A new nonclassical crystallographic group (NCG) theoretical system is set up.This system can describe infinite kinds of nonclassical periodic structures,especially for those with locally n-fold rotational symmetries f...A new nonclassical crystallographic group (NCG) theoretical system is set up.This system can describe infinite kinds of nonclassical periodic structures,especially for those with locally n-fold rotational symmetries forbidden by the rules of the classical crystallography. The formal classification of NCGs is given.展开更多
Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conju...Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).展开更多
Six kinds of nonclassical periodic lattices with locally 10-fold rotational symmetries are proposed. They can be delineated via nonclassical plane-crystallographic groups. The projections on the planes of correspondin...Six kinds of nonclassical periodic lattices with locally 10-fold rotational symmetries are proposed. They can be delineated via nonclassical plane-crystallographic groups. The projections on the planes of corresponding unit cells consisting of embedding polyhedra generate the periodic lattices, respectively. The Fourier-transform patterns of the periodiclattices have almost perfect 10-fold rotational symmetries, which are very similar to those displaying in the electron-diffraction patterns of so-called quasicrystals.展开更多
Nonclassical periodic lattices are classified and enumerated via the nonclassical crystallographic groups(NCGs). Some of the nonclassical periodic lattices might be used to construct larger unit cells to explain the s...Nonclassical periodic lattices are classified and enumerated via the nonclassical crystallographic groups(NCGs). Some of the nonclassical periodic lattices might be used to construct larger unit cells to explain the structures of so-called quasicrystals or other nonclassical crystalline substances.展开更多
Eight kinds of nonclasslcal periodic lattices with locally 8-fold rotational symmetries are introduced.They can be described via nonclassical Planc-crystallographic groups. The periodic lattices may be interpreted byt...Eight kinds of nonclasslcal periodic lattices with locally 8-fold rotational symmetries are introduced.They can be described via nonclassical Planc-crystallographic groups. The periodic lattices may be interpreted bythe projections on the plane of the corresponding unit cells consisting of embedding polyhedrons, respectively. TheFourier-transform patterns of the Periodic lattices have striking approximate'8-fold rotational symmetries', some ofwhich are similar to those displaying in the electrton-diffraction patterns of so-called quasicrystals.展开更多
Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will class...Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will classify all crystallographic groups associated with W up to conjugation in the affine group A(V).展开更多
The discoveries of so-called quasicrystals have broken through the theoretic foundation set up by the classical crystallographic group theory since 1891 and proposed new topics for study of solid structures. Electron ...The discoveries of so-called quasicrystals have broken through the theoretic foundation set up by the classical crystallographic group theory since 1891 and proposed new topics for study of solid structures. Electron diffraction patterns (EDP' s) and high-resolution microscopic (HREM) images have proved invaluable tools of studying the structures of crystals. The recognition and determination of EDP's and HREM images of a real-structure play a key role for understanding the structure. This paper will introduce some new developments about crystallographic group theory and new image processing methods on EDP's and HREM images. Contrary to popular beliefs, the research shows that quasicrystals can be understood (perturbed) complex periodic structures.展开更多
A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endo...A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces. Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups. In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.展开更多
We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they ...We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.展开更多
文摘A new nonclassical crystallographic group (NCG) theoretical system is set up.This system can describe infinite kinds of nonclassical periodic structures,especially for those with locally n-fold rotational symmetries forbidden by the rules of the classical crystallography. The formal classification of NCGs is given.
文摘Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).
文摘Six kinds of nonclassical periodic lattices with locally 10-fold rotational symmetries are proposed. They can be delineated via nonclassical plane-crystallographic groups. The projections on the planes of corresponding unit cells consisting of embedding polyhedra generate the periodic lattices, respectively. The Fourier-transform patterns of the periodiclattices have almost perfect 10-fold rotational symmetries, which are very similar to those displaying in the electron-diffraction patterns of so-called quasicrystals.
文摘Nonclassical periodic lattices are classified and enumerated via the nonclassical crystallographic groups(NCGs). Some of the nonclassical periodic lattices might be used to construct larger unit cells to explain the structures of so-called quasicrystals or other nonclassical crystalline substances.
文摘Eight kinds of nonclasslcal periodic lattices with locally 8-fold rotational symmetries are introduced.They can be described via nonclassical Planc-crystallographic groups. The periodic lattices may be interpreted bythe projections on the plane of the corresponding unit cells consisting of embedding polyhedrons, respectively. TheFourier-transform patterns of the Periodic lattices have striking approximate'8-fold rotational symmetries', some ofwhich are similar to those displaying in the electrton-diffraction patterns of so-called quasicrystals.
文摘Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will classify all crystallographic groups associated with W up to conjugation in the affine group A(V).
文摘The discoveries of so-called quasicrystals have broken through the theoretic foundation set up by the classical crystallographic group theory since 1891 and proposed new topics for study of solid structures. Electron diffraction patterns (EDP' s) and high-resolution microscopic (HREM) images have proved invaluable tools of studying the structures of crystals. The recognition and determination of EDP's and HREM images of a real-structure play a key role for understanding the structure. This paper will introduce some new developments about crystallographic group theory and new image processing methods on EDP's and HREM images. Contrary to popular beliefs, the research shows that quasicrystals can be understood (perturbed) complex periodic structures.
文摘A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces. Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups. In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.
基金The first and third authors are partially supported by Project MTM2005-01637the second is partially supported by Projects Fondecyt 1030252,1030373 UTFSM 12.05.21
文摘We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.
