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I'm just starting to study tilings in a groups and geometry module, and I'd like some confirmation of my understanding of precisely what it is which differentiates a single tile, from a tiling- when it is considered as part of a tiling. Perhaps I misunderstand, but it seems that a tiling can be considered as an arbitrarily large assemblage of tiles, and also as a single tile, whose vertices and edges are determined by the other tiles that it is in contact with?

I figured it would be best to illustrate my interpretation. So I will re-edit later, with images!

Hexagonal tiling. This is a tile (or polygon), with 6 vertices, and 6 edges. This relationship is the same regardless of whether the tile is thought of in the context of a tiling or a polygon

Brick tiling. Each tile, in this case, when considered as part of a tiling, has 6 vertices, and 6 edges.

However, it doesn't seem completely clear what exactly the distinction between a tile and a tiling is? A tiling could presumably be an arbitrarily large collection of tiles, or just one?

(I'm observing the tiling in terms of the tiles which are adjacent to a single tile)

enter image description here

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A tiling is a collection of tiles, which cover the plane without gaps or overlaps. In your first example, each hexagon is a tile; the tiling consists of infinitely many hexagons. In your second example, each rectangle is a tile; the tiling consists of infinitely many rectangles.

It does not make sense to have a tiling of one tile. The tile would have to be the entire plane.

There can be a distinction, as you note, between the vertices and edges of a tile, and the vertices and edges of the tiling. They coincide in the case of the hexagonal tiling, so there's no issue there. But in your second example, the vertices of each rectangle are its four corners, and the edges are its four edges. However, the vertices of the tiling are all the intersections of tiles which are points, and as you mention there are 6 of these on the boundary of each tile. The edges of the tiling are the line segments where two tiles intersect, and again there are 6 on the boundary of each tile. To prevent this confusion, in the book Tilings and Patterns (the reference for plane tilings), Grünbaum and Shephard refer to the corners and sides of the polygons, and vertices and edges of the tiling.

So in your second example, we'd say that each rectangle tile has four corners and four sides, but has six vertices and six edges on its boundary.

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