Tiling the Plane

Abstract

But let us return to ordinary geometry. Among the host of geometric figures around us, the regular polygons have always played a special role. A polygon (from the Greek words polys = many and gonon = angle) is a closed planar figure made up of straight line segments. A regular polygon is a polygon whose sides and angles are all equal. The simplest regular polygon is the equilateral triangle; next comes the square, followed by the pentagon, the hexagon, and so on. As we saw in Chapter 1, the Greeks were particulary interested in these regular polygons and used them to find an approximation for the number π. They knew, of course, that there exist infinitely many of these polygons; that is, for any given integer n ≥ 3, there exists a regular polygon having n sides—an “n-gon”, as mathematicians say.

Threefold is the form of space:

Length, with every restless motion,

Seeks eternity’s wide ocean;

Breadth with boundless sway extends;

Depth to unknown realms descends.

— Friedrich von Schiller (1759–1805).