staff motion research built and utilized normally via Louis Michel to the research of N-dimensional periodic lattices is the relevant topic of the booklet. assorted uncomplicated mathematical instruments presently used for the outline of lattice geometry are brought and illustrated via purposes to crystal constructions in - and 3-dimensional house, to summary multi-dimensional lattices and to lattices linked to integrable dynamical platforms. ranging from normal Delone units the authors flip to assorted symmetry and topological classifications together with specific building of orbifolds for 2- and three-d aspect and area teams. Voronoï and Delone cells including optimistic quadratic kinds and lattice description by way of root structures are brought to illustrate replacement ways to lattice geometry research. Zonotopes and zonohedral households of 2-, 3-, 4-, 5-dimensional lattices are explicitly visualized utilizing graph idea strategy. in addition to crystallographic functions, qualitative beneficial properties of lattices of quantum states showing for quantum difficulties linked to classical Hamiltonian integrable dynamical platforms are almost immediately mentioned. The presentation of the cloth is gifted via a few concrete examples with an in depth use of graphical visualization. The booklet is geared toward graduated and post-graduate scholars and younger researchers in theoretical physics, dynamical platforms, utilized arithmetic, stable nation physics, crystallography, molecular physics, theoretical chemistry, ...