# Stereohedron

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A convex polyhedron in a regular decomposition of space into equal polyhedra, i.e. convex fundamental domains of arbitrary (Fedorov) groups of motions. The number of different lattices for a regular decomposition of an $n$-dimensional space, in which the stereohedron is adjoined on all its edges (the sides of the fundamental domains), obviously depends only on the dimension $n$ of the space. For $n=3$, the number of edges of the stereohedron does not exceed 390. The classification has been made only for particular forms of stereohedra, for example, parallelohedra (cf. Parallelohedron).

#### References

 [1] , Symmetry designs , Moscow (1980) (In Russian; translated from English) [2] B.N. Delone, N.N. Sandakova, "Theory of stereohedra" Trudy Mat. Inst. Steklov. , 64 (1961) pp. 28–51 (In Russian)

#### References

 [a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) [a2] B. Grünbaum, G.C. Shephard, "Tilings with congruent tiles" Bull. Amer. Math. Soc. , 3 (1980) pp. 951–973 [a3] P. McMullen, "Convex bodies which tile space by translations" Mathematika , 27 (1980) pp. 113–121 [a4] B.N. Delone, "Proof of the fundamental theorem in the theory of stereohedra" Soviet Math. Dokl. , 2 : 3 (1961) pp. 812–817 Dokl. Akad. Nauk SSSR , 138 (1961) pp. 1270–1272 [a5] H.S.M. Coxeter, "Regular polytopes" , Macmillan (1948)
How to Cite This Entry:
Stereohedron. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stereohedron&oldid=31579
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article