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I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree 2, 3,..., 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, ..., 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?

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Do you have a definition for the degree of a tile? –  Daniel Rust Nov 10 '13 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 '13 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? –  Seraphina Nov 11 '13 at 22:13

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