I'm reading M. Omar Ali's Elementary Solid State Physics and in it, in Subsection 1.4 The Fourteen Bravais Lattices and the Seven Crystal Systems he says that "..., but one cannot place many such pentagons side by side so that they fit tightly and cover the whole area. In fact, it can be demonstrated that the requirement of translation symmetry in two dimensions restricts the number of possible lattices to only five (see the problem section at the end of this chapter)." However, the problem section does not explain it either. So, in simple terms, a Bravais lattice is just a mathematical way to describe all solid single-crystal structures. It is an idealization that depends on being able to describe every point in terms of appropriately scaled (by integers!) basis vectors. In 2D, there should be just two vectors because two noncolinear basis vectors are all that is needed to span two dimensions. The fact that there must be translational symmetry given by integer multiples of the basis vectors greatly reduces the total number of possible 2D Bravais lattices. But, how do you get 5? And in 3D, how do you get 14? I think starting with 2D for now makes more sense as I can't quickly sketch a rough proof in my head. Thanks, as always, for the help.