ABSTRACT
Symmetry of geometric figures is among the most frequently recurring themes in science. The present chapter discusses symmetry of discrete geometric structures, namely of polytopes, polyhedra, and related polytope-like figures. These structures have an outstanding history of study unmatched by almost any other geometric object. The most prominent symmetric figures, the regular solids, occur from very early times and are attributed to Plato (427-347 B.C.E.). Since then, many changes in point of view have occurred about these figures and their symmetry. With the arrival of group theory in the 19th century, many of the early approaches were consolidated and the foundations were laid for a more rigorous development of the theory. In this vein, Schläfli (1814-1895) extended the concept of regular polytopes and tessellations to higher dimensional spaces and explored their symmetry groups as reflection groups.
