ABSTRACT
Looking at a colored map of the United States in almost any atlas, we see that neighboring states are distinguished by being differently colored, and the total number of colors used is five or six. The word “ordinary” is introduced here because mathematicians often speak of another kind of plane, called the “real projective plane,” in which a map may need six colors. The problem of coloring maps is easily reduced to the problem of coloring trivalent maps. P. J. Heawood’s paper of 1890 includes the important observation that the coloring problem, which has never been solved for maps on a sphere, can be completely solved for maps on any multiply-connected surface. Many people think of mathematics as arithmetic, algebra, and geometry. The remarks serve to illustrate the fact that the same kind of logical thinking can be applied to the different subject of topology, which has very little connection with numbers, and none with measurement.
