Michel Le Bellac
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Tous droits de traduction, d'adaptation et de reproduction par tous proce ´de ´s, re ´serve ´s pour tous pays.La loi du 11 mars 1957 n'autorisant, aux termes des aline ´as 2 et 3 de l'article 41, d'une part, que les « copies ou reproductio…
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que dans les limites des conditions générales d'utilisation du site ou, le cas échéant, des conditions générales de la licence souscrite par votre établissement.
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View article: C. Notations for point and crystallographic groups
C. Notations for point and crystallographic groups Open
Notations for point and crystallographic groupsThe notation used for symmetry groups varies depending on the science domain and on the class of groups used in applications.Point symmetry groups of three-dimensional space are the most widel…
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Les Ulis Cedex A Tous droits de traduction, d'adaptation et de reproduction par tous procédés réservés pour tous pays.Toute reproduction ou représentation intégrale ou partielle, par quelque procédé que ce soit, des pages publiées dans le …
View article: E. Orbit spaces for 3D-irreducible Bravais groups
E. Orbit spaces for 3D-irreducible Bravais groups Open
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View article: 4 Lattice symmetry
4 Lattice symmetry Open
View article: 6 Lattices and positive quadratic forms
6 Lattices and positive quadratic forms Open
Lattices and positive quadratic forms 6.2 Two dimensional quadratic forms and latticesThe strata of the action of GL 2 (Z) on C + (Q 2 ) are the Bravais classes (see section 4.3 for initial definitions and chapter 8 for further details).Th…
View article: 1 Introduction
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ATous droits de traduction, d'adaptation et de reproduction par tous procédés réservés pour tous pays.Toute reproduction ou représentation intégrale ou partielle, par quelque procédé que ce soit, des pages publiées dans le présent ouvrage,…
View article: A. Basic notions of group theory with illustrative examples
A. Basic notions of group theory with illustrative examples Open
View article: 7 Root systems and root lattices
7 Root systems and root lattices Open
Chapter 7Root systems and root lattices Root systems of lattices and root latticesA hyperplane H r of a n-dimensional vector space E n is a (n -1)dimensional subspace.It is completely characterized by a normal vector r.Definition: reflecti…
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ATous droits de traduction, d'adaptation et de reproduction par tous procédés réservés pour tous pays.Toute reproduction ou représentation intégrale ou partielle, par quelque procédé que ce soit, des pages publiées dans le présent ouvrage,…
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element, 73 Gram matrix, 101, 155, 159 graph, 223
View article: 5 Lattices and their Voronoï and Delone cells
5 Lattices and their Voronoï and Delone cells Open
Lattices and their Voronoï and Delone cellsIn this section we study lattices from the point of view of their tilings by polytopes. Tilings by polytopes: some basic conceptsDefinition: polytope A polytope P is a compact body with a nonempty…
View article: 3 Delone sets and periodic lattices
3 Delone sets and periodic lattices Open
Chapter 3Delone sets and periodic lattices Delone setsWe begin our study of lattices in a more general setting.In the 1930s B.N. Delone (Delaunay) and his colleagues in Moscow began a long-term project of reconstructing mathematical crysta…
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View article: Avant-propos
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View article: B. Graphs, posets, and topological invariants
B. Graphs, posets, and topological invariants Open
Graphs, posets, and topological invariantsThe purpose of this appendix is to give a minimal required system of definitions and mathematical constructions needed to understand and to follow the discussion of the visualization of lattices by…
View article: 9 Applications
9 Applications Open
Lattices appear naturally in rather different domains of natural science and formal mathematics.The goal of the present chapter is to discuss briefly several examples of problems which are tightly related with the lattice constructions, la…
View article: 8 Comparison of lattice classifications
8 Comparison of lattice classifications Open
View article: D. Orbit spaces for plane crystallographic groups
D. Orbit spaces for plane crystallographic groups Open
D Orbit spaces for plane crystallographic groupsA standard ITC [14] representation of 2D-plane crystallographic groups is given below along with corresponding description of orbit spaces (orbifolds) and with notation suggested by Conway wh…