Isoperimetric Pentagonal Tilings
Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank, Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner
TL;DR
This paper characterizes the least-perimeter tilings of the plane using convex pentagons, specifically Cairo and Prismatic types, and proves their optimality among all convex polygon tilings with up to five sides.
Contribution
It identifies the minimal-perimeter convex pentagonal tilings and proves their optimality among all convex polygons with at most five sides.
Findings
Cairo and Prismatic pentagons tile the plane with minimal perimeter.
Infinitely many such tilings exist.
These tilings minimize perimeter among convex polygon tilings with up to five sides.
Abstract
We identify least-perimeter unit-area tilings of the plane by convex pentagons, namely tilings by Cairo and Prismatic pentagons, find infinitely many, and prove that they minimize perimeter among tilings by convex polygons with at most five sides.
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