International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 720776
https://doi.org/10.1107/97809553602060000930 Chapter 3.2. Point groups and crystal classes^{a}Institut für Kristallographie, RWTH Aachen University, 52062 Aachen, Germany,^{b}Fachbereich Chemie, PhilippsUniversität, D35032 Marburg, Germany, and ^{c}Departamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain Section 3.2.1 treats the geometric and grouptheoretical aspects of both crystallographic and noncrystallographic point groups. It includes descriptions and diagrams of the sub and supergroups of the crystallographic point groups and tabulations and diagrams of the (infinitely many) noncrystallographic point groups and their subgroups, with special emphasis on the two icosahedral point groups. All entries and terms are thoroughly described in the text. Section 3.2.2 discusses the relations between symmetry and the physical properties of crystals. The following properties and their dependence on the crystallographic pointgroup symmetries are described and presented in several tables: crystal morphology, symmetry of etch figures, optical birefringence, optical activity, enantiomorphism, pyroelectricity (ferroelectricity) and piezoelectricity. Section 3.2.3 presents an extensive tabulation of the 10 twodimensional and the 32 threedimensional crystallographic point groups, containing for each group the stereographic projections of the symmetry elements and the face poles of the general crystal form, and a table with the Wyckoff positions, their site symmetries and the coordinates of symmetryequivalent points, and, in addition, the names and Miller indices {hkl} of the general and special face and point forms. Each pointgroup table concludes with the three major projection symmetries. Section 3.2.4 discusses molecular symmetry, including noncrystallographic symmetries, the symmetry of polymeric molecules and symmetry aspects of chiral molecules and crystal structures. It includes a table of the crystallographic and noncrystallographic rod groups. 
A point group^{1} is a group of symmetry operations all of which leave at least one point unmoved. Thus, all operations containing translations are excluded. Point groups can be subdivided into crystallographic and noncrystallographic point groups. A crystallographic point group is a point group that maps a point lattice onto itself. Consequently, rotations and rotoinversions are restricted to the well known crystallographic cases 1, 2, 3, 4, 6 and (cf. Section 1.2.1 ); matrices for these symmetry operations are listed in Tables 1.2.2.1 and 1.2.2.2 . No such restrictions apply to the noncrystallographic point groups.
The numbers of the crystallographic point groups are finite: 2 for one dimension, 10 for two dimensions and 32 for three dimensions. The numbers of noncrystallographic point groups for dimensions are infinite. The two and threedimensional crystallographic point groups and their crystal systems are summarized in Tables 3.2.1.1 and 3.2.1.2. They are described in detail in Section 3.2.1.2. The two onedimensional point groups are discussed in Section 2.1.3.16 . The noncrystallographic point groups are treated in Section 3.2.1.4.


Crystallographic point groups occur:
General point groups, i.e. crystallographic and noncrystallographic point groups, occur as:
A (geometric) crystal class (pointgroup type) is the set of all crystals having the same pointgroup symmetry. The word `class', therefore, denotes a classificatory pigeonhole and should not be used as synonymous with the point group of a particular crystal. The symbol of a crystal class is that of the common point group. (For geometric and arithmetic crystal classes of space groups, see Sections 1.3.4.2 and 1.3.4.4 .)
Of particular importance for the structure determination of crystals are the 11 centrosymmetric crystallographic point groups, because they describe the possible symmetries of the diffraction record of a crystal: ; ; mmm; ; ; ; ; ; ; ; . This is due to Friedel's rule, which states that, provided anomalous dispersion is neglected, the diffraction record of any crystal is centrosymmetric, even if the crystal is noncentrosymmetric. The symmetry of the diffraction record determines the Laue class of the crystal; this is further explained in Chapter 1.6 . For a given crystal, its Laue class is obtained if a symmetry centre is added to its point group, as shown in Table 3.2.2.1.
In two dimensions, six `centrosymmetric' crystallographic point groups and hence six twodimensional Laue classes exist: 2; 2mm; 4; 4mm; 6; 6mm. These point groups are, for instance, the only possible symmetries of zerolayer Xray photographs.
Among the centrosymmetric crystallographic point groups in three dimensions, the seven lattice point groups (holohedral point groups, holohedries) are of special importance because they constitute the possible point symmetries of lattices, i.e. the site symmetries of their nodes. In three dimensions, the seven holohedries are: ; ; mmm; ; ; ; . Note that is the point symmetry of the rhombohedral lattice and the point symmetry of the hexagonal lattice; both occur in the hexagonal crystal family (cf. Chapter 2.1 ). Point groups that are, within a crystal family, subgroups of a holohedry are called merohedries; they are called specifically hemihedries for subgroups of index 2, tetartohedries for index 4 and ogdohedries for index 8.
In two dimensions, four holohedries exist: 2; 2mm; 4mm; 6mm. Note that the hexagonal crystal family in two dimensions contains only one lattice type, with point symmetry 6mm.
Another classification of the crystallographic point groups is that into isomorphism classes. Here all those point groups that have the same kind of group table appear in one class. These isomorphism classes are also known under the name of abstract point groups.
There are 18 abstract crystallographic point groups in three dimensions: the point groups in each of the following lines are isomorphous and belong to the same abstract group: In two dimensions, the ten crystallographic point groups form nine abstract groups; the groups 2 and m are isomorphous and belong to the same abstract group, the remaining eight point groups correspond to one abstract group each.
In crystallography, point groups usually are described
Descriptions (i) through (iii) are given in this section, whereas for crystal drawings and actual photographs reference is made to textbooks of crystallography and mineralogy [Buerger (1956, ch. 10) and Phillips (1971, chs. 3, 4 and 6) are particularly rich in pictures of crystal morphologies]; this also applies to the construction and the properties of the stereographic projection.
In Tables 3.2.3.1 and 3.2.3.2, the two and threedimensional crystallographic point groups are listed and described. The tables are arranged according to crystal systems and Laue classes. Within each crystal system and Laue class, the sequence of the point groups corresponds to that in the spacegroup tables of this volume: pure rotation groups are followed by groups containing reflections, rotoinversions and inversions. The holohedral point group is always given last.
In Tables 3.2.3.1 and 3.2.3.2, some point groups are described in two or three versions, in order to bring out the relations to the corresponding space groups (cf. Section 2.1.3.2 ):
The presentation of the point groups is similar to that of the space groups in Part 2 . The headline contains the short Hermann–Mauguin and the Schoenflies symbols. The full Hermann–Mauguin symbol, if different, is given below the short symbol. No Schoenflies symbols exist for twodimensional groups. For an explanation of the symbols see Sections 1.4.1 and 2.1.3.4 , and Chapter 3.3 .
Next to the headline, a pair of stereographic projections is given. The diagram on the left displays a general crystal or point form, that on the right shows the `framework of symmetry elements'. Except as noted below, the c axis is always normal to the plane of the figure, the a axis points down the page and the b axis runs horizontally from left to right. For the five trigonal point groups, the c axis is normal to the page only for the description with `hexagonal axes'; if described with `rhombohedral axes', the direction [111] is normal and the positive a axis slopes towards the observer. The conventional coordinate systems used for the various crystal systems are listed in Table 2.1.1.1 and illustrated in Figs. 2.1.3.1 to 2.1.3.10 .
In the righthand projection, the graphical symbols of the symmetry elements are the same as those used in the spacegroup diagrams; they are listed in Chapter 2.1 . Note that the symbol of a symmetry centre, a small circle, is also used for a face pole in the lefthand diagram. Mirror planes are indicated by heavy solid lines or circles; thin lines are used for the projection circle, for symmetry axes in the plane and for some special zones in the cubic system.
In the lefthand projection, the projection circle and the coordinate axes are indicated by thin solid lines, as are again some special zones in the cubic system. The dots and circles in this projection can be interpreted in two ways.

