Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 475-507 (2001)

On Three-Dimensional Space Groups

John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, William P. Thurston

Department of Mathematics, Princeton University, Princeton NJ 08544-1000, USA, e-mail:; Department of Mathematics, Bielefeld University, D-33501 Bielefeld, Germany, e-mail:; Applied and Computational Mathematics, Princeton University, Princeton NJ 08544-1000, USA, e-mail: (current address: Celera Genomics, 45 West Gude Drive, Rockville MD 20850 USA); Department of Mathematics, University of California at Davis, e-mail:

Abstract: A entirely new and independent enumeration of the crystallographic spacegroups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. For the 35 "irreducible" groups for which it is not, an independent method is used that has the advantage of elucidating their subgroup relationships. Each space group is given a short "fibrifold name" which, much like the orbifold names for two-dimensional groups, while being only specified up to isotopy, contains enough information to allow the construction of the group from the name.

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