Research
Research areas:
Tiling Theory
Knot Theory
Discrete Geometry
Papers
On the Relationship Between Minimal Cube Knots and Minimal Lattice Knots, Journal of Knot Theory and its Ramifications 14 (2005), no. 7, 841-885 (with J. McLoud)
Hyperbolic Tiles with Notched Edges, Discrete & Computational Geometry 34 (2005), 143-166
Heesch's Tiling Problem, Amer. Math.. Monthly, Vol. Vol. 111 (6), June-July 2004, 509-517
A Tile with Surround Number 2, Amer. Math. Monthly, Vol. 109 (4), April 2002, 383-388
Submitted Papers
Minimal Lattice Knots (with McCarty, McLoud, Smith, and Ranalli), provisionally accepted to The Journal of Knot Theory and its Ramifications, submission date August 2006
Distance Functions in Three-Dimensional Lattices (with McCarty, McLoud, Smith, and Ranalli), submitted to Discrete and Computational Geometry, submission date August 2006
Talks and Presentations at Professional Meetings
An Infinite Family of Prototiles with Heesch Number 5, Regional Meeting of the AMS, Lowell, Mass, April 1999
Heesch's Problem, Regional Workshop, Lincoln, Nebraska, Oct. 2000
Diffraction from
Non-Periodic Patterns and 2D Quasicrystals
Research Day 2001 for Regional Universities University of Central Oklahoma,
Edmond, OK, Nov. 2001
A Tile with Surround Number 2, Oklahoma-Arkansas Sectional Meeting of the MAA, Arkadelphia, AR, April 2002
Regular Hyperbolic Polygons with Notched Edges, Texas Sectional Meeting of the MAA, Huntsville, TX, April 2003
Regular Hyperbolic Polygons with Notched Edges, Sectional Meeting of the AMS, San Francisco, CA, May 2003
On the Relationship Between Minimal Lattice Knots and Minimal Cube Knots, Texas Sectional Meeting of the MAA, Corpus Christi, TX, April 2004
On the Relationship Between Minimal Lattice Knots and Minimal Cube Knots, ITV LSAMP Video Conference, University of Texas at Tyler, July 2004
Lattice Knots and Cell Knots, Regional Meeting of the AMS, Bowling Green, KY, March 2005
Minimal Knotting Numbers, National Joint Meeting of the AMS and MAA, New Orleans, LA, January 2007
Heesch Numbers of Polyforms with Marked Edges, Regional Meeting of the AMS, Davidson NC, March 2007
Student-Faculty Research
Emily Walker and Zack Cole (at Northeastern State University): Emily and Zack studied the diffraction patterns formed from a laser passing through nonperiodic dot patterns printed on transparent media as a two-dimensional analog of three dimensional "quasicrystals." Their work was presented as a poster display at the University of Central Oklahoma's "Research Day for Regional Universities."
Bobby Thomas: Bobby did a student-faculty research project with me on implementing an algorithm for studying Heesch's Tiling Problem during the 2003-2004 academic year. Bobby received grants through LSAMP and we were awarded the President's Faculty-Student Research Grant from UT Tyler.
Ben McCarty: Ben did research with me and Dr. Jennifer McLoud-Mann during the 2004-2005 academic year. This research concerned knot theory, and in particular minimal lattice knots. Ben was awarded grants from LSAMP for this research. Ben presented his research at the Fall 2004 LSAMP conference in El Paso, TX. He received the 3rd place award in the math/sciences division.
Jenny Tompkins: Jenny began a research project with me and Dr. Jennifer McLoud in the summer of 2005. This project concerns the application of knot theory to the study of DNA replication and recombination. This project was very ambitious for a summer project as it required much background study in knot theory, biology, and in the known literature on the interface between these two subjects. Jenny is currently writing an expository article aimed at undergraduate students on what she has learned on this topic. As we continue this project into the regular academic year, we plan to explore some original research in this area; in particular, the idea of studying the "unknotting probability" of a knotted DNA molecule via computer experiments seems worth pursuing. Jenny has presented her work twice, and received awards for both presentations. Additionally, Jenny's write up of her work from this project has been published in the Rose Hulman Journal of Undergraduate Mathematics.
Maecy Bruner: Maecy was my student for her senior thesis during the Spring 2006 semester. She did a project in tiling theory titled Squared Squares.
Jahn Veach: Jahn worked with me during the summer of 2006. He was supported by our REU grant from NSA in the amount of $3200. Jahn studied tilings of compact 2-dimensional manifolds by squares. In particular, Jahn studied the previously unstudied cases of the Klein bottle and the projective plane. He found a minimal perfect squaring of the projective plane, and studied both cases with the restriction that the squares are horizontally aligned. He wrote computer code to study these problems. I expect that Jahn and I will have publishable results from this project.