The general, special and limiting crystal forms and point forms constitute the main part of the table for each point group. The theoretical background is given below in Section 3.2.1.2.2, Crystal and point forms and an explanation of the listed data is to be found in Section 3.2.1.2.3, Description of crystal and point forms.
The last entry for each point group contains the Symmetry of special projections, i.e. the plane point group that is obtained if the threedimensional point group is projected along a symmetry direction. The special projection directions are the same as for the space groups; they are listed in Section 2.1.3.14 . The relations between the axes of the threedimensional point group and those of its twodimensional projections can easily be derived with the help of the stereographic projection. No projection symmetries are listed for the twodimensional point groups.
Note that the symmetry of a projection along a certain direction may be higher than the symmetry of the crystal face normal to that direction. For example, in point group all faces have face symmetry 1, whereas projections along any direction have symmetry 2; in point group 422, the faces (001) and have face symmetry 4, whereas the projection along [001] has symmetry 4mm.
For a point group a crystal form is a set of all symmetryequivalent faces; a point form is a set of all symmetryequivalent points. Crystal and point forms in point groups correspond to `crystallographic orbits' in space groups; cf. Sections 1.1.7 and 1.4.4.1 .
Two kinds of crystal and point forms with respect to can be distinguished. They are defined as follows:
General and special crystal and point forms can be represented by their sets of equivalent Miller indices and point coordinates x, y, z. Each set of these `triplets' stands for infinitely many crystal forms or point forms which are obtained by independent variation of the values and signs of the Miller indices h, k, l or the point coordinates x, y, z.
It should be noted that for crystal forms, owing to the well known `law of rational indices', the indices h, k, l must be integers; no such restrictions apply to the coordinates x, y, z, which can be rational or irrational numbers.
Example
In point group 4, the general crystal form {hkl} stands for the set of all possible tetragonal pyramids, pointing either upwards or downwards, depending on the sign of l; similarly, the general point form x, y, z includes all possible squares, lying either above or below the origin, depending on the sign of z. For the limiting cases l = 0 or z = 0, see below.
In order to survey the infinite number of possible forms of a point group, they are classified into Wyckoff positions of crystal and point forms, for short Wyckoff positions. This name has been chosen in analogy to the Wyckoff positions of space groups; cf. Sections 1.4.4.2 and 2.1.3.11 . In point groups, the term `position' can be visualized as the position of the face poles and points in the stereographic projection. Each `Wyckoff position' is labelled by a Wyckoff letter.
Definition
A `Wyckoff position of crystal and point forms' consists of all those crystal forms (point forms) of a point group for which the face poles (points) are positioned on the same set of conjugate symmetry elements of ; i.e. for each face (point) of one form there is one face (point) of every other form of the same `Wyckoff position' that has exactly the same face (site) symmetry.
Each point group contains one `general Wyckoff position' comprising all general crystal and point forms. In addition, up to two `special Wyckoff positions' may occur in two dimensions and up to six in three dimensions. They are characterized by the different sets of conjugate face and site symmetries and correspond to the seven positions of a pole in the interior, on the three edges, and at the three vertices of the socalled `characteristic triangle' of the stereographic projection.
Examples
It is instructive to subdivide the crystal forms (point forms) of one Wyckoff position further, into characteristic and noncharacteristic forms. For this, one has to consider two symmetries that are connected with each crystal (point) form:
Examples
The eigensymmetries and the generating symmetries of the 47 crystal forms (point forms) are listed in Table 3.2.1.3. With the help of this table, one can find the various point groups in which a given crystal form (point form) occurs, as well as the face (site) symmetries that it exhibits in these point groups; for experimental methods see Sections 3.2.2.2 and 3.2.2.3. Diagrams of the 47 crystal forms are presented in Fig. 3.2.1.1.
^{†}These limiting forms occur in three or two nonequivalent orientations (different types of limiting forms); cf. Table 3.2.3.2.
^{‡}In point groups and , the tetragonal prism and the hexagonal prism occur twice, as a `basic special form' and as a `limiting special form'. In these cases, the point groups are listed twice, as `' and as `'. 
With the help of the two groups and , each crystal or point form occurring in a particular point group can be assigned to one of the following two categories:
The importance of this classification will be apparent from the following examples.
Examples
The general forms of the 13 point groups with no, or only one, symmetry direction (`monoaxial groups') , are always noncharacteristic, i.e. their eigensymmetries are enhanced in comparison with the generating point groups. The general positions of the other 19 point groups always contain characteristic crystal forms that may be used to determine the point group of a crystal uniquely (cf. Section 3.2.2).^{4}
So far, we have considered the occurrence of one crystal or point form in different point groups and different Wyckoff positions. We now turn to the occurrence of different kinds of crystal or point forms in one and the same Wyckoff position of a particular point group.
In a Wyckoff position, crystal forms (point forms) of different eigensymmetries may occur; the crystal forms (point forms) with the lowest eigensymmetry (which is always well defined) are called basic forms (German: Grundformen) of that Wyckoff position. The crystal and point forms of higher eigensymmetry are called limiting forms (German: Grenzformen) (cf. Table 3.2.1.3). These forms are always noncharacteristic.
Limiting forms^{5} occur for certain restricted values of the Miller indices or point coordinates. They always have the same multiplicity and oriented face (site) symmetry as the corresponding basic forms because they belong to the same Wyckoff position. The enhanced eigensymmetry of a limiting form may or may not be accompanied by a change in the topology^{6} of its polyhedra, compared with that of a basic form. In every case, however, the name of a limiting form is different from that of a basic form.
The face poles (or points) of a limiting form lie on symmetry elements of a supergroup of the point group that are not symmetry elements of the point group itself. There may be several such supergroups.
Examples
Whereas basic and limiting forms belonging to one `Wyckoff position' are always clearly distinguished, closer inspection shows that a Wyckoff position may contain different `types' of limiting forms. We need, therefore, a further criterion to classify the limiting forms of one Wyckoff position into types: A type of limiting form of a Wyckoff position consists of all those limiting forms for which the face poles (points) are located on the same set of additional conjugate symmetry elements of the holohedral point group (for the trigonal point groups, the hexagonal holohedry has to be taken). Different types of limiting forms may have the same eigensymmetry and the same topology, as shown by the examples below. The occurrence of two topologically different polyhedra as two `realizations' of one type of limiting form in point groups 23, and 432 is explained below in Section 3.2.1.2.4, Notes on crystal and point forms, item (viii).
Examples
Not considered in this volume are limiting forms of another kind, namely those that require either special metrical conditions for the axial ratios or irrational indices or coordinates (which always can be closely approximated by rational values). For instance, a rhombic disphenoid can, for special axial ratios, appear as a tetragonal or even as a cubic tetrahedron; similarly, a rhombohedron can degenerate to a cube. For special irrational indices, a ditetragonal prism changes to a (noncrystallographic) octagonal prism, a dihexagonal pyramid to a dodecagonal pyramid or a crystallographic pentagondodecahedron to a regular pentagondodecahedron. These kinds of limiting forms are listed by A. Niggli (1963).
In conclusion, each general or special Wyckoff position always contains one set of basic crystal (point) forms. In addition, it may contain one or more sets of limiting forms of different types. As a rule,^{7} each type comprises polyhedra of the same eigensymmetry and topology and, hence, of the same name, for instance `ditetragonal pyramid'. The name of the basic general forms is often used to designate the corresponding crystal class, for instance `ditetragonalpyramidal class'; some of these names are listed in Table 3.2.1.4.

