proceedings of the american mathematical society Volume 89. Number 3. November 1983 A SYMMETRIC STAR POLYHEDRATHAT TILES BUT NOT AS A FUNDAMENTAL DOMAIN SÁNDOR SZABÓ Abstract. In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates tile 10-space but whose translates by a lattice do not tile it. In [8] he constructed a 5-dimensional star polyhedron whose translates tile 5-space but whose congruent copies by a group of motions do not tile it. So there is no lattice tiling by translates of this polyhedron. In the present paper we shall construct a 5-dimensional centrally-symmetric star polyhedron whose translates tile 5-space but whose congruent copies by a group of motions do not tile it. Furthermore, this phenomenon occurs at an infinitude of dimensions. 1. Preliminaries. Let % be a set in «-dimensional Euclidean space 7?" which is homeomorphic to an «-dimensional cube. If there exists a point P in % such that for every point Q in % the section PQ lies in %, then % is a star body. A system of congruent copies of %, whose union is R" and whose interiors are disjoint, is a tiling. The second part of the 18th problem of D. Hubert is the following question [2]: "Whether polyhedra also exist which do not appear as fundamental regions by groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space possible?". In [2] K. Reinhard constructed a 3-dimensional polyhedron which showed that the answer is "yes". J. Milnor gives reference for example in R2 star, not symmetric. In [8] S. K. Stein constructed a 5-dimensional star polyhedron with this property. Furthermore, all motions in his example are translations. We shall construct a 5-dimensional centrally-symmetric star polyhedron with this property. Another problem in the field of the geometry of numbers is the following question. If translates of a centrally-symmetric star body in Euclidean space can be packed with a certain density, is it possible to find a lattice packing by translates of it that is at least as dense? In [7] S. K. Stein constructed a 10-dimensional body which showed that the answer is "no". We shall construct a 5-dimensional poly- hedron with this property. 2. The Theorem. We shall use the following lemmas. Reference [9] contains proofs of these lemmas. Lemma 1 (S. K. Zaremba [11]). If q, r, s are integers such that q is a power of a prime and r > 1 and s — (qr — l)/(q — 1), then for an s-dimensional chess board of Received by the editors September 15, 1982 and, in revised form, January 20, 1983. 1980 Mathematics Subject Classification. Primary 05B45; Secondary 20K01, 52A45. Key words and phrases. Tiling, star body, factorization of finite abelian groups, groups of motions, lattice. 563 ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use