I don’t think so? If you have eg 8 vectors arranged evenly in a 2D plane (so at 45 degrees to each other) there’s a lot of orthogonality, but no tegum product. I think the key weirdness of a tegum product is that it’s a partition, where every pair in different bits of the partition is orthogonal. I could totally imagine that eg the best way to fit 2n vectors is n dimensional space is two sets of n orthogonal vectors, but at some arbitrary angle to each other.
I can believe that tegum products are the right way to maximise the number of orthogonal pairs, though that still feels a bit weird to me. (technically, I think that the optimal way to fit kn vectors in R^n is to have n orthogonal directions and k vectors along each direction, maybe with different magnitudes—which is a tegum product. It forming 2D-3D subspaces feels odd though).
I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
I don’t think so? If you have eg 8 vectors arranged evenly in a 2D plane (so at 45 degrees to each other) there’s a lot of orthogonality, but no tegum product. I think the key weirdness of a tegum product is that it’s a partition, where every pair in different bits of the partition is orthogonal. I could totally imagine that eg the best way to fit 2n vectors is n dimensional space is two sets of n orthogonal vectors, but at some arbitrary angle to each other.
I can believe that tegum products are the right way to maximise the number of orthogonal pairs, though that still feels a bit weird to me. (technically, I think that the optimal way to fit kn vectors in R^n is to have n orthogonal directions and k vectors along each direction, maybe with different magnitudes—which is a tegum product. It forming 2D-3D subspaces feels odd though).
Oh yes you’re totally right.
I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
Yeah, agreed that’s not an optimal arrangement, that was just a proof of concept for ’non tegum things can get a lot of orthogonality