Editing The Road to Reality Study Notes (section)

===2.7 Relation to physical space===
Penrose ends chapter 2 with a short discussion on the applicability of hyperbolic geometry to physical space.  He notes that the geometry works perfectly well in higher dimensions and asks the question about our universe on cosmological scale.  Is it Euclidean? Hyperbolic?

There are three types of geometry that would satisfy isotropy and homogeneity: Euclidean, hyperbolic, and elliptic.  Einstein’s general relativity shows that Euclidean, although very accurate, is only an approximation of the actual geometry.  We do not yet know the answer to the above question.  There is evidence and support for each of the three geometries, but Penrose is transparent in his preference of the hyperbolic argument.

Penrose states that ‘fortunately for those…who are attracted to the beauties of hyperbolic geometry’, the space of velocities, according to modern relativity theory, is certainly a three-dimensional hyperbolic geometry rather than Euclidean.
  
He ends the chapter by stating that the Pythagorean theorem remains vital.  The [https://en.wikipedia.org/wiki/Riemannian_geometry ‘Riemannian’ geometries] that generalize hyperbolic geometry depend on the theorem in the limit of small distances, even though it is superseded for ‘large’ distances.