Tilings with the neighborhood property and tiling lattice points with blocks

We examine tilings of the plane (plane tilings) and of 3-space that have the neighborhood property (NEBP). If N(T), the neighborhood of T, is the set of all tiles that have a nonempty intersection with T, then a tiling has NEBP if for every tile T$\sb1$ there is a tile T$\sb2$ such that N(T$\sb1$) = N(T$\sb2$). All of our tiles will be (closed) topological disks. If all the tiles are convex, then the tiling is called a convex tiling. We prove that a convex plane tiling with NEBP has only triangular tiles and each tile has a 3-valent vertex. Removing 3-valent vertices and their incident edges from such a tiling yields an edge-to-edge planar triangulation. Conversely, given any edge-to-edge planar triangulation, we can construct a convex plane tiling with NEBP. It will follow easily that a monohedral convex plane tiling with NEBP is unique. We could classify all convex plane tilings with NEBP using two prototiles if all the plane tilings by isosceles triangles were classified. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We discuss tilings of $\IR\sp3$ with NEBP and exhibit a monohedral tetrahedral tiling of $\IR\sp3$ with NEBP. In the second chapter, we investigate tilings of $\doubz\sp{\rm n}$ with blocks, where a block is a finite subset of $\doubz\sp{\rm n}$. Given a finite set ${\cal B}$ of blocks in $\doubz$, we discuss a procedure that leads to an algorithm to determine if ${\cal B}$ tiles $\doubz$. We introduce a tiling automaton A(${\cal B}$) and give a correspondence between two-way infinite walks in A(${\cal B}$) and tilings of $\doubz$ with ${\cal B}$. We also investigate block tilings of $\doubz\sp{\rm n}$ by stacks . Stacking is a technique that sometimes works to reduce a tiling problem in $\doubz\sp{\rm n}$ to a problem in a lower dimensional space. In particular, stacking may reduce a problem in $\doubz\sp2$ to a tiling problem in $\doubz$ which can be easily solved