Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Next: Modelling a BPMS Up: Constructing Quasiperiodic Minimal Surfaces Previous: Constructing Quasiperiodic Minimal Surfaces
Symmetry and BPMSs
A balance periodic minimal surface (BPMS) partitions space into two congruent continuous labyrinths. The congruence of the labyrinths implies that there exists a symmetry operation interchanging the two regions, and the surface itself must be invariant under this operation. Consequently, the unoriented surface has higher symmetry than the labyrinths. Specifically, the symmetry group of each labyrinth is a subgroup of index two in the symmetry group of the surface. For the case of a BPMS, the symmetry operation that interchanges the two labyrinths can be thought of as expanding the symmetry group of a labyrinth into the symmetry group of the surface. The symmetry group of the surface and its subgroup that leaves each labyrinth invariant is said to form a group-subgroup pair. The possible group-subgroup pairs for BPMSs in three dimensions and for icosahedrally symmetric BPMSs were worked out by Sheng[ 26 ] and independently by Fischer and Koch [ 29 ].
An alternate description which lends itself well to visualization of the labryinths is to associate a color with each of the labyrinths. The symmetry operations that preserve the labyrinth preserve color, while the symmetry operation that interchanges the labyrinths also interchanges the colors of the labyrinths. In this terminology, the group-subgroup pairs are said to be color symmetric. The subgroup operations do not interchange the colors, while the symmetry operation of order two that expands the subgroup into the full symmetry group of the unoriented surface is the color-switching operation.
Here, it suffices to indicate that the space groups of the cubic lattices in six dimensions have three basic ways in which color symmetry may be introduced. One way involves viewing a BCC lattice as two colored simple cubic lattices lattices related by a body-centered translation. Color-preserving symmetry operations map each sublattice onto itself, while a body-centered translation swaps the colors and generates the supergroup. The quasiperiodic surface studied in detail by Sheng and further studied here is of this type. A second group-subgroup pairing arises by considering a primitive lattice as comprising two FCC sublattices-one with all points whose coordinates sum to an even number, the other with coordinates summing to an odd number. The color-switching operation that interchanges the two lattices is a translation by (100000) . Finally, since the 120-element icosahedral group Y h may be factored into the 60-element icosahedral group Y and the two-element group generated by inversion, the inversion operator can be used as a color-switching agent that generates the full icosahedral group Y h .
Chris Kimmer 2011-06-01