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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
- ckj@ornl.gov

## Literature

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 (admin@geom.umn.edu).

Crystallographic Topology Home Page

*Page last revised: March 28, 1996*