EXAMPLE 1 Identifying the Regular Tilings

Which regular polygons can tile the plane, and how?

A regular hexagon has interior angles of 120°; 120 divides into 360 evenly, and three regular hexagons fit together exactly around a point. A regular 7-gon (heptagon)—or any regular polygon with more than 6 sides—has interior angles that are larger than 120° but smaller than 180°. Now, 360 divided by 120 gives 3, and 360 divided by 180 gives 2—and there aren’t any other possibilities in between. Angles between 180° and 120° divided into 360° will give a result between 2 and 3, which consequently is not an integer. So there are no regular tilings of the plane with polygons of more than six sides.