The main part of each pointgroup table in Section 3.2.3 describes the general and special crystal and point forms of that point group, in a manner analogous to the positions in a space group. The general Wyckoff position is given at the top, followed downwards by the special Wyckoff positions with decreasing multiplicity. Within each Wyckoff position, the first block of column 6 refers to the basic forms, the subsequent blocks list the various types of limiting form, if any.
The columns, from left to right, contain the following data (further details are to be found below in Section 3.2.1.2.4, Notes on crystal and point forms):
Column 1: Multiplicity of the `Wyckoff position', i.e. the number of equivalent faces and points of a crystal or point form.
Column 2: Wyckoff letter. Each general or special `Wyckoff position' is designated by a `Wyckoff letter', analogous to the Wyckoff letter of a position in a space group (cf. Sections 1.4.4.2 and 2.1.3.11 ).
Column 3: Face symmetry or site symmetry, given in the form of an `oriented pointgroup symbol', analogous to the oriented sitesymmetry symbols of space groups (cf. Sections 1.4.4.2 and 2.1.3.12 ). The face symmetry is also the symmetry of etch pits, striations and other face markings. For the twodimensional point groups, this column contains the edge symmetry, which can be either 1 or m.
Column 4: Coordinates of the symmetryequivalent points of a point form.
Column 5: Name of crystal form. If more than one name is in common use, several are listed. The names of the limiting forms are also given. The crystal forms, their names, eigensymmetries and occurrence in the point groups are summarized in Table 3.2.1.3, which may be useful for determinative purposes, as explained in Sections 3.2.2.2 and 3.2.2.3. There are 47 different types of crystal form. Frequently, 48 are quoted because `sphenoid' and `dome' are considered as two different forms. It is customary, however, to regard them as the same form, with the name `dihedron'.
Name of point form (printed in italics). There exists no general convention on the names of the point forms. Here, only one name is given, which does not always agree with that of other authors. The names of the point forms are also contained in Table 3.2.1.3. Note that the same point form, `line segment', corresponds to both sphenoid and dome.
Column 6: Miller indices (hkl) for the symmetryequivalent faces (edges) of a crystal form. In the trigonal and hexagonal crystal systems, when referring to hexagonal axes, Bravais–Miller indices (hkil) are used, with .
With a few exceptions, the triplets of Miller indices (hkl) and point coordinates are arranged in such a way as to show analogous sequences; they are both based on the same set of generators, as described in Sections 1.4.3 and 2.1.3.10 . For all point groups, except those referred to a hexagonal coordinate system, the correspondence between the (hkl) and the triplets is immediately obvious.^{8}
The sets of symmetryequivalent crystal faces also represent the sets of equivalent reciprocallattice points, as well as the sets of equivalent Xray (neutron) reflections. This important aspect is treated in Klapper & Hahn (2010).
Examples

Further discussion of the data in Tables 3.2.3.1 and 3.2.3.2 as far as molecular symmetry is concerned can be found in Section 3.2.4.3.

Several different sets of names have been devised for the 32 crystal classes. Their use, however, has greatly declined since the introduction of the international pointgroup symbols. As examples, two sets (both translated into English) that are frequently found in the literature are given in Table 3.2.1.4. To the name of the class the name of the system has to be added: e.g. `tetragonal pyramidal' or `tetragonal tetartohedry'.
Note that Friedel (1926) based his nomenclature on the point symmetry of the lattice. Hence, two names are given for the five trigonal point groups, depending whether the lattice is hexagonal or rhombohedral: e.g. `hexagonal ogdohedry' and `rhombohedral tetartohedry'.
In this section, the sub and supergroup relations between the crystallographic point groups are presented in the form of a `family tree'.^{12} Figs. 3.2.1.2 and 3.2.1.3 apply to two and three dimensions. The sub and supergroup relations between two groups are represented by solid or dashed lines. For a given point group of order the lines to groups of lower order connect with all its maximal subgroups with orders ; the index [i] of each subgroup is given by the ratio of the orders . The lines to groups of higher order connect with all its minimal supergroups with orders ; the index [i] of each supergroup is given by the ratio . In other words: if the diagram is read downwards, subgroup relations are displayed; if it is read upwards, supergroup relations are revealed. The index is always an integer (theorem of Lagrange) and can be easily obtained from the group orders given on the left of the diagrams. The highest index of a maximal subgroup is [3] for two dimensions and [4] for three dimensions.

