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

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  • $\begingroup$ Do you have a definition for the degree of a tile? $\endgroup$
    – Dan Rust
    Nov 10, 2013 at 13:35
  • $\begingroup$ 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? $\endgroup$
    – MvG
    Nov 10, 2013 at 21:32
  • $\begingroup$ 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? $\endgroup$
    – Seraphina
    Nov 11, 2013 at 22:13

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The degree of a tile is not determined by its shape. Consider the following two tilings of $1\times 2$ rectangles:

enter image description here

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.

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