Crystallographic and noncrystallographic point groups

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000520

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 10.1, pp. 762-803
https://doi.org/10.1107/97809553602060000520

Chapter 10.1. Crystallographic and noncrystallographic point groups

Th. Hahn^a ^* and H. Klapper^a ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtl.rwth-aachen.de

Chapter 10.1 treats the geometric and group-theoretical aspects of both crystallographic and noncrystallographic point groups. The central part of the chapter is an extensive tabulation of the 10 two-dimensional and the 32 three-dimensional 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 names and Miller indices {hkl} of the general and special face and point forms. Each point-group table concludes with the three major projection symmetries. The section continues with descriptions and diagrams of the sub- and supergroups of the crystallographic point groups and with 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.

Keywords: crystallographic point groups; noncrystallographic point groups; point groups; crystal systems; stereographic projections; crystal forms; point forms; Wyckoff positions; eigensymmetries; crystal classes; subgroups; supergroups; icosahedral point groups.

10.1.1. Introduction and definitions

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A point group¹ 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 $[\overline{1},\ \overline{2} = m, \overline{3}, \overline{4}, \overline{6}]$ (cf. Chapter 1.3 ); matrices for these symmetry operations are listed in Tables 11.2.2.1 and 11.2.2.2 . 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 $[n \geq 2]$ are infinite. The two- and three-dimensional crystallographic point groups and their crystal systems are summarized in Tables 10.1.1.1 and 10.1.1.2. They are described in detail in Section 10.1.2 . The two one-dimensional point groups are discussed in Section 2.2.17 . The noncrystallographic point groups are treated in Section 10.1.4 .

Table 10.1.1.1| top | pdf |
The ten two-dimensional crystallographic point groups, arranged according to crystal system

The horizontal rule extending across the columns `Square' and `Hexagonal' separates point groups with different Laue classes within one crystal system.

General symbol	Crystal system
General symbol	Oblique (top) Rectangular (bottom)		Square	Hexagonal
n	1	2	4	3	6
nmm	m	2mm	4mm	3m	6mm

Table 10.1.1.2| top | pdf |
The 32 three-dimensional crystallographic point groups, arranged according to crystal system (cf. Chapter 2.1 )

Full Hermann–Mauguin (left) and Schoenflies symbols (right). The horizontal rule extending across the columns 'Tetragonal' to 'Cubic' separates point groups with different Laue classes within one crystal system.

General symbol	Crystal system
General symbol	Triclinic		Monoclinic (top) Orthorhombic (bottom)		Tetragonal		Trigonal		Hexagonal		Cubic
n	1	$[C_{1}]$	2	$[C_{2}]$	4	$[C_{4}]$	3	$[C_{3}]$	6	$[C_{6}]$	23	T
$[\overline{n}]$	$[\overline{1}]$	$[C_{i}]$	$[m \equiv \overline{2}]$	$[C_{s}]$	$[\overline{4}]$	$[S_{4}]$	$[\overline{3}]$	$[C_{3{i}}]$	$[\overline{6} \equiv 3/m]$	$[C_{3h}]$	–	–
				$[C_{2h}]$		$[C_{4h}]$	–	–		$[C_{6h}]$	$[2/m\overline{3}]$	$[T_{h}]$
n22			222	$[D_{2}]$	422	$[D_{4}]$	32	$[D_{3}]$	622	$[D_{6}]$	432	O
nmm			mm2	$[C_{2v}]$	4mm	$[C_{4v}]$	3m	$[C_{3v}]$	6mm	$[C_{6v}]$	–	–
$[\overline{n}2m]$			–	–	$[\overline{4}2m]$	$[D_{2d}]$	$[\overline{3}2/m]$	$[D_{3d}]$	$[\overline{6}2m]$	$[D_{3h}]$	$[\overline{4}3m]$	$[T_{d}]$
$[n/m\; 2/m\; 2/m]$			$[2/m\; 2/m\; 2/m]$	$[D_{2h}]$	$[4/m\; 2/m\; 2/m]$	$[D_{4h}]$	–	–	$[6/m\; 2/m\; 2/m]$	$[D_{6h}]$	$[4/m\; \overline{3}\; 2/m]$	$[O_{h}]$

Crystallographic point groups occur:

(i) in vector space as symmetries of the external shapes of crystals, i.e. of the set of vectors normal to the crystal faces (morphological symmetry);
(ii) in point space as site symmetries of points in lattices or in crystal structures and as symmetries of atomic groups and coordination polyhedra.

