Combinatorial Geometry

Abstract

The development of geometry, as inspired by the deep results of Bernhard Riemann in the second half of the nineteenth century, has meant that scientific work in this field moved quite far from the “naive” or elementary geometry practiced by the Greek mathematicians of around the beginning of our era, and their numerous successors in later times. Classically, the main focus of geometry has been on the proofs or constructions connected with properties of basic geometrical objects (points, straight lines, circles, triangles, half-planes, tetrahedra, etc.), that is, problems that can be visualized. On the other hand, a paradoxical characteristic of contemporary scientific works in “pure” geometry is the fact that the majority of them are completely devoid of pictures; or the fact that “geometrical intuition” is more often required by specialists in mathematical analysis or algebra than by mathematicians who consider themselves geometers. The apparent dissatisfaction of a number of mathematicians with this situation has led to new directions of research in geometry; this is mainly in response to modern problems in optimization. As examples of such geometrical optimization problems one can consider the problem of filling a plane or a space (or some of their parts) with some system of geometrical objects, or covering parts of a plane or a space with the smallest possible number of copies of a geometrical object. These and several other reasons have led, especially in the past half century, to the rapid development of several nontraditional branches of geometry (or areas closely related to geometry), among them combinatorial geometry.