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[ <i>τ</i>]. The number <i>τ</i> = (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 <i>mmm</i>-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 <i>mmm</i>-extensions of monoclinic point groups divided into arithmetic classes. It is established that the <i>mmm</i>-extensions of point groups from the 2/ <i>m</i> <sub>α</sub>-class divided into two arithmetic classes <i>m</i> <sub>α</sub> <i>m</i> <sub>α</sub> <i>m</i> <sub>α</sub> and <i>m</i> <sub>α</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> for ℜ = Z and ℜ = Z[ <i>τ</i>], and the <i>mmm</i>-extensions of point groups from the 2/ <i>m</i> <sub>α</sub>-class divided into three arithmetic classes <i>m</i> <sub>α</sub> <i>m</i> <sub>α</sub> <i>m</i> <sub>α</sub>, <i>m</i> <sub>α</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> and <i>m</i> <sub>α</sub> <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub> for ℜ = Z[√2] and ℜ = Z[√3], and the <i>mmm</i>-extensions of point groups from the 2/ <i>m</i> <sub>β</sub>-class divided into three arithmetic classes <i>m</i> <sub>α</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub>, * <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> and <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub>* for ℜ = Z and ℜ = Z[√ <i>τ</i>], and the <i>mmm</i>-extensions of point groups from the 2/ <i>m</i> <sub>β</sub>-class divided into five arithmetic classes <i>m</i> <sub>α</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub>, * <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub>, <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub>*, * <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>γ</sub> and <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>γ</sub>* for ℜ = Z[√2] and ℜ = Z[√3], and the <i>mmm</i>-extensions of point groups from the 2/ <i>m</i> <sub>γ</sub>-class divided into five arithmetic classes <i>m</i> <sub>α</sub> <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub>, * <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>γ</sub>, <i>m</i> <sub>β</sub> <i>m</i> <sub>β</sub> <i>m</i> <sub>γ</sub>*, * <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub> and <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub> <i>m</i> <sub>γ</sub>* for ℜ = Z[√2] and ℜ = Z[√3]. The complete lists of the arithmetic classes of point <i>mmm</i>-group are composed. In the next our article we will list the arithmetic classes of point <i>mm</i>2 -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.