Tilings with congruent tiles

Abstract

Introduction. The purpose of this paper is to survey recent results related to the second part of Hubert's eighteenth problem (see Hilbert [1900]). This problem, which is concerned with tilings of Euclidean space by congruent polyhedra, will be stated below after the necessary terminology has been introduced. Although Hubert's original question was answered by one of his assistants (Karl Reinhardt) more than fifty years ago, there remain many unsolved problems in this area of mathematics; a description of recent results therefore seems to us to be of considerable interest.