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"Regular Production Systems and Aperiodicity in the Hyperbolic Plane"Algebra & Discrete Mathematics
|Speaker: ||Prof. Chaim Goodman-Strauss, Univ. of Arkansas|
|Location: ||693 Kerr|
|Start time: ||Thu, Feb 20 2003, 12:10PM|
We introduce the use of "regular production systems"--- a certain
generalization of symbolic substitution systems--- as a tool for analyzing
tilings in general. These systems precisely capture the combinatorial
structure of any set of tiles residing on a two-dimensional surface, though
in this talk we are particularly interested in tilings of the hyperbolic
We briefly discuss a number of applications, such as the construction of the
first known "strongly aperiodic" set of tiles in the hyperbolic plane. As
another application, we conjecture necessary and sufficient conditions under
which we may tile the sphere, hyperbolic or Euclidean plane by copies of a
given triangle, and prove the conjecture on all but a measure-zero set in
the space of all triangles. We give a new proof of Poincare's Triangle
theorem as an aside.
We also show most triangles that do tile are "weakly aperiodic"; that is,
they admit tilings, and admit tilings that are invariant under some infinite
cyclic symmetry, but do not admit tilings with a compact fundamental domain.
Decidability and rigidity play an interesting role.
Join seminar with the Geometry Topology seminar.