General point groups , i.e. crystallographic and noncrystallographic point groups, occur as:

(iii) symmetries of (rigid) molecules (molecular symmetry):
(iv) symmetries of physical properties of crystals (e.g. tensor symmetries); here noncrystallographic point groups with axes of order infinity are of particular importance, as in the symmetries of spheres or rotation ellipsoids:
(v) approximate symmetries of the local environment of a point in a crystal structure, i.e. as local site symmetries. Examples are sphere-like atoms or ions in crystals, as well as icosahedral atomic groups. These noncrystallographic symmetries, however, are only approximate, even for the close neighbourhood of a site.

A (geometric) crystal class is the set of all crystals having the same point-group symmetry. The word `class', therefore, denotes a classificatory pigeon-hole 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 8.2.3 and 8.2.4 .)

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: $[\overline{1}]$ ; [2/m] ; mmm; [4/m] ; [4/mmm] ; $[\overline{3}]$ ; $[\overline{3}m]$ ; [6/m] ; [6/mmm] ; $[m\overline{3}]$ ; $[m\overline{3}m]$ . 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 Part 3 . For a given crystal, its Laue class is obtained if a symmetry centre is added to its point group, as shown in Table 10.2.1.1 .

In two dimensions, six `centrosymmetric' crystallographic point groups and hence six two-dimensional Laue classes exist: 2; 2mm; 4; 4mm; 6; 6mm. These point groups are, for instance, the only possible symmetries of zero-layer X-ray photographs.

Among the centrosymmetric crystallographic point groups in three dimensions, the lattice point groups (holohedral point groups, holohedries) are of special importance because they constitute the seven possible point symmetries of lattices, i.e. the site symmetries of their nodes. In three dimensions, the seven holohedries are: $[\overline{1}]$ ; [2/m] ; mmm; [4/mmm] ; $[\overline{3}m]$ ; [6/mmm] ; $[m\overline{3}m]$ . Note that $[\overline{3}m]$ is the point symmetry of the rhombohedral lattice and [6/mmm] 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: $[\matrix{\hbox{Order }{\hbox to 1pt{}} 1:1\hfill &\hbox{Order }\phantom{\;} 8:422, 4mm, \overline{4}2m\hfil\cr\phantom{Order\ } 2:\overline{1},2,m \hfill&\phantom{Order\ 1} 12:6/m\hfill\cr\phantom{Order\ } 3:3 \hfill&\phantom{Order\ 1} 12:\overline{3}m, 622, 6mm, \overline{6}2m\hfill\cr\phantom{Order\ } 4:2/m, 222, mm2 \hfill&\phantom{Order\ 1} 12:23\hfill\cr\phantom{Order\ } 4:4, \overline{4} \hfill&\phantom{Order\ 1} 16:4/mmm\hfill\cr\phantom{Order\ } 6:\overline{3}, 6, \overline{6} \hfill&\phantom{Order\ 1} 24:6/mmm\hfill\cr\phantom{Order\ } 6:32, 3m \hfill&\phantom{Order\ 1} 24:m\overline{3}\hfill\cr\phantom{Order\ } 8:mmm \hfill&\phantom{Order\ 1} 24:432, \overline{4}3m\hfill\cr\phantom{Order\ } 8:4/m \hfill&\phantom{Order\ 1} 48:m\overline{3}m.\hfill\cr}]$ 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.

10.1.2. Crystallographic point groups

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10.1.2.1. Description of point groups

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In crystallography, point groups usually are described

(i) by means of their Hermann–Mauguin or Schoenflies symbols;
(ii) by means of their stereographic projections;
(iii) by means of the matrix representations of their symmetry operations , frequently listed in the form of Miller indices (hkl) of the equivalent general crystal faces;
(iv) by means of drawings of actual crystals, natural or synthetic.

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; this also applies to the construction and the properties of the stereographic projection.

In Tables 10.1.2.1 and 10.1.2.2 , the two- and three-dimensional 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 space-group tables of this volume: pure rotation groups are followed by groups containing reflections, rotoinversions and inversions. The holohedral point group is always given last.

Table 10.1.2.1| top | pdf |
The ten two-dimensional crystallographic point groups

General, special and limiting edge forms and point forms (italics), oriented edge and site symmetries , and Miller indices (hk) of equivalent edges [for hexagonal groups Bravais–Miller indices (hki) are used if referred to hexagonal axes]; for point coordinates see text.