Maximal subgroups and minimal supergroups of the twodimensional crystallographic point groups. Solid lines indicate maximal normal subgroups; double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. The group orders are given on the left. 

Maximal subgroups and minimal supergroups of the threedimensional crystallographic point groups. Solid lines indicate maximal normal subgroups; double or triple solid lines mean that there are two or three maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. The group orders are given on the left. Full Hermann–Mauguin symbols are used. 
Two important kinds of subgroups, namely sets of conjugate subgroups and normal subgroups, are distinguished by dashed and solid lines. They are characterized as follows:
The subgroups of a group are conjugate subgroups if are symmetryequivalent in , i.e. if for every pair at least one symmetry operation of exists which maps onto : ; cf. Sections 1.1.5 and 1.1.8 .
Examples

The subgroup of a group is a normal (or invariant) subgroup if no subgroup of exists that is conjugate to in . Note that this does not imply that is also a normal subgroup of any supergroup of . Subgroups of index [2] are always normal and maximal (cf. Section 1.1.5 ). (The role of normal subgroups for the structure of space groups is discussed in Sections 1.3.3 and 1.4.2.3 .)
Examples

Figs. 3.2.1.2 and 3.2.1.3 show that there exist two `summits' in both two and three dimensions from which all other point groups can be derived by `chains' of maximal subgroups. These summits are formed by the square and the hexagonal holohedry in two dimensions and by the cubic and the hexagonal holohedry in three dimensions.
The sub and supergroups of the point groups are useful both in their own right and as a basis of the translationengleiche or t subgroups and supergroups of space groups (cf. Section 1.7.1 ). Tables of the sub and supergroups of the plane groups and space groups are contained in Volume A1 of International Tables for Crystallography (2010). A general discussion of sub and supergroups of crystallographic groups, together with further explanations and examples, is given in Section 1.7.1 .
In Sections 3.2.1.2 and 3.2.1.3, only the 32 crystallographic point groups (crystal classes) are considered. In addition, infinitely many noncrystallographic point groups exist that are of interest as possible symmetries of molecules and of quasicrystals and as approximate local site symmetries in crystals. Crystallographic and noncrystallographic point groups are collected here under the name general point groups. They are reviewed in this section and listed in Tables 3.2.1.5 and 3.2.1.6.
Because of the infinite number of these groups only classes of general point groups (general classes)^{13} can be listed. They are grouped into general systems, which are similar to the crystal systems. The `general classes' are of two kinds: in the cubic, icosahedral, circular, cylindrical and spherical system, each general class contains one point group only, whereas in the 4Ngonal, gonal and gonal system, each general class contains infinitely many point groups, which differ in their principal nfold symmetry axis, with for the 4Ngonal system, for the gonal system and for the gonal system.
Furthermore, some general point groups are of order infinity because they contain symmetry axes (rotation or rotoinversion axes) of order infinity^{14} (∞fold axes). These point groups occur in the circular system (two dimensions) and in the cylindrical and spherical systems (three dimensions).
The Hermann–Mauguin and Schoenflies symbols for the general point groups follow the rules of the crystallographic point groups (cf. Sections 1.4.1 , 2.1.3.4 and 3.3.1 ). This extends also to the infinite groups where symbols like or are immediately obvious.
In two dimensions (Table 3.2.1.5), the eight general classes are collected into three systems. Two of these, the 4Ngonal and the gonal systems, contain only point groups of finite order with one nfold rotation point each. These systems are generalizations of the square and hexagonal crystal systems. The circular system consists of two infinite point groups, with one ∞fold rotation point each.
^{†}A rotating circle has no mirror lines; there exist two enantiomorphic circles with opposite senses of rotation. A stationary circle has infinitely many mirror lines through its centre.

