Dynamical Systems Seminar- On Penrose tilings and tiling dynamical systems: I

 

Abstract: Penrose tilings were discovered around 1976 by Sir Roger Penrose as he was trying to see how close he could come to tiling the plane by regular pentagons. Penrose tilings are aperiodic tilings that are nevertheless in some sense almost periodic, and they inherit a pentagonal pseudo-symmetry from Penrose’s pentagons.  Penrose tilings were popularized by Martin Gardner, who reported on some work on them by John H. Conway. The big advance in understanding these remarkable tilings, however, came with N. G. de Bruijn’s 1981 papers “Algebraic Theory of non-periodic tilings of the plane I & II”, which showed how to interpret Penrose tilings as a 2-dimensional slice through 5-dimensional space. The mural across from the math office is a piece of Penrose tiling. 

 

In the late 1980’s Penrose tilings were proposed by the U. Penn physicists Levine and Steinhardt as a model for a newly discovered state of matter called quasicrystals. Like quasicrystals, Penrose tilings can have a 5-fold rotational symmetry that is forbidden for ordinary crystals. I learned about Penrose tilings as a postdoc at Penn, and realized many of the ideas in the theory have a dynamical systems interpretation. My 1996 Transactions paper “The dynamical properties of Penrose tilings” showed how to use de Briujn's structure theorem to model Penrose dynamics as a total rotation action.

 

In this talk (and the next few weeks in the seminar) I will present the work of Penrose and de Bruijn, as well as my own work on Penrose dynamics. These talks are based on a series of lectures I gave about 20 years ago at Tsuda College in Tokyo. But work on aperiodic tilings, their dynamics and the relation to quasicrystals continues to be active research area under the title of “aperiodic order”.

 

Once we finish Penrose tilings in a few weeks, the seminar will revert to the subject of the year: Dynamics on surfaces (I will even tell you how to fit Penrose tilings into this context).  We will start working through the paper The conformal geometry of billiards, Laura de Marco, BAMS 48, 2011, http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01322-7/home.html As before, the seminar will include talks by participants as well as several outside speakers.