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 90o rotation has antisymmetry, but the second order 180o 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 90o to 45o, 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.
Related diagrams for the cubic bicolor elliptic 2-orbifolds and the tricolor elliptic 2-orbifolds are shown in Figs. B.4 and B.5, respectively.
Appendix A. Subgroup Families from Color Groups
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