Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional

periodic lattices is the central subject of the book. Different basic mathematical tools

currently used for the description of lattice geometry are introduced and illustrated through

applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional

lattices and to lattices associated with integrable dynamical systems. Starting from general Delone

sets the authors turn to different symmetry and topological classifications including explicit construction

of orbifolds for two- and three-dimensional point and space groups.

Voronoï and Delone celles together with positive quadratic forms and lattice description by root

systems are introduced to demonstrate alternative approaches to lattice geometry study. Zonotopes

and zonohedral families of 2-, 3-, 4-, 5-dimensional lattices are explicitly visualized using

graph theory approach. Along with crystallographic applications, qualitative features of lattices of

quantum states appearing for quantum problems associated with classical Hamiltonian integrable

dynamical systems are shortly discussed.

The presentation of the material is presented through a number of concrete examples with an extensive

use of graphical visual zation. The book is aimed at graduated and post-graduate students and

young researchers in theoretical physics, dynamical systems, applied mathematics, solid state physics,

crystallography, molecular physics, theoretical chemistry, ...

Book series edited by Michèle Leduc and Michel Le Bellac.