Section 10.1.4. Noncrystallographic point groups

In Sections 10.1.2 and 10.1.3, 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 10.1.4.1 to 10.1.4.3.

Table 10.1.4.1| top | pdf | Classes of general point groups in two dimensions (N = integer 0) General Hermann–Mauguin symbol Order of group General edge form General point form Crystallographic groups 4N-gonal system (n-fold rotation point with ) n n Regular n-gon Regular n-gon 4 nmm 2n Semiregular di-n-gon Truncated n-gon 4mm -gonal system (n-fold or -fold rotation point with ) Regular -gon Regular -gon 1, 3 n Semiregular di--gon Truncated -gon m, 3m n n Regular n-gon Regular n-gon 2, 6 nmm 2n Semiregular di-n-gon Truncated n-gon 2mm, 6mm Circular system† ∞ ∞ Rotating circle Rotating circle – ∞m ∞ Stationary circle Stationary circle – †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.

Table 10.1.4.2| top | pdf | Classes of general point groups in three dimensions (N = integer 0) Short general Hermann–Mauguin symbol, followed by full symbol where different Schoenflies symbol Order of group General face form General point form Crystallographic groups 4N-gonal system (single n-fold symmetry axis with ) n n n-gonal pyramid Regular n-gon 4 n -gonal streptohedron -gonal antiprism 2n n-gonal dipyramid n-gonal prism n22 2n n-gonal trapezohedron Twisted n-gonal antiprism 422 nmm 2n Di-n-gonal pyramid Truncated n-gon 4mm 2n n-gonal scalenohedron -gonal antiprism sliced off by pinacoid 4n Di-n-gonal dipyramid Edge-truncated n-gonal prism -gonal system (single n-fold symmetry axis with ) n n n-gonal pyramid Regular n-gon 1, 3 2n n-gonal streptohedron n-gonal antiprism n2 2n n-gonal trapezohedron Twisted n-gonal antiprism 32 nm 2n Di-n-gonal pyramid Truncated n-gon 3m 4n Di-n-gonal scalenohedron n-gonal antiprism sliced off by pinacoid -gonal system (single n-fold symmetry axis with ) n n n-gonal pyramid Regular n-gon 2, 6 n -gonal dipyramid -gonal prism 2n n-gonal dipyramid n-gonal prism n22 2n n-gonal trapezohedron Twisted n-gonal antiprism 222, 622 nmm 2n Di-n-gonal pyramid Truncated n-gon mm2, 6mm 2n Di--gonal dipyramid Truncated -gonal prism 4n Di-n-gonal dipyramid Edge-truncated n-gonal prism mmm, Cubic system (for details see Table 10.1.2.2) 23 T 12 Pentagon-tritetrahedron Snub tetrahedron 23 24 Didodecahedron Cube & octahedron & pentagon-dodecahedron 432 O 24 Pentagon-trioctahedron Snub cube 432 24 Hexatetrahedron Cube truncated by two tetrahedra 48 Hexaoctahedron Cube truncated by octahedron and by rhomb-dodecahedron Icosahedral system† (for details see Table 10.1.4.3) 235 I 60 Pentagon-hexecontahedron Snub pentagon-dodecahedron – 120 Hecatonicosahedron Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron – Cylindrical system‡ ∞ ∞ Rotating cone Rotating circle – ∞ Rotating double cone Rotating finite cylinder – ∞2 ∞ `Anti-rotating' double cone `Anti-rotating' finite cylinder – ∞m ∞ Stationary cone Stationary circle – ∞ Stationary double cone Stationary finite cylinder – Spherical system§ K ∞ Rotating sphere Rotating sphere – ∞ Stationary sphere Stationary sphere – †The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and (see text). ‡Rotating and `anti-rotating' 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: One single rotating cone can be regarded as a right-handed or left-handed screw, depending 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 `anti-rotating' 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 `anti-rotating' 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 right-handed or left-handed senses of rotation around the axes (cf. Section 10.2.4 , 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.

