Thanks. The simplex is exactly what I was aiming for. Tom Andy Latto writes:
On Mon, Aug 30, 2010 at 10:34 PM, Tom Karzes <karzes@sonic.net> wrote:
Here's an obvious followup question: In n dimensions, is the maximum number of mutually-obtuse rays with a common origin always n+1?
Yes, my proof generalizes. If one ray passes through (0, 0, 0, ... 1), then the last coordinate of all the other rays is negative. Project them all onto the plane perpendicular to the first ray. By the inductive hypothesis, the angle between two of the projections is non-obtuse. The angle between the rays before being projected is smaller (because the final coordinate makes a positive contribution to the dot product), so is not obtuse.
Here's a suggested construction that I believe generates canonical, symmetrical solutions:
Just place a simplex with its center at the origin, and use the rays through its vertices. The symmetrical way to express this is by embedding the n-dimensional space as a hyperplane in n+1 dimensional space, specifically the hyperplane where the coordinates sum to 1. Put the center point at the point whose coordinates are all 1/(n+1), and use the rays from there to the n+1 points that have one coordinate 1 and the rest 0.
Andy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun