Summary
We present a generalization of the Voderberg tile, which, in addition to admitting periodic and nonperiodic spiral tilings of the plane, has the property that just two copies can surround 1 or 2 copies of the tile. We construct a generalization of this tile that admits periodic and nonperiodic spiral tilings of the plane while enjoying the property that any number of copies of the tile can be surrounded by just 2 copies. In doing so, we solve two open problems posed in the classic book Tilings and Patterns by Grünbaum and Shephard.
Acknowledgments
The authors were supported by NSF grant DMS 1460699. They also thank the University of Washington Bothell for its support.
Additional information
Notes on contributors
Jadie Adams
JADIE ADAMS earned her Bachelor of Science degree in math from Westminster College before going on to work in automatic speech recognition. She is currently working toward a Ph.D. in computing at the University of Utah, where she her research focuses on machine learning and statistical shape modeling for medical image analysis.
Gabriel Lopez
GABRIEL LOPEZ earned his Bachelor of Science in mathematics at California State University, San Bernardino. As of this publication, he is a graduate student in mathematics at the University of Colorado, Boulder. He hopes to use his training to not only go into higher education, but to work in outreach and create research opportunities for students who are interested in the mathematical sciences, in particular to those from historically underrepresented groups in the community.
Casey Mann
CASEY MANN is a mathematics professor at the University of Washington Bothell. His research interests include tilings and knot theory. He thinks it is impactful to engage undergraduate students in the process of mathematical discovery, as exemplified in this article and the accomplishments of his coauthors!
Nhi Tran
NHI TRAN earned her Bachelor of Science in Mathematics from the University of Washington Bothell in 2017.