In three dimensions (Table 3.2.1.6), the 33 general classes are collected into seven systems. Three of these, the 4Ngonal, the gonal and the gonal systems,^{15} contain only point groups of finite order with one principal nfold symmetry axis each. These systems are generalizations of the tetragonal, trigonal and hexagonal crystal systems (cf. Table 3.2.3.2). The five cubic groups are well known as crystallographic groups. The two icosahedral groups of orders 60 and 120, characterized by special combinations of twofold, threefold and fivefold symmetry axes, are discussed in more detail below. The groups of the cylindrical and the spherical systems are all of order infinity; they describe the symmetries of cylinders, cones, rotation ellipsoids, spheres etc.^{16}
^{†}The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and (see text).
^{‡}Rotating and `antirotating' forms in the cylindrical system have no `vertical' mirror planes, whereas stationary forms have infinitely many vertical mirror planes. In classes ∞ and , enantiomorphism occurs, i.e. forms with opposite senses of rotation. Class exhibits no enantiomorphism due to the centre of symmetry, even though the double cone is rotating in one direction. This can be understood as follows: The handedness of a rotating cone depends on the sense of rotation with respect to the axial direction from the base to the tip of the cone. Thus, the rotating double cone consists of two cones with opposite handedness and opposite orientations related by the (single) horizontal mirror plane. In contrast, the `antirotating' double cone in class consists of two cones of equal handedness and opposite orientations, which are related by the (infinitely many) twofold axes. The term `antirotating' means that upper and lower halves of the forms rotate in opposite directions. ^{§}The spheres in class of the spherical system must rotate around an axis with at least two different orientations, in order to suppress all mirror planes. This class exhibits enantiomorphism, i.e. it contains spheres with either righthanded or lefthanded senses of rotation around the axes (cf. Section 3.2.2.4, Optical properties). The stationary spheres in class contain infinitely many mirror planes through the centres of the spheres. Group is sometimes symbolized by ; group by or . The symbols used here indicate the minimal symmetry necessary to generate the groups; they show, furthermore, the relation to the cubic groups. The Schoenflies symbol K is derived from the German name Kugelgruppe. 
It is possible to define the threedimensional point groups on the basis of either rotoinversion axes or rotoreflection axes . The equivalence between these two descriptions is apparent from the following examples: In the present tables, the standard convention of using rotoinversion axes is followed.
Tables 3.2.1.5 and 3.2.1.6 contain for each class its general Hermann–Mauguin and Schoenflies symbols, the group order and the names of the general face form and its dual, the general point form.^{17} Special and limiting forms are not given, nor are `Miller indices' (hkl) and point coordinates x, y, z. They can be derived easily from Tables 3.2.3.1 and 3.2.3.2 for the crystallographic groups.^{18}
The two point groups 235 and of the icosahedral system (orders 60 and 120) are of particular interest among the noncrystallographic groups because of the occurrence of fivefold axes and their increasing importance as symmetries of molecules (viruses), of quasicrystals, and as approximate local site symmetries in crystals (alloys, icosahedron). Furthermore, they contain as special forms the two noncrystallographic platonic solids, the regular icosahedron (20 faces, 12 vertices) and its dual, the regular pentagondodecahedron (12 faces, 20 vertices).
The icosahedral groups (cf. diagrams in Table 3.2.3.3) are characterized by six fivefold axes that include angles of . Each fivefold axis is surrounded by five threefold and five twofold axes, with angular distances of between a fivefold and a threefold axis and of between a fivefold and a twofold axis. The angles between neighbouring threefold axes are , between neighbouring twofold axes . The smallest angle between a threefold and a twofold axis is .
Each of the six fivefold axes is perpendicular to five twofold axes; there are thus six maximal conjugate pentagonal subgroups of types 52 (for 235) and (for ) with index [6]. Each of the ten threefold axes is perpendicular to three twofold axes, leading to ten maximal conjugate trigonal subgroups of types 32 (for 235) and (for ) with index [10]. There occur, furthermore, five maximal conjugate cubic subgroups of types 23 (for 235) and (for ) with index [5].
The two icosahedral groups are listed in Table 3.2.3.3, in a form similar to the cubic point groups in Table 3.2.3.2. Each group is illustrated by stereographic projections of the symmetry elements and the general face poles (general points); the complete sets of symmetry elements are listed below the stereograms. Both groups are referred to a cubic coordinate system, with the coordinate axes along three twofold rotation axes and with four threefold axes along the body diagonals. This relation is well brought out by symbolizing these groups as 235 and instead of the customary symbols 532 and .
The table contains also the multiplicities, the Wyckoff letters and the names of the general and special face forms and their duals, the point forms, as well as the oriented face and sitesymmetry symbols. In the icosahedral `holohedry' , the special `Wyckoff position' 60d occurs in three realizations, i.e. with three types of polyhedra. In 235, however, these three types of polyhedra are different realizations of the limiting general forms, which depend on the location of the poles with respect to the axes 2, 3 and 5. For this reason, the three entries are connected by braces; cf. Section 3.2.1.2.4, Notes on crystal and point forms, item (viii).
Not included are the sets of equivalent Miller indices and point coordinates. Instead, only the `initial' triplets (hkl) and for each type of form are listed. The complete sets of indices and coordinates can be obtained in two steps^{19} as follows:

This sequence of matrices ensures the same correspondence between the Miller indices and the point coordinates as for the crystallographic point groups in Table 3.2.3.2.
The matrices^{20} are with^{21} These matrices correspond to counterclockwise rotations of 72, 144, 216 and around a fivefold axis parallel to .
The resulting indices h, k, l and coordinates x, y, z are irrational but can be approximated closely by rational (or integral) numbers. This explains the occurrence of almost regular icosahedra or pentagondodecahedra as crystal forms (for instance pyrite) or atomic groups (for instance icosahedron).
Further descriptions (including diagrams) of noncrystallographic groups are contained in papers by Nowacki (1933) and A. Niggli (1963) and in the textbooks by P. Niggli (1941, pp. 78–80, 96), Shubnikov & Koptsik (1974) and Vainshtein (1994). For the geometry of polyhedra, the well known books by H. S. M. Coxeter (especially Coxeter, 1973) are recommended.
In Figs. 3.2.1.4 to 3.2.1.6, the subgroup and supergroup relations between the twodimensional and threedimensional general point groups are illustrated. It should be remembered that the index of a group–subgroup relation between two groups of order infinity may be finite or infinite. For the two spherical groups, for instance, the index is [2]; the cylindrical groups, on the other hand, are subgroups of index [] of the spherical groups.

Subgroups and supergroups of the twodimensional general point groups. Solid lines indicate maximal normal subgroups, double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. For the finite groups, the orders are given on the left. Note that the subgroups of the two circular groups are not maximal and the diagram applies only to a specified value of N (see text). For complete examples see Fig. 3.2.1.5. 

The subgroups of the twodimensional general point groups 16mm (4Ngonal system) and 18mm [gonal system, including the gonal groups]. Compare with Fig. 3.2.1.4 which applies only to one value of N. 

