Department of Mathematics

Combinatorics and Graph Theory

  •  James Propp, University of Massachusetts, Lowell
  •  A Pentagonal Number Theorem for Tribone Tilings
  •  11/02/2022
  •  3:00 PM - 3:50 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Bruce E Sagan (

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.



Department of Mathematics
Michigan State University
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