Brenna Chasney | 2026 I.S. Symposium
Name: 叠谤别苍苍补听颁丑补蝉苍别测
Title: Tiling the Plane, Breaking the Rules: Aperiodicity and Infinity in Tiling Theory
Major: Mathematics
Minor: Religious Studies
Advisor: Pamela Pierce
This Independent Study explores a fascinating question in mathematics: how simple shapes can cover an infinite surface without ever repeating a pattern. Specifically, it focuses on aperiodic tilings of the Euclidean plane and the recent discovery of an 鈥渆instein鈥 tile鈥攁 single shape that can tile the plane only in a non-repeating way. What makes this problem especially compelling is how it connects visual patterns with deep mathematical ideas about symmetry, logic, and infinity. My project begins by introducing the basic principles of tiling and explaining the difference between periodic (repeating) and aperiodic (non-repeating) patterns in an accessible way. It then traces the historical development of the field, highlighting key breakthroughs such as Wang tiles, Penrose tilings, and Taylor鈥揝ocolar tiles. These examples show how mathematicians gradually moved closer to answering a long-standing question: can one simple shape force a pattern that never repeats? What excites me most about this topic is how something that looks artistic and intuitive鈥攍ike tiling patterns鈥攃an reveal complex and surprising mathematical structure. The discovery of the einstein tile demonstrates that even in a well-studied field, entirely new phenomena can still emerge. Through this project, I found that aperiodic tilings challenge our assumptions about order and predictability, showing that structure does not always require repetition. Future research could explore aperiodic tilings in three dimensions or investigate how these ideas relate to computation and undecidability. Overall, this project highlights how abstract mathematical questions can have broad and unexpected implications.
Posted in Symposium 2026 on May 1, 2026.
