Holey Coronas A Solution of the Grünbaum-Shephard Conjecture on Convex Isohedral Tilings

Abstract

In an article published recently in the section of Unsolved Problems of this MONTHLY, Branko Griunbaum [1] proposes the following conjecture: If S9 is an isohedral tiling of the Euclidean d-dimensional space by copies of a convex tile, then the corona of each tile is topological ball. The same conjecture is discussed as an open question in the article of Geoffrey C. Shephard [3], in which the author proves its stronger version for d = 2. We shall therefore refer to it as the Griinbaum-Shephard Conjecture. A tiling S9' is a collection of closed topological balls (disks, for d = 2) with mutually disjoint interiors (the tiles of S) such that the union of all tiles is the whole space. If all tiles are congruent, then S9 is monohedral, and their common shape is said to be the prototile of Y A tiling is isohedral provided its symmetry group acts transitively on the tiles; it is convex provided each of its tiles is convex. A lattice (of vectors) in R d is the collection of linear combinations with integer coefficients of a basis for R d. A typical example of a lattice is the collection of all vectors with integer coordinates, called the integer lattice (every lattice is affinely equivalent to the integer lattice). A monohedral tiling consisting of translates of the prototile by the vectors of a lattice is called a lattice tiling. Obviously, every lattice tiling is isohedral, while the converse is not true even in the plane. The corona of a tile T of S is the union of all tiles that meet T, including T itself. For example, in the usual (face-to-face) integer-lattice tiling of d-dimensional space by unit cubes, the corona of each tile is a cube of edge-length 3. Obviously, in an isohedral tiling, the coronas of any two tiles are congruent. See Griinbaum and Shephard [2] for a nearly encyclopedic survey of the theory of plane tilings; also see Schulte [4] and Schattschneider and Senechal [5], which deal with tilings in higher dimensional spaces. Grunbaum [1] observes that a confirmation of his conjecture for d = 2 (even without the assumption of convexity) can be established by inspection, since all types of isohedral tilings of the plane are known [2, Chapter 6]. He also conjectures that in the plane even more is true, namely that in every monohedral tiling of the plane the corona of every tile is simple (meaning simply-connected, hence homeomorphic to a disk). Shephard [3] found a non-convex counterexample, but also proved that in convex monohedral plane tilings every corona is simple [3, p. 358].