OBLIQUE SYSTEM
1
1	a	1	Single edge	(hk)
			Single point (a)
2
2	a	1	Pair of parallel edges	$[\matrix{(hk) &(\bar{h}\bar{k})\cr}]$
			Line segment through origin (e)
RECTANGULAR SYSTEM
m
2	b	1	Pair of edges	$[\matrix{(hk) &(\bar{h}k)\cr}]$
			Line segment (c)
			Pair of parallel edges	$[\matrix{(10) &(\bar{1}0)\cr}]$
			Line segment through origin
1	a	.m.	Single edge	(01) or $[(0\bar{1})]$
			Single point (a)
2mm
4	c	1	Rhomb	$[\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{h}k) &(h\bar{k})\cr}]$
			Rectangle (i)
2	b	.m.	Pair of parallel edges	$[\matrix{(01) &(0\bar{1})\cr}{\hbox to 4.6pc{}}]$
			Line segment through origin (g)
2	a	..m	Pair of parallel edges	$[\matrix{(10) &(\bar{1}0)\cr}{\hbox to 4.6pc{}}]$
			Line segment through origin (e)
SQUARE SYSTEM
4
4	a	1	Square	$[\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{k}h) &(k\bar{h})\cr}]$
			Square (d)
4mm
8	c	1	Ditetragon	$[\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{k}h) &(k\bar{h})\cr}]$
			Truncated square (g)	$[\matrix{(\bar{h}k) &(h\bar{k}) &(kh) &(\bar{k}{\hbox to .5pt{}}\bar{h})\cr}]$
4	b	..m	Square	$[\matrix{(11) &(\bar{1}\bar{1}) &(\bar{1}1) &(1\bar{1})\cr}]$
			Square (f)
4	a	.m.	Square	$[\matrix{(10) &(\bar{1}0) &(01) &(0\bar{1})\cr}]$
			Square (d)
HEXAGONAL SYSTEM
3
3	a	1	Trigon	$[\matrix{(hki) &(ihk) &(kih)\cr}]$
			Trigon (d)
3m1
6	b	1	Ditrigon	$[{\hbox to 9pt{}}\matrix{(hki) &{\hbox to 2pt{}}(ihk) &{\hbox to 3pt{}}(kih)\hfill\cr}]$
			Truncated trigon (e)	$[{\hbox to 12pt{}}\matrix{(\bar{k}\bar{h}\bar{i\hskip-2pt\phantom l}) &{\hbox to 2pt{}}(\bar{\phantom l\hskip-2.5pt i}\bar{k}\bar{h}) &{\hbox to 3pt{}}(\bar{h}\bar{\hskip-2.5pt\phantom li}\bar{k})\cr}]$
			Hexagon	$[{\hbox to 11pt{}}\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]$
			Hexagon	$[{\hbox to 11pt{}}\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]$
3	a	.m.	Trigon	$[{\hbox to 11.5pt{}}\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]$
			Trigon (d)	or $[\matrix{(\bar{1}01) &(1\bar{1}0) &(01\bar{1})\cr}]$
31m
6	b	1	Ditrigon	$[{\hbox to 8pt{}}\matrix{(hki) {\hbox to 13pt{}}(ihk) {\hbox to 12pt{}}(kih)\cr}]$
			Truncated trigon (d)	$[{\hbox to 8pt{}}\matrix{(khi) {\hbox to 13pt{}}(ikh) {\hbox to 12pt{}}(hik)\cr}]$
			Hexagon	$[{\hbox to 11pt{}}\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]$
			Hexagon	$[{\hbox to 11pt{}}\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]$
3	a	..m	Trigon	$[{\hbox to 11pt{}}\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]$
			Trigon (c)	or $[\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]$
6
6	a	1	Hexagon	$[\matrix{(hki) &(ihk) &(kih)\cr}]$
			Hexagon (d)	$[\matrix{(\bar{h}\bar{k}\bar{\phantom l\hskip-2.5pti}) &(\bar{\phantom l\hskip-2.5pti}\bar{h}\bar{k}) &(\bar{k}\bar{\phantom l\hskip-2.5pti}\bar{h})\cr}]$
6mm
12	c	1	Dihexagon	$[\matrix{(hki){\hbox to 2.5pt{}}& (ihk)&{\hbox to 2.5pt{}} (kih)}]$
			Truncated hexagon (f)	$[\matrix{(\bar{h}\bar{k}\bar{\phantom l\hskip-2.5pti}){\hbox to 2pt{}} &(\bar{\phantom l\hskip-2.5pti}\bar{h}\bar{k})\hfill &{\hbox to 2pt{}}(\bar{k}\bar{\phantom l\hskip-2.5pti}\bar{h})\cr}\hfill]$
				$[\matrix{(\bar{k}\bar{h}\bar{\phantom l\hskip-2.5pti}){\hbox to 2pt{}}&(\bar{\phantom l\hskip-2.5pt i}\bar{k}\bar{h}) &{\hbox to 2pt{}}(\bar{h}\bar{\phantom l\hskip-2.5pti}\bar{k})\cr}{\hbox to 2pt{}}]$
				$[\matrix{(khi){\hbox to 2.5pt{}} &(ikh) &{\hbox to 2.5pt{}}(hik)\cr\hfill}]$
6	b	.m.	Hexagon	$[\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]$
			Hexagon (e)	$[\matrix{(\bar{1}01) &(1\bar{1}0) &(01\bar{1})\cr}]$
6	a	..m	Hexagon	$[\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]$
			Hexagon (d)	$[\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]$

