Crystallographic Topology
Crystallographic Orbifold Covers

The International Tables for Crystallography (ITCr) is the standard reference for the 230 crystallographic space groups which characterize the symmetry of crystals. We are compiling an unofficial supplement on crystallographic topology which illustrates and characterizes the singular set of the Euclidean 3-orbifold that results from wrapping up the asymmetric unit (fundamental domain) of each space group into a manifold with singularities. We find that more systematic methods are needed to correlate related orbifolds. One possibility is illustrating the mappings between a 3-Euclidean orbifold and its n-sheeted covers (n=2,3,4,6), which provides the orbifold analogue of the group vs. normal subgroups information in the ITCr.

We noticed that the color crystallographic groups provide a convenient mechanism for describing group vs. subgroup relationships and wondered what the orbifold for a color group would look like. A bicolor (black and white) orbifold is based on the fundamental domain of the normal subgroup which contains only the regular (black) symmetry elements within the bicolor group since folding (orbi-folding) cannot occur on antisymmetry (white) elements. Thus the antisymmetry elements describe the sheeting character of the two-sheeted cover. Similar relationships hold for the general color groups. There are 1561 bicolor 3-Euclidean crystallographic groups which are called either Shubnikov, Heesch, or magnetic space groups in the crystallographic literature.

The elements of the singular set (i.e. Wyckoff set) within a 3-Euclidean bicolor orbifold come from the set of 58 bicolor 2-elliptic orbifolds which we derived from the crystallographic bicolor point groups. There is an additional set of at least 6 tricolor 2-elliptic orbifolds, and an unknown number of related tricolor 3-Euclidean orbifolds needed to compile a Crystallographic Orbifold Covers atlas which should include at least the 2- and 3-sheeted covers.

We would appreciate comments on the approach described and alternate possibilities for correlating orbifolds. Pointers to related ongoing research and literature references would also be appreciated.

Mr. Carroll K. Johnson, PhD
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37831-6197, USA


Color groups are described briefly in A. V. Shubnikov and V. A. Koptsik, "Symmetry in Science and Art", Plenum Press (1974). A full color atlas for the bicolor space groups is in V. A. Koptsik's book "Shubnikov Groups" (in Russian) Izd. MGU, Moscow (1966). The classic reference on orbifolds is W. P. Thurston's 1979 class notes on "The Geometry and Topology of Three-Manifolds", which are unpublished but available from The Geometry Center (

Crystallographic Topology Home Page

Page last revised: March 28, 1996