For the standard sphere in D dimensions. Among the 2^D directions (+-1, +-1, +-1, ..., +-1) choose a D^2-element subset forming a "Kerdock code." One exists if D>1 happens to be a power of 4. Also choose these 2D additional directions: the positive & negative coordinate axes. What is so amazing about this set of points on the sphere? 1. The number of points is N=(D+2)*D. 2. The minimum angular separation between points is theta where sec(theta)=sqrt(D). 3. Each point has exactly (D^2)/2 nearest neighbors (at angle exactly theta to it). 4. The points form a 5-design, i.e. averaging F(x) on this point set yields the same result as averaging F over the whole sphere surface, whenever F is a polynomial of degree<=5. 5. A theorem of Delsarte-Goethals-Seidel states that any 5-design must have at least (D+1)*D points, hence ours is optimal up to a subleading term. (I suspect it is optimal, period, and also that it is "rigid.") This nodecount also is optimal up to subleading term for all numerical integration rules (with unequal weights and points off the sphere surface, and perhaps even complex, allowed). This is one of the (nicer & conciser) results of mine in a paper I'm working on about numerical integration... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)