Motivation

Nebe, G.

doi:10.1107/97809553602060000541

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 29 | 1 | 2 |

Section 1.5.2.1. Motivation

Gabriele Nebe^a ^*

^a Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

1.5.2.1. Motivation

| top | pdf |

The aim of this section is to give a mathematical model for the `point space' (also known in crystallography as `direct space' or `crystal space') which the positions of atoms in crystals (the so-called `points') occupy. This allows us in particular to describe the symmetry groups of crystals and to develop a formalism for calculating with these groups which has the advantage that it works in arbitrary dimensions. Such higher-dimensional spaces up to dimension 6 are used, e.g., for the description of quasicrystals and incommensurate phases. For example, the more than 29 000 000 crystallographic groups up to dimension 6 can be parameterized, constructed and identified using the computer package [CARAT]: Crystallographic AlgoRithms And Tables, available from http://wwwb.math.rwth-aachen.de/carat/index.html .

As well as the points in point space, there are other objects, called `vectors'. The vector that connects the point P to the point Q is usually denoted by $[{\overrightarrow {P\,Q}}]$ . Vectors are usually visualized by arrows, where parallel arrows of the same length represent the same vector.

Whereas the sum of two points P and Q is not defined, one can add vectors. The sum $[{\bf v}+{\bf w}]$ of two vectors $[{\bf v}]$ and $[{\bf w}]$ is simply the sum of the two arrows. Similarly, multiplication of a vector $[{\bf v}]$ by a real number can be defined.

All the points in point space are equally good, but among the vectors one can be distinguished, the null vector $[{\bf o}]$ . It is characterized by the property that $[{\bf v} + {\bf o} = {\bf v}]$ for all vectors $[{\bf v}]$ .

Although the notion of a vector seems to be more complicated than that of a point, we introduce vector spaces before giving a mathematical model for the point space, the so-called affine space, which can be viewed as a certain subset of a higher-dimensional vector space, where the addition of a point and a vector makes sense.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 29