Table 10.1.2.2| top | pdf |
The 32 three-dimensional crystallographic point groups

General, special and limiting face forms and point forms (italics), oriented face and site symmetries , and Miller indices (hkl) of equivalent faces [for trigonal and hexagonal groups Bravais–Miller indices (hkil) are used if referred to hexagonal axes]; for point coordinates see text.

TRICLINIC SYSTEM
1	$[C_{1}]$
1	a	1	Pedion or monohedron	(hkl)
			Single point (a)
Symmetry of special projections
Along any direction
1
$[\bar{1}]$	$[C_{i}]$
2	a	1	Pinacoid or parallelohedron	$[\matrix{(hkl) &(\bar{h}\bar{k}\bar{l})\cr}]$
			Line segment through origin (i)
Symmetry of special projections
Along any direction
2
MONOCLINIC SYSTEM
2	$[C_{2}]$
				Unique axis b	Unique axis c
2	b	1	Sphenoid or dihedron	$[\matrix{(hkl) &(\bar{h}k\bar{l})\cr}]$	$[\matrix{(hkl) &(\bar{h}\bar{k}l)\cr}]$
			Line segment (e)
			Pinacoid or parallelohedron	$[\matrix{(h0l) &(\bar{h}0\bar{l})\cr}]$	$[\;\;\matrix{(hk0) &(\bar{h}\bar{k}0)\cr}]$
			Line segment through origin
1	a	2	Pedion or monohedron	$[\quad(010) \hbox{ or } (0\bar{1}0)]$	$[\;\quad(001) \hbox{ or } (00\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{&&&&\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr \hbox{Unique axis } b&&&&m&&2&&m\cr}c&&&&m&&m&&2}]$
m	$[C_{s}]$
				Unique axis b	Unique axis c
2	b	1	Dome or dihedron	$[\matrix{\;(hkl) &\;(h\bar{k}l)\cr}]$	$[\matrix{\;(hkl) &\;(hk\bar{l})\cr}]$
			Line segment (c)
			Pinacoid or parallelohedron	$[\;\;\matrix{(010) &(0\bar{1}0)\cr}]$	$[\;\;\matrix{(001) &(00\bar{1})\cr}]$
			Line segment through origin
1	a	m	Pedion or monohedron	(h0l)	(hk0)
			Single point (a)
Symmetry of special projections
$[\!\matrix{&&&&{\hbox to 2pt{}}\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr \hbox{Unique axis } b&&&&{\hbox to 2pt{}}m&&1&&m\cr}c&&&&{\hbox to 2pt{}}m&&m&&1\cr}]$
	$[C_{2h}]$
				Unique axis b	Unique axis c
4	c	1	Rhombic prism	$[{\hbox to 2pc{}}\matrix{(hkl) &(\bar{h}k\bar{l}) &(\bar{h}\bar{k}\bar{l}) &(h\bar{k}l)\cr}]$	$[{\hbox to 2pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}\bar{k}\bar{l}) &(hk\bar{l})\cr}]$
			Rectangle through origin (o)
2	b	m	Pinacoid or parallelohedron	$[{\hbox to 2pc{}}\matrix{(h0l) &(\bar{h}0\bar{l})\cr}]$	$[{\hbox to 2pc{}}\matrix{(hk0) {\hbox to 7.5pt{}}(\bar{h}\bar{k}0)\cr}]$
			Line segment through origin (m)
2	a	2	Pinacoid or parallelohedron	$[{\hbox to 2pc{}}\matrix{(010) {\hbox to 7.5pt{}}(0\bar{1}0)\cr}]$	$[{\hbox to 2pc{}}\matrix{(001) {\hbox to 7pt{}}(00\bar{1})\cr}]$
			Line segment through origin (i)
Symmetry of special projections
$[\matrix{&&&{\hbox to 5pt{}}\hbox{Along } [100]&&&\hbox{Along } [010]&&&\hbox{Along }[001]\cr \hbox{Unique axis }b&&&{\hbox to 5pt{}}2mm&&&2&&&2mm\cr \hbox{}c&&&{\hbox to 5pt{}}2mm&&&2mm&&&2}]$
ORTHORHOMBIC SYSTEM
222	$[D_{2}]$
4	d	1	Rhombic disphenoid or rhombic tetrahedron	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]$
			Rhombic tetrahedron (u)
			Rhombic prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to 7.5pt{}}(\bar{h}\bar{k}0) {\hbox to 7.5pt{}}(\bar{h}k0) {\hbox to 9pt{}}(h\bar{k}0)\cr}]$
			Rectangle through origin
			Rhombic prism	$[{\hbox to 8pc{}}\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]$
