# The geometry of Penrose tilings: projection and renormalization.

#### by Dr Jeroen Lamb

Institution: Imperial College
Date: Wed 27th July 2005
Time: 11:00 AM
Location: Russell Love Theatre, Richard Berry Building

Abstract: In the early 1970's, R. Penrose constructed a set of two
tiles that can tile the plane only nonperiodically. His proof uses a
renormalization argument based on the existence of substitution rules. In
1981, N. de Bruijn showed that Penrose tiling also can be obtained by
projection of a discrete plane in R5 (with vertices in Z5) to the nearest
two-dimensional hyperplane. In this talk we show that projection tilings
of this type admit (a countable infinity of different) substitution rules
if and only if there exists a "quadratic" hyperbolic lattice automorphism
that fixes the projection hyperspace. As the latter condition is very easy
to verify, we obtain a simple characterization (and many new examples) of
such renormalizable projection tilings. This is joint work with Edmund
Harriss.