I guess a rigorous proof does require a lemma. Which is perhaps a bit surprising. ____________________________________________________________________ Lemma: ------ Suppose vectors v_1,...,v_n belong to some Euclidean space and have all pairwise angular separations equal to a constant, call it theta. Then the maximum such theta can be realized in R^(n-1). ____________________________________________________________________ Proof left as an exercise. —Dan
On Nov 4, 2015, at 3:37 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 04/11/2015 23:37, Dan Asimov wrote:
Though I would add that only R^3 is necessary to find the maximum common angular separation of 4 vectors.
Taking every other vertex of the cube [-1,1]^3 gets an example.
I came very close to just writing "Obviously it's -1/3 because the right picture is a regular tetrahedron" but thought that might be not quite rigorous enough :-).
(Not quite rigorous enough to be sure I wasn't completely wrong, as well as not quite rigorous enough to satisfy Dan.)
(:-)>