Optimal Pentagonal Tilings
In 2001 Thomas Hales proved that hexagons provide the least-perimeter way to tile the plane with unit areas. Of course, among hexagons, the regular one is best. Similarly, the best quadrilateral is square and the best triangle is equilateral. But what is the best pentagonal tile? Unfortunately, the regular pentagon does not tile. But as announced in 2015 there are 15 types of pentagons that do tile. Some were discovered by amateur mathematician Marjorie Rice, to be featured in Doris Schattschneider’s 2021 MathFest talk. The pentagonal champion requires a trip to Cairo, Egypt. We’ll discuss open questions as well as results, some by undergraduates.
Frank Morgan is Atwell Professor of Mathematics Emeritus and founder of the SMALL undergraduate research project at Williams College and inaugural winner of the MAA Haimo national teaching award. His six books range from his advanced text on Geometric Measure Theory to his Math Chat Book, based on his popular live, call-in TV show and column. He has served as vice-president of the MAA and of the AMS.