# Dive into the Mind-Boggling Math of Tessellating Pentagons

Children’s blocks lie scattered on the floor. You start playing with them—squares, rectangles, triangles and hexagons—moving them around, flipping them over, seeing how they fit together. You feel a primal satisfaction from arranging these shapes into a perfect pattern, an experience you’ve probably enjoyed many times. But of all the blocks designed to lie flat on a table or floor, have you ever seen any shaped like pentagons?

#### Quanta Magazine

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

People have been studying how to fit shapes together to make toys, floors, walls and art—and to understand the mathematics behind such patterns—for thousands of years. But it was only this year that we finally settled the question of how five-sided polygons “tile the plane.” Why did pentagons pose such a big problem for so long?

To understand the problem with pentagons, let’s start with one of the simplest and most elegant of geometric structures: the regular tilings of the plane. These are arrangements of regular polygons that cover flat space entirely and perfectly, with no overlap and no gaps. Here are the familiar triangular, square and hexagonal tilings. We find them in floors, walls and honeycombs, and we use them to pack, organize and build things more efficiently.

These are the easiest tilings of the plane. They are “monohedral,” in that they consist of only one type of polygonal tile; they are “edge-to-edge,” meaning that corners of the polygons always match up with other corners; and they are “regular,” because the one tile being used repeatedly is a regular polygon whose side lengths are all the same, as are its interior angles. Our examples above use equilateral triangles (regular triangles), squares (regular quadrilaterals) and regular hexagons.

Remarkably, these three examples are the only regular, edge-to-edge, monohedral tilings of the plane: No other regular polygon will work. Mathematicians say that no other regular polygon “admits” a monohedral, edge-to-edge tiling of the plane. And this far-reaching result is actually quite easy to establish using only two simple geometric facts.

First, there’s the fact that in a polygon with n sides, where n must be at least 3, the sum of an n-gon’s interior angles, measured in degrees, is