Stereohedron

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)