Which Coronas are Simple?
A tile in d-dimensional Euclidean space is a convex closed topological disk. A tiling of d-dimensional Euclidean space is a collection of tiles with disjoint interiors. The tiling is monohedral if all tiles are congruent under rigid motions of the ambient Euclidean space. The tiling is isohedral if the symmetry group acts transitively on the tiles.
The corona of a tile T is the union of all tiles that meet T, including T (this includes tiles that meet T in only a single point). For example, the corona of any tile in the usual infinite grid in the plane consists of 9 tiles in a 3 x 3 subgrid.
Conjecture: [G] In any isohedral tiling of d-dimensional space, the corona of each tile a topological disk.
All isohedral tilings of the plane are known, and by checking them the conjecture is known for d = 2. This holds even if the tile is non-convex.
Grunbaum says it may be worthwile to assume the tiling is face-to-face, that is, the intersection of any family of tiles is a face, edge, or vertex of the participating tiles. The corresponding simpler question is open, even in the 3-dimensional case. He also mentions that the Voderberg tile [1, p. 123] shows that the proof for monohedral tilings may not be completely straightforward, even in the case of the plane.
[G] B. Grunbaum, Which coronas are simple?, in `Unsolved problemsí, Amer. Math. Month.
[GS] †B. Grunbaum
and G.C. Shephard, Tilings
Submitted by: Dan Archdeacon, Dept. of Math. and Stat.,
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