2

My question is exactly as above. If there are $n$-vectors having the same origin in an $m$-dimensional Euclidean space then what is the maximum angle of separation that you can achieve.

Example:

In $2D$ Given $2$ vectors the max angle of separation is $180°$. For $3$ vectors it is $120°$; for $4$ it is $90°$; and so on (here we simply divide $360$ by $n$).

In $3D$ for $4$ vectors it $109.5°$ (I know this from chemistry, the arrangement of carbon compounds) and for $6$ vectors it is $90°$.

So if we are to generalize this to $n$ vectors in $m$ dimension what's the result? Can you provide at least an approximate solution to this problem or some resources that pursue it?

KReiser
  • 65,137
Rishi rao
  • 23
  • I did find a solution to the 3D case here: https://math.stackexchange.com/questions/297209/how-to-determine-maximum-angles-between-vectors

    and few more leads here: https://math.stackexchange.com/questions/3052520/maximum-angle-between-vectors

    The first question gave an approximation in terms of the surface area of sphere and spherical cap, but I don't think this can be generalized to m dimensons.

     – Rishi rao Aug 03 '20 at 09:54
  • What do you mean by maximum angle? For 3 vectors in 2D, one of the angles between "consecutive" vectors could be $(180°,90°,90°)$, not just the regular( $\frac{360°}{n}$) case. –  Aug 03 '20 at 10:30

1 Answers1

1

This is a very difficult question in general. I will assume unit vectors throughout.

In high dimensions, it's natural to talk about inner products and restrict them. Let $\epsilon>0,$ then given dimension $m$ one might ask what is the maximum number of unit vectors $v_1,\ldots,v_n\in \mathbb{R}^m,$ so that the mutual inner product between any two distinct vectors is no larger than $\epsilon.$

It turns out, even if you restrict yourself to vectors of the form $$(\pm 1,\pm 1,\ldots,\pm 1)/\sqrt{m},$$ you can have as many as $n=c 2^{\epsilon m}$ for large $m.$

In small dimensions there are some constructions, but general solutions attaining optima are unknown.

Look up Welch lower bounds for more information about the inner product formulation. Another keyword is "spherical codes". For example, the points at the centres of the pentagons and hexagons of a soccer ball form a spherical code in 3 dimensions. How many pentagons on a soccer ball?

See https://www.quantamagazine.org/a-new-path-to-equal-angle-lines-20170411/ for some information about the case that you restrict all pairwise angles to be the same.

kodlu
  • 9,338