#20: Coincidence Isometries of 3D Lattices — Presentation Detail — Celebration of Scholars

Additional Information

More information is available at carthage.edu/celebration-scholars/. The following are members of the Research, Scholarship, and Creativity Committee who are eager to listen to ideas and answer questions:

Thomas Carr
Katherin Hilson
Kim Instenes
John Kirk
Sarah Terrill

#20: Coincidence Isometries of 3D Lattices

Name: Theodore Reimer
Major: Mathematics, Religion
Hometown: Appleton, WI
Faculty Sponsor: Diana Thomson
Other Sponsors:
Type of research: SURE
Funding: SURE

Abstract

Crystal structures are important to the material and geographic sciences, so it is reasonable to desire a mathematical model for their study. We do this using lattices, which are integer linear combinations of a set of basis vectors which we will represent as a single matrix. Rotating, shifting, and otherwise altering these lattices can help us represent crystal defects, which occur where two different crystals meet. In this work we review coincidence site lattices, that is, new lattices made of points where a lattice and the image of its transformation intersect. Our primary concern is determining when a rotation actually results in a coincidence site lattice. We cover the matrices for three dimensional crystals with two complex eigenvalues and one real eigenvalue, and we prove that we can consider this case as a rotation of lattices with a real diagonal basis matrix.

Poster file

Submit date: March 29, 2023, 10:41 a.m.