When I found this nice nonagon and star tiling, I couldn't get They Might Be Giants out of my head. The underlying symmetry comes from hexagons though:
The relationship described in the previous post tells us that the next regular polygon that can be put around a hexagon like this is the dodecagon (see the line k = 6):
And sure enough, these dodecagons can also form a nice hexagon-based tiling, with somewhat pointier stars.
The k = 6 line above gives us a whole family of polygon and star hexagonal tilings that follow the relationship n = 3m + 3 - the stars don't start to appear until m = 2