			Rectangle through origin
			Rhombic prism	$[{\hbox to 8pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]$
			Rectangle through origin
2	c	..2	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(001) {\hbox to 7.5pt{}}(00\bar{1})\cr}]$
			Line segment through origin (q)
2	b	.2.	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(010) {\hbox to 7.5pt{}}(0\bar{1}0)\cr}]$
			Line segment through origin (m)
2	a	2..	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(100) {\hbox to 7.5pt{}}(\bar{1}00)\cr}]$
			Line segment through origin (i)
Symmetry of special projections
$[\matrix{\hbox{Along }[100]&\hbox{Along }[010]&\hbox{Along }[001]\cr 2mm&2mm&2mm\cr}]$
mm2	$[C_{2v}]$
4	d	1	Rhombic pyramid	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(h\bar{k}l) &(\bar{h}kl)\cr}]$
			Rectangle (i)
			Rhombic prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0)\cr}]$
			Rectangle through origin
2	c	m..	Dome or dihedron	$[{\hbox to 8pc{}}\matrix{(0kl) &(0\bar{k}l)\cr}]$
			Line segment (g)
			Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]$
			Line segment through origin
2	b	.m.	Dome or dihedron	$[{\hbox to 8pc{}}\matrix{(h0l) &(\bar{h}0l)\cr}]$
			Line segment (e)
			Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]$
			Line segment through origin
1	a	mm2	Pedion or monohedron	$[{\hbox to 8pc{}}(001) \hbox{ or } (00\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr m&&m&&2mm\cr}]$
$[\openup 3pt\matrix{m\ m\ m\cr{\displaystyle{2 \over m}\ {2 \over m}\ {2 \over m}}\cr}]$	$[D_{2h}]$
8	g	1	Rhombic dipyramid	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]$
			Quad (α)	$[{\hbox to 8pc{}}\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl)\cr}]$
4	f	..m	Rhombic prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0)\cr}]$
			Rectangle through origin (y)
4	e	.m.	Rhombic prism	$[{\hbox to 8pc{}}\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]$
			Rectangle through origin (w)
4	d	m..	Rhombic prism	$[{\hbox to 8pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]$
			Rectangle through origin (u)
2	c	mm2	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (q)
2	b	m2m	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]$
			Line segment through origin (m)
2	a	2mm	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]$
			Line segment through origin (i)
Symmetry of special projections
$[\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr 2mm&&2mm&&2mm\cr}]$
TETRAGONAL SYSTEM
4	$[C_{4}]$
4	b	1	Tetragonal pyramid	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]$
			Square (d)
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]$
			Square through origin
1	a	4..	Pedion or monohedron	$[{\hbox to 8.1pc{}}(001) \hbox{ or } (00\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&m&&m\cr}]$
$[\bar{4}]$	$[S_{4}]$
4	b	1	Tetragonal disphenoid or tetragonal tetrahedron	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]$
			Tetragonal tetrahedron (h)
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(k\bar{h}0) {\hbox to .65pc{}}(\bar{k}h0)\cr}]$
			Square through origin
2	a	2..	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&m&&m\cr}]$
	$[C_{4h}]$
8	c	1	Tetragonal dipyramid	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]$
