>How many quanta can travel at light speed relative to each other simultaneously?

I know the concept of being "simultaneous" breaks down at a distance so let's assume for the sake of argument that all of the quanta pass within an arbitrarily close distance of each other at some "moment in time".

In classical mechanics, we can ask the slightly different question of "how many *paths* can be orthogonal to each other simultaneously?" the answer seems fairly obviously "three" because of the three dimensions we perceive.

But I'd like to understand whether there are different answers in other representations of the universe, such as quantum theory, general relativity, the standard model and supersymmetry.  What's the maximum number of quanta which can all be all travelling orthogonal to each other, and does this change with the added condition that they be travelling at light speed?