Tilings with the neighborhood property

Abstract

The neighborhood N(T) of a tile T is the set of all tiles which meet T in at least one point. If for each tile T there is a different tile T1 such that N(T) = N(T1) then we say the tiling has the neighborhood property (NEBP). Grünbaum and Shepard conjecture that it is impossible to have a monohedral tiling of the plane such that every tile T has two different tiles T1, T2 with N(T) = N(T1) = N(T2). If all tiles are convex we show this conjecture is true by characterizing the convex plane tilings with NEBP. More precisely 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 the incident edges from such a tiling yields an edge‐to‐edge planar triangulation. Conversely, given any edge‐to‐edge planar triangulation followed by insertion of a vertex and three edges that triangulate each triangle yields a convex plane tiling with NEBP. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We briefly discuss tilings of R3 with NEBP and exhibit a monohedral tetrahedral tiling of R3 with NEBP.