Multiplicity of Hexagon Numbers

Summary

Imagine you have an unlimited supply of congruent equilateral triangles. Polygon numbers are the number of these triangles used to tile a convex polygon. For example, triangle numbers are square integers n², where the positive integer n is the side length of a tiled equilateral triangle. Hexagon numbers are the number of triangles used to tile a convex hexagon, and can be realized by removing corners from a tiled equilateral triangle. The number of ways (multiplicity up to congruence) that a given hexagon number can be constructed geometrically from such tiles is the subject of our paper.

Acknowledgments

The authors wish to thank the UAB Mathematics BS/MS Fast-Track Program for support and encouragement. We are also indebted to discussions with Dr. Frank Patane (Samford University, Birmingham, AL) and postings to Mathematics Stack Exchange, in particular, one posting of Dr. Jyrki Lahtonen (University of Turku, Finland), for our understanding of the results on sums of squares.

Additional information

Notes on contributors

Cameron G. Hale

Cameron Hale ([email protected]) is a recently graduated MS student in mathematics at the University of Alabama at Birmingham. Prior to his undertaking undergraduate research on hexagon tilings in the UAB Mathematics BS/MS Fast-Track Program, he received an AS degree from Lawson State Community College in Alabama. His master’s research has been in laminations of the unit disk, a tool used in the study of holomorphic (complex analytic) dynamics. He currently works at Cadence Bank in the IT department.

Jonathan R. Kelleher

Jonathan Kelleher ([email protected]) is completed his BS and MS in mathematics at the University of Alabama at Birmingham in the Mathematics BS/MS Fast-Track Program. Jonathan is now in the Ph.D. program in mathematics at Auburn University (Alabama).

John C. Mayer

John Mayer ([email protected]) is a professor of mathematics at the University of Alabama at Birmingham. He earned his BS in philosophy at Randolph-Macon College in Virginia, and his MA and Ph.D. in philosophy and Ph.D. in mathematics at the University of Florida. His main areas of research are holomorphic dynamics, continuum theory (a branch of topology), and the teaching and learning of mathematics. He regularly mentors students in mathematical research at many levels, including: doctoral, masters, undergraduate, and high school.