International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 1, Pages 33-37
doi:10.1155/S0161171296000063
Tilings with the neighborhood property
1Division of Mathematics, Midwestern State University, 3410 Taft Boulevard, Wichita Falls 76308-2099, TX, USA
2Department of Mathematics, University of Nebraska, Lincoln 68588-0323, Nebraska, USA
Received 11 January 1994
Copyright © 1996 Linda S. Fosnaugh and Earl S. Kramer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.