Celebration of Scholars
A Study of Coincidence Reflections in the Plane
Name:
Calvin Raymond
Major: Mathematics
Hometown: Plymouth, IN
Faculty Sponsor: Diana Thomson
Other Sponsors:
Type of research: SURE
Funding: SURE
Name:
Victoria Wheeler
Major: Mathematics
Hometown: Kenosha, WI
Faculty Sponsor: Diana Thomson
Other Sponsors:
Type of research: SURE
Funding: SURE
Abstract
In this presentation, we demonstrate the process for finding coincidence reflections for lattices defined by matrices in 2-D. This research works towards applications of crystallography specifically used to model the atomic structures of crystalline solids. This allows us to get a better understanding and compare crystals defects and theoretical crystalline structures. One of the methods for modeling physical crystalline structures involves seeing how they behave under reflections. We expand on previous work in theorizing when coincidence reflections exist. The group of coincidence reflections is a subgroup of the group of coincidence isometries. We will be looking at standard forms that are representative of the conjugacy classes of 2x2 of invertible matrices. A given matrix can be considered a structure matrix of a lattice with an orthogonal matrix acting on it as a linear transformation. Furthermore, the intersection of a lattice and its image will be a coincidence isometry if the intersection forms a sublattice of the lattice and its image. We give vectors of specific forms that define coincidence reflections of the lattice and analyze how these reflections behave in the plane.
Submit date: Feb. 25, 2021, 7:49 p.m.