Mathematical Maps

Tom Wieting
Department of Mathematics, Reed College

Abstract: To prepare a planar map of a spherical surface, one must practice compromise. For more than two thousand years, map makers have designed such maps to meet one or another of an ever more complex array of objectives. In general, they have based their designs upon the familiar terrestrial/celestial network of mutually parallel circles of constant latitude (the parallels) and mutually confocal semicircles of constant longitude (the meridians).

Indeed, they have portrayed their designs by projecting a generic network of parallels and meridians upon the planes of the maps, calling the projected networks graticules -- for instance, the Sinusoidal Graticule:

In practice, they have sought to preserve angles or to preserve areas. Very early, they found that they could not do both. In this lecture, we will apply the elements of differential geometry to describe a variety of maps representative of the achievements of map makers, exquisitely clever in design and useful to the practice of navigation and astronomy.