Subgroups and supergroups of the threedimensional general point groups. Solid lines indicate maximal normal subgroups, double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. For the finite groups, the orders are given on the left and on the right. Note that the subgroups of the five cylindrical groups are not maximal and that the diagram applies only to a specified value of N (see text). Only those crystallographic point groups are included that are maximal subgroups of noncrystallographic point groups, cf. Fig. 3.2.1.3. Full Hermann–Mauguin symbols are used. 
Fig. 3.2.1.4 for two dimensions shows that the two circular groups ∞m and ∞ have subgroups of types nmm and n, respectively, each of index []. The order of the rotation point may be or . In the first case, the subgroups belong to the 4Ngonal system, in the latter two cases, they belong to the gonal system. [In the diagram of the gonal system, the gonal groups appear with the symbols and .] The subgroups of the circular groups are not maximal because for any given value of N there exists a group with which is both a subgroup of the circular group and a supergroup of the initial group.
The subgroup relations, for a specified value of N, within the 4Ngonal and the gonal system, are shown in the lower part of the figure. They correspond to those of the crystallographic groups. A finite number of further maximal subgroups is obtained for lower values of N, until the crystallographic groups (Fig. 3.2.1.2) are reached. This is illustrated for both systems in Fig. 3.2.1.5.
Fig. 3.2.1.6 for three dimensions illustrates that the two spherical groups and each have one infinite set of cylindrical maximal conjugate subgroups, as well as one infinite set of cubic and one infinite set of icosahedral maximal finite conjugate subgroups, all of index [].
Each of the two icosahedral groups 235 and has one set of five cubic, one set of six pentagonal and one set of ten trigonal maximal conjugate subgroups of indices [5], [6] and [10], respectively (cf. Section 3.2.1.4.2, The two icosahedral groups); they are listed on the right of Fig. 3.2.1.6. For the crystallographic groups, Fig. 3.2.1.3 applies. The subgroup types of the five cylindrical point groups are shown on the upper left part of Fig. 3.2.1.6. As explained above for two dimensions, these subgroups are not maximal and of index []. Depending upon whether the main symmetry axis has the multiplicity 4N, or , the subgroups belong to the 4Ngonal, gonal or gonal system.
The subgroup and supergroup relations within these three systems are displayed in the lower left part of Fig. 3.2.1.6. They are analogous to the crystallographic groups. To facilitate the use of the diagrams, the gonal and the gonal systems are combined, with the consequence that the five classes of the gonal system now appear with the symbols and . Again, the diagrams apply to a specified value of N. A finite number of further maximal subgroups is obtained for lower values of N, until the crystallographic groups (Fig. 3.2.1.3) are reached (cf. the twodimensional examples in Fig. 3.2.1.5).
In the previous section (Section 3.2.1), the crystallographic and noncrystallographic point groups are treated under geometrical aspects only. In the present section the pointgroup symmetries of the physical properties are considered. Among the physical properties, those represented by tensors are accessible to a mathematical treatment of their symmetries, which are apparent by the invariance of tensor components under symmetry operations. For more details the reader is referred to Nye (1957, 1985), Paufler (1986), Schwarzenbach & Chapuis (2006) and Shuvalov (1988). A similar comprehensive treatment of nontensorial properties, such as cleavage, plasticity, hardness or crystal growth does not exist, but these properties are, of course, also governed by the crystallographic pointgroup symmetry.
Neumann's principle (Neumann, 1885) describes the relation between the symmetry of a physical property and the crystallographic point group of a crystal. It states that the symmetry of any physical property of a crystal is higher than, or at least equal to, its crystallographic pointgroup symmetry, or in the language of groups, the symmetry of any physical property of a crystal is a proper or improper supergroup of its crystallographic symmetry : . This is easily illustrated for polar secondrank tensors, which are represented by ellipsoids and hyperboloids. The representation surfaces of triclinic, monoclinic and orthorhombic crystals are general ellipsoids or hyperboloids of symmetry , which is a proper supergroup of all triclinic, monoclinic and orthorhombic point groups with the only exception of the orthorhombic holohedry , for which the two symmetries are the same. For the uniaxial crystals of the trigonal, tetragonal and hexagonal systems the representation ellipsoids and hyperboloids have rotation symmetries with the cylindrical point group = (see Table 3.2.1.6), which is a proper supergroup of all uniaxial crystallographic point groups. For cubic crystals, secondrank tensors are isotropic and represented by a sphere with pointgroup symmetry (), a proper supergroup of all cubic crystallographic point groups. For more details the reader is referred to Paufler (1986) and Authier (2014). A short resume is given by Klapper & Hahn (2005).
As a consequence of the invariance of tensor components under a symmetry operation (or alternatively: under a transformation of the coordinate system to a symmetryequivalent one), some of the tensor components are equal or even zero. The number of independent components decreases when the symmetry of the crystal increases. Thus, an increase of the pointgroup symmetry from 1 (triclinic) to (cubic) or to the sphere group (i.e. isotropy) reduces the number of tensor components for symmetrical secondrank tensors from 6 to 1. Even more drastic is this reduction for all tensors of odd rank (such as pyroelectricity and piezoelectricity) or axial tensors of second rank (e.g. optical activity): all components are zero if an inversion centre is present, i.e. properties described by these tensors do not exist in centrosymmetric crystals (see the textbooks of tensor physics mentioned above). These properties, which exist only in noncentrosymmetric crystals are, as a rule, the most important ones, not only for physical applications but also for structure determination, because they allow a proof of the absence of a symmetry centre.
For the description of noncentrosymmetric crystals and their specific properties, certain notions are of importance and these are explained in the following two sections.
Curie's principle (Curie, 1894) describes the crystallographic symmetry of a macroscopic crystal which is subject to an external influence F, for example to an electric field E, to uniaxial stress σ_{ii}, to a temperature change ΔT etc. For this treatment, the pointgroup symmetries of the external influences (Curie groups) are defined as follows (see Authier, 2014, p. 11; Paufler, 1986, p. 29):

According to Curie's principle, the pointgroup symmetry of the crystal under the external field F is the intersection symmetry of the two point groups: of the crystal without field and of the field without crystal: ; i.e. is a (proper or improper) subgroup of both groups and .
As examples we consider the effect of an electric field ( ) and of a uniaxial stress () along one of the (four) threefold rotoinversion axes of cubic crystals with point groups and .
Electric field parallel to [111]:
Uniaxial stress parallel to [111]:
Note that, in contrast to uniaxial stress, the electric field destroys centrosymmetry, leading to pyroelectricity and even optical gyration. The polarity and the gyration sense are reversed upon reversal of the electric field. If the electric field and the uniaxial stress were applied to an arbitrary (nonsymmetry) direction of the above cubic crystals, point groups 1 and would result in the two cases.
If a scalar influence () (centrosymmetric sphere group) is applied to a crystal, its symmetry is not changed, provided that no phase transition is induced.
For further reading on tensor properties of crystals the textbooks in the references and International Tables for Crystallography, Volume D (2014) are recommended.
All these terms refer to the same symmetry restriction, the absence of improper rotations (rotoinversions, rotoreflections) in a crystal or in a molecule. This implies in particular the absence of a centre of symmetry, , and of a mirror plane, , but also of a axis. As a consequence, such chiral crystals or molecules can occur in two different forms, which are related as a right and a left hand; hence, they are called righthanded and lefthanded forms. These two forms of a molecule or a crystal are mirrorrelated and not superimposable (not congruent). Thus, the only symmetry operations that are allowed for chiral objects are proper rotations. Such objects are also called dissymmetric, in contrast to asymmetric objects, which have no symmetry.
The terms enantiomerism and chirality are mainly used in chemistry and applied to molecules, whereas the term enantiomorphism is preferred in crystallography if reference is made to crystals.
Enantiomorphic crystals, as well as solutions or melts containing chiral molecules of one handedness, exhibit optical activity (cf. Section 3.2.2.4.2). Crystals and molecules of the other handedness show optical activity with the opposite sense of rotation. For this reason, two molecules of opposite chirality are also called optical isomers.
Chiral molecules form enantiomorphic crystals of the corresponding handedness. These crystals belong, therefore, to one of the 11 crystal classes allowing enantiomorphism (Table 3.2.2.1). Racemic mixtures (containing equal amounts of molecules of each chirality), however, may crystallize in nonenantiomorphic or even centrosymmetric crystal classes. Racemization (i.e. the switching of molecules from one chirality to the other) of an optically active melt or solution can occur in some cases during crystallization, leading to nonenantiomorphic crystals.

