# What is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree $$2, 3, \ldots, n$$ (i.e. a tile of an arbitrary degree) would be? Asked another way- what is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

By degree of a tile, I mean the number of other tiles which are adjacent to it, and by adjacency, I mean tiles which share a common boundary.

And I am referring to any tiling of the plane initially - though maybe it would be simpler to look at a regular tesselation of the plane, first. Then, would there be a finite or infinite number of ways of dividing the plane to acquire tiles of degree $$2, 3, \ldots, n$$? A related question could be, to what extent does the type of tiling affect my question? Presumably the degree of a tile and the symmetry group must be interconnected?

• Do you have a definition for the degree of a tile? Nov 10, 2013 at 13:35
• What kind of tiling are you talking about? Euclidean geometry? Do you know the symmetry group of the tiling? Can you rely on tiles being polygons, or being convex?
– MvG
Nov 10, 2013 at 21:32
• I'm referring to any tiling of the plane initially- I suppose, an additional question would be, to what extent does the type of tiling affect my question? Presumably the degree of a tile, and the symmetry group must be interconnected? Nov 11, 2013 at 22:13

The degree of a tile is not determined by its shape. Consider the following two tilings of $$1\times 2$$ rectangles:
One of these has degree $$4$$ at every tile, and one has degree $$6$$ at every tile.
If all tiles meet along some portion of their border, and not just at a corner, then (so long as there are finitely many shapes of tile used) the average degree will always be $$6$$; this is a consequence of Euler's formula for planar graphs.