Chapter 9.1 Bases, lattices, Bravais lattices and other classifications

Space‐group symmetry

First Online Edition (2006)

Part 9. Crystal lattices

  1. H. Burzlaff1,
  2. H. Zimmermann2

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000517

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Burzlaff, H. and Zimmermann, H. 2006. Bases, lattices, Bravais lattices and other classifications. International Tables for Crystallography. A:9:9.1:742–749.

Author Information

  1. 1

    Universität Erlangen–Nürnberg, Robert‐Koch‐Strasse 4a, D‐91080 Uttenreuth, Germany

  2. 2

    Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D‐91054 Erlangen, Germany

Publication History

  1. Published Online: 1 JAN 2006



In this chapter, the symmetry and some topological aspects of lattices are considered. Different classifications of lattices into five Voronoi types, 24 Symmetrische Sorten and 14 Bravais lattices are described. Conventional lattice bases and their cells are defined and illustrated. Extensive tabulations for the two‐ and three‐dimensional Bravais lattices are provided. Further classifications use reduction theory; details of the Delaunay reduction are given.


  • Symmetrische Sorten;
  • bases;
  • Delaunay reduction;
  • Bravais lattices;
  • topological properties of lattices;
  • Voronoi types