Enantiomorphic crystals can also be built up from achiral molecules or atom groups. In these cases, the achiral molecules or atom groups form chiral configurations in the structure. The best known examples are quartz and NaClO_{3}. For details, reference should be made to Rogers (1975).
A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis. Only proper rotation or screw axes can be polar. The polar and nonpolar directions in the 21 noncentrosymmetric point groups are listed in Table 3.2.2.2.

The terms polar point group or polar crystal class are used in two different meanings. In crystal physics, a crystal class is considered as polar if it allows the existence of a permanent dipole moment, i.e. if it is capable of pyroelectricity (cf. Section 3.2.2.5). In crystallography, however, the term polar crystal class is frequently used synonymously with noncentrosymmetric crystal class. The synonymous use of polar and acentric, however, can be misleading, as is shown by the following example. Consider an optically active liquid. Its symmetry can be represented as a righthanded or a lefthanded sphere (cf. Sections 3.2.1.4 and 3.2.2.4). The optical activity is isotropic, i.e. magnitude and rotation sense are the same in any direction and its counterdirection. Thus, no polar direction exists, although the liquid is noncentrosymmetric with respect to optical activity.
According to Neumann's principle, information about the point group of a crystal may be obtained by the observation of physical effects. Here, the term `physical properties' includes crystal morphology and etch figures. The use of any of the techniques described below does not necessarily result in the complete determination of symmetry but, when used in conjunction with other methods, much information may be obtained. It is important to realize that the evidence from these methods is often negative, i.e. that symmetry conclusions drawn from such evidence must be considered as only provisional.
In the following sections, the physical properties suitable for the determination of symmetry are outlined briefly. For more details, reference should be made to the monographs by Bhagavantam (1966), Nye (1957) and Wooster (1973).
If a crystal shows well developed faces, information on its symmetry may be derived from the external form of the crystal. By means of the optical goniometer, faces related by symmetry can be determined even for crystals far below 1 mm in diameter. By this procedure, however, only the eigensymmetry (cf. Section 3.2.1.2.2) of the crystal morphology (which may consist of a single form or a combination of forms) can be established. The determination of the point group is unique in all cases where the observed eigensymmetry group is compatible with only one generating group.
Column 6 in Table 3.2.1.3 lists all point groups for which a given crystal form (characterized by its name and eigensymmetry) can occur. In 19 cases, the point group can be uniquely determined because only one entry appears in column 6. These crystal forms are always characteristic general forms, for which eigensymmetry and generating pointgroup symmetry are identical. They belong to the 19 point groups with more than one symmetry direction.
If a crystal exhibits a combination of forms which by themselves do not permit unambiguous determination of the point group, those generating point groups are possible that are common to all crystal forms of the combination. The mutual orientation of the forms, if variable, has to be taken into account, too.
Example
Two tetragonal pyramids, each of eigensymmetry 4mm, rotated with respect to each other by an angle , determine the point group 4 uniquely because the eigensymmetry of the combination is only 4.
In practice, however, unequal or incomplete development of the faces of a form often simulates a symmetry that is lower than the actual crystal symmetry. In such cases, or when the morphological analysis is ambiguous, the crystallization of a small amount of the compound on a seed crystal, ground to a sphere, is useful. By this procedure, faces of additional forms (and often of the characteristic general form) appear as small facets on the sphere and their interfacial angles can be measured.
In favourable cases, even the space group can be derived from the morphology of a crystal: this is based on the socalled Bravais–Donnay–Harker principle. A textbook description is given by Phillips (1971, ch. 13).
Furthermore, measurements of the interfacial angles by means of the optical goniometer permit the determination of the relative dimensions of a `morphological unit cell' with good accuracy. Thus, for instance, the interaxial angles α, β, γ and the axial ratio a:b:c of a triclinic crystal may be derived. The ratio a:b:c, however, may contain an uncertainty by an integral factor with respect to the actual cell edges of the crystal. This means that any one unit length may have to be multiplied by an integer in order to obtain correspondence to the `structural' unit cell.
Additional information on the point group of a crystal can be gained from the face symmetry, which is usually determined by observation of etch figures, striations and other face markings. Etch pits are produced by heating the crystal in vacuum (evaporation from the surface) or by attacking it with an appropriate reagent, which should not be optically active. The etch pits generally appear at the end points of dislocation lines on the face. They exhibit the symmetry of one of the ten twodimensional point groups which, in general,^{22} corresponds to the symmetry of the crystal face under investigation.
The observation of etch figures is important when the morphological analysis is ambiguous (cf. Section 3.2.2.2). For instance, a tetragonal pyramid, which is compatible with point groups 4 and 4mm, can be uniquely attributed to point group 4 if its face symmetry is found to be 1. For face symmetry m, group 4mm would result. The (oriented) face symmetries of the 47 crystal forms in the various point groups are listed in column 6 of Table 3.2.1.3 and in column 3 of Table 3.2.3.2.
In noncentrosymmetric crystals, the etch pits on parallel but opposite faces, even though they have the same symmetry, may be of different size or shape, thus proving the absence of a symmetry centre. Note that the orientation of etch pits with respect to the edges of the face is significant (cf. Buerger, 1956), as well as the mutual arrangement of etch pits on opposite faces. Thus, for a pinacoid with face symmetry 1, the possible point groups , 2 and m of the crystal can be distinguished by the mutual orientation of etch pits on the two faces. Moreover, twinning by merohedry and the true symmetry of the two (or more) twin partners (`twin domains') may be detected.
The method of etching can be applied not only to growth faces but also to cleavage faces or arbitrarily cut faces.
Optical studies provide good facilities with which to determine the symmetry of transparent crystals. The following optical properties may be used.
The dependence of the refractive index on the vibration direction of a planepolarized light wave travelling through the crystal can be obtained from the optical indicatrix. This surface is an ellipsoid, which can degenerate into a rotation ellipsoid or even into a sphere. Thus, the lowest symmetry of the property `refraction' is 2/m 2/m 2/m, the point group of the general ellipsoid. According to the three different forms of the indicatrix, three categories of crystal systems have to be distinguished (Table 3.2.2.3).

