Aperiodic Tilings: The Relationship between Spectrum and "Bubbles"

Presenter Information

Ashely Rand, Minnesota State University, Mankato

Location

CSU 284

Start Date

21-4-2008 8:00 AM

End Date

21-4-2008 10:00 AM

Student's Major

Mathematics and Statistics

Student's College

Science, Engineering and Technology

Mentor's Name

Brian Martensen

Mentor's Department

Mathematics and Statistics

Mentor's College

Science, Engineering and Technology

Description

An n-dimensional tiling is formed by laying tiles, chosen from a finite collection of shapes (prototiles), with their boundaries touching and filling n-dimensional euclidean space. This tiling is aperiodic if any sliding (translation) produces a different tiling. Aperiodic tilings appear in Physics in the study of quasi-crystals and their spectrum, in Biology and Computer Science in the study of neural networks, and in Mathematics in the coding of attractors. Analyzing the structure of aperiodic tilings yields information relevant to these applications. Given a tiling, we form an associated continuum on which translation is induced. This tiling contains "bubbles" which combinatorially take the form of translational disagreements and allow us to distinguish between multiple tilings. We also use them to establish a relationship between the geometry of the continuum and the dynamics of the translation operation. Our research demonstrates a correspondence between "bubbles" and what are know as balanced tiles, expanding on the current research by considering the geometry of the tiles instead of merely their combinatorics. In this way, it is shown that geometrically balanced tiles determine the dynamics and spectral properties of a tiling. In particular, our "bubbles" can lead directly to the x-ray diffraction patterns of the associated quasi-crystals.