On 05/11/2015 20:46, Dan Asimov wrote:
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). [...] Is there a more immediate way to see that this number is -1/n ???
Dunno whether it's much easier -- it's obviously more or less equivalent -- but if you start by putting n+1 vertices at n,-1,-1,...,-1 -1,n,-1,...,-1 -1,-1,n,...,-1 ... -1,-1,...,-1,n then their centroid is obviously at 0, their norms are all sqrt(n^2+n), and their inner products are all -2n+(n-1) [at this point it's obvious, if it wasn't already, that they are the vertices of a regular n-simplex], so cos theta = (-n-1)/[n(n+1)] = -1/n. -- g