Subtended Angles

We consider the following question. Suppose that d≥2 and n are fixed, and that θ_1,θ_2,…,θ_n are n specified angles. How many points do we need to place in ℝ^d to realise all of these angles? A simple degrees of freedom argument shows that m points in ℝ^2 cannot realise more than 2m-4 general angles. We give a construction to show that this bound is sharp when m≥ 5. In d dimensions the degrees of freedom argument gives an upper bound of dm-d+12-1 general angles. However, the above result does not generalise to this case; surprisingly, the bound of 2m-4 from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of 2m-3 of angles that cannot be realised by m points in any dimension.