Is it a Euclidean space, but without distances and angles; or without any functions that transform distances or angles, neither being defined? So there's no inner product, but apparently in an affine space, parallel lines still have the same 'structure' as in Euclidean space. So parallelograms are defined and there are volume-preserving functions. Distance isn't defined, but affine distance is (how, though?). What affine distance actually means, mathematically, is an object which is "like distance", but has nowhere to define a unit of distance. Or should I be saying, correct me if I'm wrong about this, but . . .