International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.5, pp. 29-31
Section 1.5.2. The affine space^{a}Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany |
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 . 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 of two vectors and is simply the sum of the two arrows. Similarly, multiplication of a vector 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 . It is characterized by the property that for all vectors .
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.
We shall now exploit the advantage of being independent of the dimensionality. The following definitions are independent of the dimension by replacing the specific dimensions 2 for the plane and 3 for the space by an unspecified integer number . Although we cannot visualize four- or higher-dimensional objects, we can describe them in such a way that we are able to calculate with such objects and derive their properties.
Algebraically, an n-dimensional (real) vector v can be represented by a column of n real numbers. The n-dimensional real vector space is then (In crystallography n is normally 3.) The entries are called the coefficients of the vector . On one can naturally define an addition, where the coefficients of the sum of two vectors are the corresponding sums of the coefficients of the vectors. To multiply a vector by a real number, one just multiplies all its coefficients by this number. The null vector can be distinguished, since for all .
The identification of a concrete vector space with the vector space can be done by choosing a basis of . A basis of is any tuple of n vectors such that every vector of can be written uniquely as a linear combination of the basis vectors: . Whereas a vector space has many different bases, the number n of vectors of a basis is uniquely determined and is called the dimension of . The isomorphism (see Section 1.5.3.4 for a definition of isomorphism) between and maps the vector to its coefficient column with respect to the chosen basis . The mapping respects addition of vectors and multiplication of vectors with real numbers. Moreover, is a bijective mapping, which means that for any coefficient column there is a unique vector with . Therefore one can perform all calculations using the coefficient columns.
An important concept in mathematics is the automorphism group of an object. In general, if one has an object (here the vector space ) together with a structure (here the addition of vectors and the multiplication of vectors with real numbers), its automorphism group is the set of all one-to-one mappings of the object onto itself that preserve the structure.
A bijective mapping of the vector space into itself satisfying for all and for all real numbers and all vectors is called a linear mapping and the set of all these linear mappings is the linear group of . To know the image of under a linear mapping it suffices to know the images of the basis vectors under , since . Writing the coefficient columns of the images of the basis vectors as columns of a matrix [i.e. , ], then the coefficient column of with respect to the chosen basis is just . Note that the matrix of a linear mapping depends on the basis of . The matrix that corresponds to the composition of two linear mappings is the product of the two corresponding matrices. We have thus seen that the linear group of a vector space of dimension n is isomorphic to the group of all invertible matrices via the isomorphism that associates to a linear mapping its corresponding matrix (with respect to the basis ). This means that one can perform all calculations with linear mappings using matrix calculations.
In crystallography, the translation-vector space has an additional structure: one can measure lengths and angles between vectors. An n-dimensional real vector space with such an additional structure is called a Euclidean vector space, . Its automorphism group is the set of all (bijective) linear mappings of onto itself that preserve lengths and angles and is called the orthogonal group of . If one chooses the basis to be the unit vectors (which are orthogonal vectors of length 1), then the isomorphism above maps the orthogonal group onto the set of all matrices A with , the unit matrix. ^{T} denotes the transposition operator, which maps columns to rows and rows to columns.
In this section we build up a model for the `point space'. Let us first assume . Then the affine space may be imagined as an infinite sheet of paper parallel, let us say, to the (, ) plane and cutting the axis at in crystallographic notation. The points of have coordinates which are the coefficients of the vector from the origin to the point.
This observation is generalized by the following:
If then the vector is defined as the difference (computed in the vector space ). The set of all with forms an n-dimensional vector space which is called the underlying vector space . Omitting the last coefficient, we can identify with . As the coordinates already indicate, the sets as well as can be viewed as subsets of . Computed in , the sum of two elements in is again in , since the last coefficient of the sum is and the sum of a point and a vector is again a point in (since the last coordinate is ), but the sum of two points does not make sense.
The affine group of geometry is the set of all mappings of the point space which fulfil the conditions
In the mathematical model, the affine group is the automorphism group of the affine space and can be viewed as the set of all linear mappings of that preserve .
Definition 1.5.2.4.1. The affine group is the subset of the set of all linear mappings with . The elements of are called affine mappings.
Since is linear, it holds that Hence an affine mapping also maps into itself.
Since the first n basis vectors of the chosen basis lie in and the last one in , it is clear that with respect to this basis the affine mappings correspond to matrices of the formThe linear mapping induced by on which is represented by the matrix will be referred to as the linear part of . The image of a point P with coordinates can easily be found as
If one has a way to measure lengths and angles (i.e. a Euclidean metric) on the underlying vector space , one can compute the distance between P and Q as the length of the vector and the angle determined by P, Q and R with vertex Q is obtained from . In this case, is the Euclidean affine space, .
An affine mapping of the Euclidean affine space is called an isometry if its linear part is an orthogonal mapping of the Euclidean space . The set of all isometries in is called the Euclidean group and denoted by . Hence is the set of all distance-preserving mappings of onto itself. The isometries are the affine mappings with matrices of the form where the linear part W belongs to the orthogonal group of .
Special isometries are the translations, the isometries where the linear part is , with matrix The group of all translations in is the translation subgroup of and is denoted by . Note that composition of two translations means addition of the translation vectors and is isomorphic to the translation vector space .