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?