I had heard some time ago of the tricky problem of finding the largest regular simplex in projective space P^(d-1). I.e., the largest number of lines through the origin of R^d that can all make the same angle with each other. For d = 2 the number is clearly 3. It's a bit surprising that for d = 3 the number is 6. (A fun exercise if you haven't seen it.) Also surprising that for d = 4 the number is still 6 (which I just learned today). (It's OEIS A002853 <http://oeis.org/A002853>OEIS A <http://oeis.org/A002853>002853.) It happens that P^3 is topologically the rotation group SO(3), so you'd think that maybe its symmetry would help, but apparently not. Here's a summary of the first few values (ranges for N(d) mean the exact value is not known): d 1 2 3 4 5 6 7<=d<=13 14 15 16 N(d) 1 3 6 6 10 16 28 28-30 36 40-42 d-1 0 1 2 3 4 5 6<=d-1<=12 13 14 15 The d-1 is in case you prefer to think of the problem as one of points in P^(d-1) instead of lines in R^d. Kind of surprising that you can't do any better than P^2 in P^3 !!! A little like maximizing the minimum distance for 5 points on S^2 does no better than 6 points (i.e., pi/2). —Dan