Topological principles in cartography

CHARACTER OF A MAP A map may be described as a linear graph embedded in a orientable, two-dimen sional manifold. A manifold, aside from being a two-dimensional continuum, is locally flat, that is, each point of the manifold is contained in a neighborhood homeomorphic to a disk. A linear graph so embedded will, if it contains 1-circuits, delineate a number of simply connected domains. The collection of 0-cells and 1-cells of the graph, together with the 2-cells so delineated forms a two-dimensional complex, having the special property of local flatness. The structure of this two-dimensional complex is completely specified by a pair of relations among the cells. These are the oriented incidence relations between 0-cells and 1-cells, and those between 1-cells and 2-cells,, The local flatness condition implies that there is a form of symmetry, known as duality, between these two relations. A widely used model of this relational system, the DIME encoding, illustrates this symmetry in a perspicuous manner. This representation consists of a quadruple, or pair of ordered pairs of cell identifiers,