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Abstract

The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of area π/3 for 3≤n≤10. However, for n>10, it is difficult to rule out non-convex n-gons that tile irregularly.

Author Bio

Leo Digiosia is a third year graduate student studying contact homological invariants of low dimensional manifolds. He spends his time biking around the bayous of Houston when not doing math.

Jahangir Habib is a mathematician from Los Angeles who found his way to the Purple Valley of Williams College through his love of math. Though he finds beauty in all types of math, he is partial to abstract algebra and algebraic geometry. He currently works as a Business Intelligence Analyst for the New York Mets and hopes to continue working in sports.

Jack Hirsch is a junior from Oakland, California double majoring in math and economics. He loves to play tennis and ponder riddles, and he once fished his frisbee out of a wild shark's mouth.

Lea Kenigsberg is a graduate student at Columbia University.

Kevin Li is a junior at Yale University. He plans to go to graduate school in math and is particularly interested in geometry and differential equations. He is also grateful for Frank Morgan's mentorship throughout the project.

Dylanger Pittman is a graduate student at Emory.

Jackson Petty is a double major in mathematics and linguistics at Yale University. He is interested in topology and information theory.

Christopher Xue is a math major who enjoys geometry and algebra and who in his free time enjoys playing Tetris.

Weitao Zhu is a graduate student at Columbia University.

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