			Tetragonal prism (l)	$[{\hbox to 8pc{}}\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]$
4	b	m..	Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]$
			Square through origin (j)
2	a	4..	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&2mm&&2mm\cr}]$
422	$[D_{4}]$
8	d	1	Tetragonal trapezohedron	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]$
			Twisted tetragonal antiprism (p)	$[{\hbox to 8pc{}}\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]$
			Ditetragonal prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]$
			Truncated square through origin	$[{\hbox to 8pc{}}\matrix{(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .65pc{}}(kh0) {\hbox to .65pc{}}(\bar{k}\bar{h}0)\cr}]$
			Tetragonal dipyramid	$[{\hbox to 8pc{}}\matrix{(h0l) {\hbox to .8pc{}}(\bar{h}0l) {\hbox to .8pc{}}(0hl) {\hbox to .8pc{}}(0\bar{h}l)\cr}]$
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(\bar{h}0\bar{l}) {\hbox to .8pc{}}(h0\bar{l}) {\hbox to .8pc{}}(0h\bar{l}) {\hbox to .8pc{}}(0\bar{h}\bar{l})\cr}]$
			Tetragonal dipyramid	$[{\hbox to 8pc{}}\matrix{(hhl) {\hbox to .8pc{}}(\bar{h}\bar{h}l) {\hbox to .8pc{}}(\bar{h}hl) {\hbox to .8pc{}}(h\bar{h}l)\cr}]$
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(\bar{h}h\bar{l}) {\hbox to .8pc{}}(h\bar{h}\bar{l}) {\hbox to .8pc{}}(hh\bar{l}) {\hbox to .8pc{}}(\bar{h}\bar{h}\bar{l})\cr}]$
4	c	.2.	Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .6pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]$
			Square through origin (l)
4	b	..2	Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .6pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]$
			Square through origin (j)
2	a	4..	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&2mm&&2mm\cr}]$
4mm	$[C_{4v}]$
8	d	1	Ditetragonal pyramid	$[{\hbox to 8pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]$
			Truncated square (g)	$[{\hbox to 8pc{}}\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]$
			Ditetragonal prism	$[{\hbox to 8pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]$
			Truncated square through origin	$[{\hbox to 8pc{}}\matrix{(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(\bar{k}\bar{h}0) {\hbox to .65pc{}}(kh0)\cr}]$
4	c	.m.	Tetragonal pyramid	$[{\hbox to 8pc{}}\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]$
			Square (e)
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .65pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]$
			Square through origin
4	b	..m	Tetragonal pyramid	$[{\hbox to 8pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]$
			Square (d)
			Tetragonal prism	$[{\hbox to 8pc{}}\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .65pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]$
			Square through origin
1	a	4mm	Pedion or monohedron	$[{\hbox to 8.15pc{}}(001) \hbox{ or } (00\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&m&&m\cr}]$
$[\bar{4}2m]$	$[D_{2d}]$
8	d	1	Tetragonal scalenohedron	$[{\hbox to 7pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]$
			Tetragonal tetrahedron cut off by pinacoid (o)	$[{\hbox to 7pc{}}\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(\bar{k}\bar{h}l) &(khl)\cr}]$
