One of the recent questions from an example sheet in Metric and Topological Spaces gave rise to the following more interesting general question:

What is the greatest number $N_d(\theta)$ of unit vectors in $\mathbb{R}^d$ such that the angle between any two is greater than or equal to angle $\theta$?

I don’t want to ruin this question by posting a solution, or even a sketch solution on this blog, but there is some very interesting behaviour revealed once you look into it. Anyone who wishes to investigate fully independently should now stop reading. I am going to post a few interesting remarks that break down the kind of behaviour the problem exhibits (without giving any proofs).

Firstly, it is almost obvious that if $\theta \geq \frac{\pi}{2}$ then $N_d(\theta) \leq 2d$. Unfortunately, the ‘obvious’ proof of this is a proof by ‘worst case scenario’ (an orthogonal set and its negation is clearly the best example), which unfortunately doesn’t really hold water. However, this does turn out to be true, and my process of finding proof gave me an interesting perspective on what ‘really spread out’ vectors look like.

Now we get a bizarre phenomenon: acute angles seem to behave frighteningly differently to right and obtuse angles. Any sort of inductive bound one tries to find seems to fail, yet coming up with reasonable explicit constructions is tricky. In fact it turns out that a sudden jump is made from there being linearly many unit vectors to their being exponentially many unit vectors. Getting the upper bound for this is quite simple. A lower bound is harder.