One way of looking at these is as infinitely large (and hence totally flat) versions of the five Platonic solids and thirteen Archimedean solids.
We can also think of this as a game with bathroom tiles. However, there are two rules. First the tiles must be laid edge-to-edge, so that corners (vertices) meet up with corners. The following tiling is ruled out, because some corners meet halfway along the edges of the larger (yellow) tiles:
The second condition (“vertex transitivity”) is that all corners must be equivalent. This tiling is also ruled out, because corners where three tiles meet differ from corners where four tiles meet:
More specifically, it should be possible to map any corner in the tiling onto any other corner by a rotation, reflection, translation, or glide reflection, leaving the tiling looking identical. For this example pair of corners, either a rotation or a glide reflection does the job:
In particular, equivalence of corners on the same tile means that all the tiles must be regular convex polygons.
Exploring all possibilities for tiles meeting at a corner (we need only consider one corner, since all corners are equivalent) shows that only eleven tilings satisfy the conditions. The first three tilings use only a single kind of tile; the other eight mix different tiles. One of these eight also comes in a mirror-image form.
M. C. Escher produced some interesting non-vertex-transitive tilings, as well as some beautiful variations on the basic square and hexagonal themes: