Hilbert's Eighteenth Problem (crystallographic groups) (Q2736242)

The author describes the history of Hilbert's eighteenth problem (Building up of Space from Congruent Polyhedra): Is there in \(n\)-dimensional Euclidean space also only a finite number of essentially different kinds of groups of motions with a [compact] fundamental region? In an other part of the original text Hilbert asks whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space is possible.NEWLINENEWLINENEWLINEThe author presents also a history of this question, pointing out suitable references.NEWLINENEWLINEFor the entire collection see [Zbl 0902.00028].