-1/(n+1)) / ((1-1/(n+1))^2 + n/(n+1)^2) = -1/n. __________________________________________________________________________________ Question: --------- Is there a more immediate way to see that this number is -1/n ??? —Dan
On Nov 5, 2015, at 12:10 PM, Warren D Smith <warren.wds@gmail.com> wrote:
The optimal solution was unit vectors that are the N+1 vertices of a regular simplex in N-dimensional space centered at the origin. (N=3 for Asimov's problem, regular tetrahedron.)
These are anticorrelated somewhat, dot product of any two is -1/N minimizing correlation by making it maximally negative. My solution with 4 mutually orthogonal 8-vectors was optimal if we want zero correlation, which minimizes correlation if you meant the absolute value of the correlation...