Arithmetic classes of point symmetry groups of three-dimensional generalized regular lattices. II. Monoclinic and orthorhombic singonies
A generalized regular lattice is pointed as an object of investigation of this article. The lattice was generated by an arbitrary basis over a numerical set ℜ in 3-d Euclidean space. The set ℜ is one from the Euclidean rings of Z, Z[√2], Z[√3] and Z[ τ]. The number τ = (1 + √5)/2 is the golden ratio. The article is a continuation of the work on the classification of point symmetry groups of 3-d generalized regular lattices. It was previously listed arithmetic classes of monoclinic point groups. The point groups representing these classes are written. It was determined infinite set of mmm-extensions for each arithmetic class of monoclinic point groups. The purpose of research is the determination of arithmetic classes of point symmetry groups of 3-d generalized regular lattices. In this article only the monoclinic and orthorhombic point groups are discussed. In the main part of the paper the previously obtained mmm-extensions of monoclinic point groups divided into arithmetic classes. It is established that the mmm-extensions of point groups from the 2/ m α-class divided into two arithmetic classes m α m α m α and m α m β m β for ℜ = Z and ℜ = Z[ τ], and the mmm-extensions of point groups from the 2/ m α-class divided into three arithmetic classes m α m α m α, m α m β m β and m α m γ m γ for ℜ = Z[√2] and ℜ = Z[√3], and the mmm-extensions of point groups from the 2/ m β-class divided into three arithmetic classes m α m β m β, * m β m β m β and m β m β m β* for ℜ = Z and ℜ = Z[√ τ], and the mmm-extensions of point groups from the 2/ m β-class divided into five arithmetic classes m α m β m β, * m β m β m β, m β m β m β*, * m β m β m γ and m β m β m γ* for ℜ = Z[√2] and ℜ = Z[√3], and the mmm-extensions of point groups from the 2/ m γ-class divided into five arithmetic classes m α m γ m γ, * m β m β m γ, m β m β m γ*, * m γ m γ m γ and m γ m γ m γ* for ℜ = Z[√2] and ℜ = Z[√3]. The complete lists of the arithmetic classes of point mmm-group are composed. In the next our article we will list the arithmetic classes of point mm2 -group and 222-group, and we will give total classification of point symmetry groups of lower singonies of 3-d generalized regular lattices. The results of the article can be applied in physics of solid for describing the structure of materials withnon-crystallographic symmetry.