Crystallographic Topology

Crystallographic Topology 101
Appendix B
Elliptic 2-Orbifold Covers

Figs. B.1-B.5 are related to those shown in Sect. 2 but include the details of the elliptic orbifold cover, L, that unfolds the orbifold, G, to the suborbifold, H. The antisymmetry (white operators) give the details of the cover folding. In all cases, the orbifolds G and H are also shown to clarify in the following figures the graphic symbolism. Sets of these color orbifolds can be used to construct the singular set of the Euclidean 3-orbifold cover associated with the Euclidean group/subgroup transformations. We anticipate that they will mainly be helpful for the more complex situations where the covering details are sometimes difficult to understand such as in certain order-3 cubic to orthorhombic transformations. At present, we have not worked out suitable graphical conventions for drawing axes of the full three-dimensional cover when there are mixed black and white orders such as the 4-fold white and 2-fold black combination.

Figs. B.1, B.2, and B.3 illustrate 17 examples for the 15 families of bicolor elliptic 2-orbifolds, based on the 7 topology families belonging to the cyclic (d,e,f) and dihedral (g,h,i,j) point groups. Each member of the bicolor point group represents a group/normal-subgroup (G/H) pair of regular point groups, with the regular orbifold arising from the point group G.

One group representation from the orthorhombic-tetrahedral cluster (i.e., from the bottom layer of Fig. A.2) is shown on the left of Figs. B.1-B.3, and one to three regular orbifolds (H), from corresponding subgroups, on the top or bottom of the figures. The bicolor orbifold within the table is labeled G:H and has both black (symmetry) and white (antisymmetry) components. The H suborbifold is seen by ignoring the white (i.e., outlined) orbifold elements, and the G orbifold is seen by considering all elements to be equally valid.

The simplest examples are [hd] 4'4':44 in Fig. B.2 and [jg] 4'2'2':422 in Fig. B.3 for which only the mirrors have antisymmetry. Next in complexity are [dd] 44:22 and [ff] 41':21' in Fig. B.1 where a 90^o rotation has antisymmetry, but the second order 180^o has regular symmetry. This also occurs with [hh] 4'4':2'2' of Fig. B.3 except that in this case the angle between the mirrors reduces from 90^o to 45^o, and you simply have to ignore one of the original outside mirror symbols. Similarly, with [gg] 422:222 of Fig. B.3 where a new 2-fold axis becomes active, we ignore one of the original black 2-folds.

In practice, we do not simply "ignore" the doubled symbols since the chief application is analyzing the changes in the singular set in pairs of Euclidean 3-orbifolds. The doubling of the orbifold symbol accompanies a doubling of the original fundamental domain in the group to subgroup transformation.

The most complex symbols involve the projective plane [fe] 41':20 in Fig. B.1 and [ie] 22':20 in Fig. B.2, but the detailed symbolism can be deduced from the triads of symbols.