ABSTRACT
The cylinder is a two-dimensional space that is like a circle in one direction and like a line in the perpendicular direction. A creature living in the surface of the cylinder would feel that it was living in a plane, except that the plane was periodic in one direction. The freedom to replace ordinary lines by periodic lines enables us to construct surfaces or spaces that are periodic in more than one direction. Periodicity in time and in circular motion have always been with us, in the apparent rotation of the sun, moon, and stars in the sky. The sine wave is virtually the whole story of one-dimensional periodicity. There is another interesting periodic plane defining a torus divided into seven mutually adjacent hexagons. Periodicity in two dimensions, or on the complex numbers, is a comparatively discovery.
