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?
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.