Crystallographic Groupoids: Past, Present, and
Future. Carroll K. Johnson, Oak Ridge National Laboratory, Chemical and
Analytical Sciences Division, Oak Ridge, TN 37831-6197
Groupoids, and inverse semigroups, are useful for algebraic
description of partial and local symmetries which fall outside group theory.
Sporadic Crystallographic application of groupoids began shortly after H.
Brandt’s groupoid paper in Math. Ann. 96, 360, 1926.
Crystallographic examples to be discussed include (a) order-disorder
structures; (b) quasicrystals; (c) symmetry of isolated unit
cells, asymmetric units, and orbifolds; (d) generalized pseudosymmetry
classification; and (e) integrated crystal + surface structure
Our current research centers on item (c). The portion
of a standard extended Hermann-Mauguin space group symbol following the lattice
symbol P, I, F, etc. describes a screw & glide groupoid (S&GG). The 157
nonsymmo morphic space group symbols contain 122 true S&GGs while the 73
symmo morphic space group symbols contain the 32 point groups. Jaswon and Rose
derive and characterize both the space groups and the color groups in a very
concise form using S&GGs. We plan to extend their results to the corre
sponding crystallographic orbifolds we discussed previously. Items (d)
and (e) provide suggestions for future crystallographic
(a) K. Dornberger-Schiff and H. Grell, Acta Cryst.
A 38, 491-498, 1982;
J. Grell, Acta Applic. Math. 52, 261-269, 1998.
(b) J. Kellendonk and M.V. Lawson, J. Algebra
224, 140-150, 2000.
(c) M.A. Jaswon and M.A. Rose, Crystal Symmetry:
Theory of Colour Crystallography, Horwood, 1983.
(d) M.V. Lawson, Inverse Semigroups: The Theory of
World Scientific, 1998.
(e) A. Weinstein, Notices Am. Math. Assoc.,