Tessellation

Remarks

Example: the squares $T_{a,b} = [a, a+1] \times [b, b+1]$ with $(a, b) \in \Z^2$ form a tiling of $\R^2$. Each square is closed with nonempty interior, every point $(x, y)$ lies in $T_{\floor{x}, \floor{y}}$, and two distinct open squares $(a, a+1) \times (b, b+1)$ do not meet.

Non-example: the closed unit disks centred at the points of $\Z^2$ cover $\R^2$, but the disks centred at $(0, 0)$ and $(1, 0)$ have overlapping interiors, so clause 3 fails.

Many authors also require each tile to be homeomorphic to a closed ball, so that tiles are "topological disks" in the plane. A prototile set of a tiling is a set of shapes such that each tile is congruent to one of them. In dimension $3$ and higher, tilings by polyhedra are also called honeycombs.