Park City Mathematics Institute
Modeling Materials Discretely
Project Abstract

Jim King, Cristina Cardone, Jennifer Chen, Miriam Cukier, Leah Jackman, Dayna Soares

Our working group has attended the Undergraduate Summer School course on Modeling Materials Discretely, given by Marjorie Senechal of Smith College. In addition, we spend an hour after each lecture debriefing concepts and working on problem sets with Jim King, SSTP staff.

The goal of the course is to explore aperiodic structures, both mathematical and in the physical world of crystals. The first week was devoted to polyhedra (including symmetries and visualization by Schlegel diagrams) and the lattice paradigm for crystallography, leading up to x-ray diffraction.

Then the course made a connection to the SSTP math course when the topic became Penrose tiling – though with a different flavor, using rhombs instead of the equivalent kites and darts!

Next, we moved on to on to higher dimensions. Including a special Zome building session with Jean Taylor of the Courant Institute at NYU. The Zome models were used to produce the big model of an Yt-Cd quasi-crystal, with its atoms in the right places.

The third week continued with a more detailed exploration of the nature of aperiodicity, from one dimension to three, noting that some of the shapes in two and three dimensions can be produced as projections from periodic tessellations in higher dimensions.

The course included a lot of ideas in a short time, but there were many problems and ideas that were very rich and rewarding to work on in our second hour. And the considerable number of connections with the SSTP math course added another layer of interest.

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IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America

This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.