			Ditetragonal prism	$[{\hbox to 7pc{}}\matrix{(hk0) {\hbox to 0.65pc{}}(\bar{h}\bar{k}0) {\hbox to 0.65pc{}}(k\bar{h}0) {\hbox to 0.7pc{}}(\bar{k}h0)\cr}]$
			Truncated square through origin	$[{\hbox to 7pc{}}\matrix{(\bar{h}k0) {\hbox to 0.65pc{}}(h\bar{k}0) {\hbox to 0.65pc{}}(\bar{k}\bar{h}0) {\hbox to 0.65pc{}}({\it kh}0)\cr}]$
			Tetragonal dipyramid	$[{\hbox to 7pc{}}\matrix{(h0l) &(\bar{h}0l) {\hbox to 0.75pc{}}(0\bar{h}\bar{l}) {\hbox to 0.85pc{}}(0h\bar{l})\cr}]$
			Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(\bar{h}0\bar{l}) {\hbox to 0.8pc{}}(h0\bar{l}) {\hbox to 0.8pc{}}(0\bar{h}l) {\hbox to 0.8pc{}}(0hl)\cr}]$
4	c	..m	Tetragonal disphenoid or tetragonal tetrahedron	$[{\hbox to 7pc{}}\matrix{(hhl) {\hbox to 0.8pc{}}(\bar{h}\bar{h}l) {\hbox to 0.8pc{}}(h\bar{h}\bar{l}) {\hbox to 0.8pc{}}(\bar{h}h\bar{l})\cr}]$
			Tetragonal tetrahedron (n)
			Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(110) {\hbox to 0.65pc{}}(\bar{1}\bar{1}0) {\hbox to 0.65pc{}}(1\bar{1}0) {\hbox to 0.65pc{}}(\bar{1}10)\cr}]$
			Square through origin
4	b	.2.	Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(100) {\hbox to 0.65pc{}}(\bar{1}00) {\hbox to 0.65pc{}}(0\bar{1}0) {\hbox to 0.65pc{}}(010)\cr}]$
			Square through origin (i)
2	a	2.mm	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(001) {\hbox to 0.65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&2mm&&m\cr}]$
$[\bar{4}m2]$	$[D_{2d}]$
8	d	1	Tetragonal scalenohedron	$[{\hbox to 7pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]$
			Tetragonal tetrahedron cut off by pinacoid (l)	$[{\hbox to 7pc{}}\matrix{(h\bar{k}l) {\hbox to .8pc{}}(\bar{h}kl) {\hbox to .85pc{}}(kh\bar{l}) {\hbox to .9pc{}}(\bar{k}\bar{h}\bar{l})\cr}]$
			Ditetragonal prism	$[{\hbox to 7pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(k\bar{h}0) {\hbox to .7pc{}}(\bar{k}h0)\cr}]$
			Truncated square through origin	$[{\hbox to 7pc{}}\matrix{(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(kh0) {\hbox to .7pc{}}(\bar{k}\bar{h}0)\cr}]$
			Tetragonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hhl) {\hbox to .75pc{}}(\bar{h}\bar{h}l) {\hbox to .85pc{}}(h\bar{h}\bar{l}) {\hbox to .85pc{}}(\bar{h}h\bar{l})\cr}]$
			Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(h\bar{h}l) {\hbox to .8pc{}}(\bar{h}hl) {\hbox to .8pc{}}(hh\bar{l}) {\hbox to .85pc{}}(\bar{h}\bar{h}\bar{l})\cr}]$
4	c	.m.	Tetragonal disphenoid or tetragonal tetrahedron	$[{\hbox to 7pc{}}\matrix{(h0l) {\hbox to .8pc{}}(\bar{h}0l) {\hbox to .8pc{}}(0\bar{h}\bar{l}) {\hbox to .85pc{}}(0h\bar{l})\cr}]$
			Tetragonal tetrahedron (j)
			Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(100) &{\hbox to -2pt{}}(\bar{1}00) &{\hbox to -3pt{}}(0\bar{1}0) &{\hbox to -1.5pt{}}(010)\cr}]$
			Square through origin
4	b	..2	Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(110) &{\hbox to -2pt{}}(\bar{1}\bar{1}0) &{\hbox to -3pt{}}(1\bar{1}0) &{\hbox to -1.5pt{}}(\bar{1}10)\cr}]$
			Square through origin (h)
2	a	2mm.	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(001) &{\hbox to -2pt{}}(00\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&m&&2mm\cr}]$
$[\matrix{4/mmm\hfill\cr{\displaystyle{4 \over m}{2 \over m}{2 \over m}}\hfill}]$	$[D_{4h}]$

