Introduction To Lattice Geometry Through Group Action

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. Di_ erent 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 di_ erent symmetry and topological classi_ cations
including explicit construction

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

Voronoï and Delone cells 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 visualization. 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, ...