The orientation of the indicatrix is related to the symmetry directions of the crystal. In tetragonal, trigonal and hexagonal crystals, the rotation axis of the indicatrix (which is the unique optic axis) is parallel to the main symmetry axis. For orthorhombic crystals, the three principal axes of the indicatrix are oriented parallel to the three symmetry directions of the crystal. In the monoclinic system, one of the axes of the indicatrix coincides with the monoclinic symmetry direction, whereas in the triclinic case, the indicatrix can, in principle, have any orientation relative to a chosen reference system. Thus, in triclinic and, with restrictions, in monoclinic crystals, the orientation of the indicatrix can change with wavelength λ and temperature T (orientation dispersion). In any system, the size of the indicatrix and, in all but the cubic system, its shape can also vary with λ and T.
When studying the symmetry of a crystal by optical means, note that strain can lower the apparent symmetry owing to the high sensitivity of optical properties to strain.
The symmetry information obtained from optical activity is quite different from that given by optical refraction. Optical activity is in principle confined to the 21 noncentrosymmetric classes but it can occur in only 15 of them (Table 3.2.2.1). In the 11 enantiomorphism classes, a single crystal is either right or lefthanded. In the four nonenantiomorphous classes and , optical activity may also occur; here directions of both right and lefthanded rotations of the plane of polarization exist in the same crystal. In the other six noncentrosymmetric classes, , , optical activity is not possible.
In the two cubic enantiomorphous classes 23 and 432, the optical activity is isotropic and can be observed along any direction.^{23} For the other optically active crystals, the rotation of the plane of polarization can, in practice, be detected only in directions parallel (or approximately parallel) to the optic axes. This is because of the dominating effect of double refraction. No optical activity, however, is present along an inversion axis or along a direction parallel or perpendicular to a mirror plane. Thus, no activity occurs along the optic axis in crystal classes and . In classes m and mm2, no activity can be present along the two optic axes if these axes lie in m. If they are not parallel to m, they show optical rotation(s) of opposite sense.
Light waves passing through a noncentrosymmetric crystal induce new waves of twice the incident frequency. This secondharmonic generation is due to the nonlinear optical susceptibility. The secondharmonic coefficients form a thirdrank tensor, which is subject to the same symmetry constraints as the piezoelectric tensor (see Section 3.2.2.6). Thus, only 20 noncentrosymmetric crystals, except those of class 432, can show the secondharmonic effect; cf. Table 3.2.2.1.
Secondharmonic generation is a powerful method of testing crystalline materials for the absence of a symmetry centre. With an appropriate experimental device, very small amounts (less than 10 mg) of powder are sufficient to detect the secondharmonic signals, even for crystals with small deviations from centrosymmetry (Dougherty & Kurtz, 1976).
In principle, pyroelectricity can only exist in crystals with a permanent electric dipole moment. This moment is changed by heating and cooling, thus giving rise to electric charges on certain crystal faces, which can be detected by simple experimental procedures.
An electric dipole moment can be present only along a polar direction that has no symmetryequivalent directions.^{24} Such polar directions occur in the following ten classes: 6mm, 4mm, and their subgroups 6, 4, 3m, 3, mm2, 2, m, 1 (cf. Table 3.2.2.1). In point groups with a rotation axis, the electric moment is along this axis. In class m, the electric moment is parallel to any direction in the mirror plane (direction [u0w]). In class 1, any direction [uvw] is possible. In point groups 1 and m, besides a change in magnitude, a directional variation of the electric moment can also occur during heating or cooling.
In practice, it is difficult to prevent strains from developing throughout the crystal as a result of temperature gradients in the sample. This gives rise to piezoelectrically induced charges superposed on the true pyroelectric effect. Consequently, when the development of electric charges by a change in temperature is observed, the only safe deduction is that the specimen must lack a centre of symmetry. Failure to detect pyroelectricity may be due to extreme weakness of the effect, although modern methods are very sensitive.
A crystal is ferroelectric if the direction of the permanent electric dipole moment can be changed by an electric field. Thus, ferroelectricity can only occur in the ten pyroelectric crystal classes, mentioned above.
In piezoelectric crystals, an electric dipole moment can be induced by compressional and torsional stress. For a uniaxial compression, the induced moment may be parallel, normal or inclined to the compression axis. These cases are called longitudinal, transverse or mixed compressional piezoeffect, respectively. Correspondingly, for torsional stress, the electric moment may be parallel, normal or inclined to the torsion axis.
The piezoelectricity is described by a thirdrank tensor, the moduli of which vanish for all centrosymmetric point groups. Additionally, in class 432, all piezoelectric moduli are zero owing to the high symmetry. Thus, piezoelectricity can only occur in 20 noncentrosymmetric crystal classes (Table 3.2.2.1).
The piezoelectric point groups 422 and 622 show the following peculiarity: there is no direction for which a longitudinal component of the electric moment is induced under uniaxial compression. Thus, no longitudinal or mixed compressional effects occur. The moment is always normal to the compression axis (pure transverse compressional effect). This means that, with the compression pistons as electrodes, no electric charges can be found, since only transverse compressional or torsional piezoeffects occur. In all other piezoelectric classes, there exist directions in which both longitudinal and transverse components of the electric dipole moment are induced under uniaxial compression.
An electric moment can also develop under hydrostatic pressure. This kind of piezoelectricity, like pyroelectricity, can be represented by a firstrank tensor (vector), whereby the hydrostatic pressure is regarded as a scalar. Thus, piezoelectricity under hydrostatic pressure is subject to the same symmetry constraints as pyroelectricity.
Like `secondharmonic generation' (Section 3.2.2.4.3), the piezoelectric effect is very useful for testing crystals for the absence of a symmetry centre. There exist powerful methods for testing powder samples or even small single crystals. In the old technique of Giebe & Scheibe (cf. Wooster & Brenton, 1970), the absorption and emission of radiofrequency energy by electromechanical oscillations of piezoelectric particles are detected. In the more modern method of observing `polarization echoes', radiofrequency pulses are applied to powder samples. By this procedure, electromechanical vibration pulses are induced in piezoelectric particles, the echoes of which can be detected (cf. Melcher & Shiren, 1976).
The crystallographic pointgroup types are listed in Tables 3.2.3.1 and 3.2.3.2 for twodimensional and for threedimensional space, respectively. No listings are presented for the noncrystallographic pointgroup types (i.e. having axes of orders other than 1, 2, 3, 4 and 6), but their symbols can be found in Tables 3.2.1.5, 3.2.1.6 and 3.2.3.3 (cf. Section 3.2.1.4 for a review of noncrystallographic point groups). The two icosahedral point groups 235 and are treated in detail in Section 3.2.1.4.2, while their crystallographic data are shown in Table 3.2.3.3.

