** 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.

**References: **

[G] B. Grunbaum, Which coronas are simple?, in `Unsolved problems’, *Amer. Math. Month.*

[GS] B. Grunbaum
and G.C. Shephard, Tilings
and patterns,

Submitted by: Dan Archdeacon, Dept. of Math. and Stat.,

Send comments to *dan.archdeacon@uvm.edu*

November, 2003