Conway and Lagarias used combinatorial group theory to show that certain
roughly triangular regions in the hexagonal grid cannot be tiled by the
shapes Thurston later dubbed tribones. The ideas of Conway, Lagarias, and
Thurston have found many applications in the study of tilings in the plane.
Today I'll discuss a two-parameter family of roughly hexagonal regions in
the hexagonal grid I call benzels. A variant of Gauss’ shoelace formula
allows one to compute the signed area (aka algebraic area) enclosed by a
closed polygonal path, and by “twisting” the formula one can compute the
values of the Conway-Lagarias invariant for all benzels. It emerges that the
(a,b)-benzel can be tiled by tribones if and only if a and b are the paired
pentagonal numbers k(3k+1)/2, k(3k-1)/2. This is joint work with Jesse Kim.