Abstract
A central problem in classical geometry is the classification of all regular polytopes and tessellations in spherical, euclidean or hyperbolic space. When asked within the theory of abstract regular polytopes, the classification problem must necessarily take a different form, because a priori an abstract polytope is not embedded into the geometry of an ambient space. The appropriate substitute now calls for the classification of abstract regular polytopes by their local or global topological type. The classical theory of regular polytopes is concerned with, and solves, the spherical case. In recent years, much work has been done on the toroidal case and a complete classification is now within reach.
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