Table 10.1.4.3| top | pdf | The two icosahedral point groups General, special and limiting face forms and point forms, oriented face- and site-symmetry symbols, and `initial' values of (hkl) and x, y, z (see text). 235 I   60 d 1 Pentagon-hexecontahedron (hkl)       Snub pentagon-dodecahedron (= pentagon-dodecahedron + icosahedron + pentagon-hexecontahedron) x, y, z       30 c 2..           20 b .3. 12 a ..5 Symmetry of special projections       120 e l Hecatonicosahedron or hexaicosahedron (hkl)       Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron x, y, z 60 d m.. 30 c 2mm.. Rhomb-triacontahedron (100)       Icosadodecahedron (= pentagon-dodecahedron icosahedron) x, 0, 0 20 b 3m (m3.) Regular icosahedron (111)       Regular pentagon-dodecahedron x, x, x 12 a 5m (m.5) Symmetry of special projections

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 4N-gonal, -gonal and -gonal system, each general class contains infinitely many point groups, which differ in their principal n-fold symmetry axis, with for the 4N-gonal 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 infinity¹⁴ (∞-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. Section 2.2.4 and Chapter 12.1 ). This extends also to the infinite groups where symbols like or are immediately obvious.

In two dimensions (Table 10.1.4.1), the eight general classes are collected into three systems. Two of these, the 4N-gonal and the -gonal systems, contain only point groups of finite order with one n-fold 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.

In three dimensions (Table 10.1.4.2), the 28 general classes are collected into seven systems. Three of these, the 4N-gonal, the -gonal and the -gonal systems,¹⁵ contain only point groups of finite order with one principal n-fold symmetry axis each. These systems are generalizations of the tetragonal, trigonal, and hexagonal crystal systems (cf. Table 10.1.1.2). 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.

It is possible to define the three-dimensional 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 10.1.4.1 and 10.1.4.2 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.¹⁶ 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 10.1.2.1 and 10.1.2.2 for the crystallographic groups.¹⁷

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 pentagon-dodecahedron (12 faces, 20 vertices).

The icosahedral groups (cf. diagrams in Table 10.1.4.3) 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 10.1.4.3, in a form similar to the cubic point groups in Table 10.1.2.2. 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 site-symmetry 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 10.1.2.4, 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 x, y, z for each type of form are listed. The complete sets of indices and coordinates can be obtained in two steps¹⁸ 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 10.1.2.2.

The matrices¹⁹ are with²⁰ These matrices correspond to counter-clockwise 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 pentagon-dodecahedra 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. 10.1.4.1 to 10.1.4.3, the subgroup and supergroup relations between the two-dimensional and three-dimensional 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.

Figure 10.1.4.1| top | pdf | Subgroups and supergroups of the two-dimensional 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. 10.1.4.2.

Figure 10.1.4.2| top | pdf | The subgroups of the two-dimensional general point groups 16mm (4N-gonal system) and 18mm [-gonal system, including the -gonal groups]. Compare with Fig. 10.1.4.1 which applies only to one value of N.

Figure 10.1.4.3| top | pdf | Subgroups and supergroups of the three-dimensional 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. Full Hermann–Mauguin symbols are used.

Fig. 10.1.4.1 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 4N-gonal 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 4N-gonal 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. 10.1.3.1) are reached. This is illustrated for the case in Fig. 10.1.4.2.

Fig. 10.1.4.3 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 10.1.4.2, The two icosahedral groups); they are listed on the right of Fig. 10.1.4.3. For crystallographic groups, Fig. 10.1.3.2 applies. The subgroup types of the five cylindrical point groups are shown on the left of Fig. 10.1.4.3. 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 4N-gonal, -gonal or -gonal system.

The subgroup and supergroup relations within these three systems are displayed in the lower left part of Fig 10.1.4.3. 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. 10.1.3.2) are reached (cf. the two-dimensional examples in Fig. 10.1.4.2).