A "direction" is a 3-vector (x,y,z) obeying either x>0, or x=0 and y>0, or x=y=0 and z>0, and x^2+y^2+z^2=1. The point of that is (a) to forbid 0,0,0 and (b) I want to regard negative directions and rescaled directions as same thing as original directions, so only allow one. Let f(N) be the maximum possible number of (unordered) triples of mutually-orthogonal directions in a set of N distinct directions. What can be said about f(N)? Here is a table of lower bounds L and upper bounds U on f(N) for small N. Which probably are both pretty weak. N=3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 L=1 1 2 2 3 3 4 5 5 6 6 7 8 9 9 10 11 11 12 13 14 14 15 15 16 17 18 18 19 19 U=1 1 2 4 7 8 12 13 17 20 26 28 35 37 44 48 57 60 70 73 83 88 100 104 117 121 -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)