Has anyone considered a collider in space? It seems like you could have huge distances. And just let the particles fly in the vacuum between acceleration points?
So space doesn't really help: You'll still need a circle of magnets. Approximating the circle with a polygon means you split up a smaller circle with straight segments. But these straight segments are not really useful.
It's essentially a HEP-style(ish) particle detector, mounted on the ISS, to study cosmic rays.
But the basic idea in everything but linear accelerators is the same: You want to reuse the accelerating structure multiple times be recirculating the beam through it many times. Either in a ring/closed loop, like LHC, or in some form of wound up (open) path (Race track mictrotrons, for example)
Well, in any case, so much for the circumsolar collider described in the Three Body trilogy. I guess it might still be doable/useful with some hypothetical future technology, though.