Symmetry, Subgroups And Supergroups Of Quasicrystallographic Point Groups

By using group theory, two-dimensional quasicrystal point groups and their symmetry operation, generating operation, order of group, direct product groups or semidirect product groups were derived and drawn. The pentagonal system, decagonal system, octagonal system and dodecagonal system were all included in two-dimensional quasicrystal. As well as all symmetry operation were derived, the point groups could be obtained. The rest symmetry operation of point groups could be derived from its genceration operation. The order of point group could be used to understand its maximal subgroup order. The maximal invariant subgroup of direct product groups or semidirect product groups could be directly derived from their relation formulas.By using crystallography theory, stereographic projection of twenty-six point groups in two-dimensional quasicrystal were derived and drawn. Adding known stereographic projection of crystal thirty-two point groups and icosahedral two point groups, stereographic projection of total sixty crystallographic and quasi-crystallographic point groups were drawn in this article. These stereographic projection, not only indicating symmetry of each point group, but also could be used to symmetry operation and degeneration deduction.The group multiplication tables of three greatest proper point group in two-dimensional quasicrystal, i.e. 10 22 822 and 12 22, were filled in. By using these three group multiplication tables, roto-inversion operation n|- and reciprocal rotation operation n, each multiplication of point group's two symmetry operation could be derived, then matrices for point-group symmetry operations could be derived easily.Severally, in three-dimension right-angle coordinate system and decagonal coordinate system, two series symmetry transformation matrices of pentagonal and decagonal crystal system's all symmetry operation (total forty) were derived, their traces tr(W) and determinant det(W) were calculated. In three-dimension right-angle coordinate system, matrix element...