Is the Universe infinite or is it just really big?

The global geometry of the Universe is in principle as observable an attribute as local curvature. Previous studies have established that if the Universe is wrapped into a flat hypertorus, the simplest compact space, then the fundamental domain must be at least 0.4 times the diameter of the observable Universe. We derive the spectrum of temperature fluctuations for the other five compact, orientable flat spaces and show that the same bound holds for all, although prominent geometric features in sky maps can, in principle, distinguish between these spaces. Our analysis provides the first limits on any of the twisted tori and the hexagonal prisms.