16	g	1	Ditetragonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]$
			Edge-truncated tetragonal prism (u)	$[{\hbox to 7pc{}}\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]$
8	f	.m.	Tetragonal dipyramid	$[{\hbox to 7pc{}}\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]$
			Tetragonal prism (s)	$[{\hbox to 7pc{}}\matrix{(\bar{h}0\bar{l}) &(h0\bar{l}) &(0h\bar{l}) &(0\bar{h}\bar{l})\cr}]$
8	e	..m	Tetragonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]$
			Tetragonal prism (r)	$[{\hbox to 7pc{}}\matrix{(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) &(hh\bar{l}) &(\bar{h}\bar{h}\bar{l})\cr}]$
8	d	m..	Ditetragonal prism	$[{\hbox to 7pc{}}\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]$
			Truncated square through origin (p)	$[{\hbox to 7pc{}}\matrix{(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .7pc{}}(kh0) {\hbox to .65pc{}}(\bar{k}\bar{h}0)\cr}]$
4	c	m2m.	Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .7pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]$
			Square through origin (l)
4	b	m.m2	Tetragonal prism	$[{\hbox to 7pc{}}\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .7pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]$
			Square through origin (j)
2	a	4mm	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox {Along }[100]&&\hbox{\ Along }[110]\cr 4mm&&2mm&&2mm\cr}]$
TRIGONAL SYSTEM
3	$[C_{3}]$
HEXAGONAL AXES
3	b	1	Trigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hkil) &(ihkl) &(kihl)\cr}]$
			Trigon (d)
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]$
			Trigon through origin
1	a	3..	Pedion or monohedron	$[{\hbox to 7.1pc{}}(0001) \hbox{ or } (000\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3&&1&&1\cr}]$
3	$[C_{3}]$
RHOMBOHEDRAL AXES
3	b	1	Trigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hkl) &(lhk) &(klh)\cr}]$
			Trigon (b)
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]$
			Trigon through origin
1	a	3.	Pedion or monohedron	$[{\hbox to 7.1pc{}}(111) \hbox{ or } (\bar{1}\bar{1}\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3&&1&&1\cr}]$
$[\bar{3}]$	$[C_{3i}]$
HEXAGONAL AXES
6	b	1	Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hkil) &(ihkl) &(kihl)\cr}]$
			Trigonal antiprism (g)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{i}\bar{l}) {\hbox to .8pc{}}(\bar{i}\bar{h}\bar{k}\bar{l}) {\hbox to .8pc{}}(\bar{k}\bar{i}\bar{h}\bar{l})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{i}0) {\hbox to .6pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
2	a	3..	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) {\hbox to .4pc{}}(000\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&2&&2\cr}]$
$[\bar{3}]$	$[C_{3i}]$
RHOMBOHEDRAL AXES
6	b	1	Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hkl) &(lhk) &(klh)\cr}]$
			Trigonal antiprism (f)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{l}) &(\bar{l}\bar{h}\bar{k}) &(\bar{k}\bar{l}\bar{h})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\! +\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}(h\!+\!k)) &((h\! +\! k)\bar{h}\bar{k}) &(\bar{k}(h\! +\! k)\bar{h})\cr}]$
2	a	3.	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 6&&2&&2\cr}]$
321	$[D_{3}]$
HEXAGONAL AXES
6	c	1	Trigonal trapezohedron	$[{\hbox to 7pc{}}\matrix{(hkil) {\hbox to 1.45pc{}}(ihkl) {\hbox to 1.5pc{}}(kihl)\cr}]$
			Twisted trigonal antiprism (g)	$[{\hbox to 7pc{}}\matrix{(khi\bar{l}) {\hbox to 1.45pc{}}(hik\bar{l}) {\hbox to 1.45pc{}}(ikh\bar{l})\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to 1.3pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(khi0) {\hbox to 1.3pc{}}(hik0) {\hbox to 1.3pc{}}(ikh0)\cr}]$
			Trigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]$
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}{\hbox to 1pt{}}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l})\cr}]$
			Rhombohedron	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) {\hbox to 1.25pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]$
			Trigonal antiprism	$[{\hbox to 7pc{}}\matrix{(0h\bar{h}\bar{l}) {\hbox to 1.25pc{}}(h\bar{h}0\bar{l}) {\hbox to 1.25pc{}}(\bar{h}0h\bar{l})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to 1.05pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(01\bar{1}0) {\hbox to 1.05pc{}}(1\bar{1}00) {\hbox to 1.1pc{}}(\bar{1}010)\cr}]$
3	b	.2.	Trigonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to 1.05pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]$
			Trigon through origin (e)	$[{\hbox to 6pc{}}\hbox{or}\ \matrix{(\bar{1}\bar{1}20) {\hbox to 1.1pc{}}(2\bar{1}\bar{1}0) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0)\cr}]$
2	a	3..	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) {\hbox to 1.05pc{}}(000\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&2&&1\cr}]$
312	$[D_{3}]$
HEXAGONAL AXES
6	c	1	Trigonal trapezohedron	$[{\hbox to 7pc{}}\matrix{(hkil) {\hbox to 1.5pc{}}(ihkl) {\hbox to 1.45pc{}}(kihl)\cr}]$
			Twisted trigonal antiprism (l)	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}\bar{l}) {\hbox to 1.45pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.45pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to 1.35pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.3pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.25pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
			Trigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) {\hbox to 1.3pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]$
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(0\bar{h}h\bar{l}) {\hbox to 1.3pc{}}(\bar{h}h0\bar{l}) {\hbox to 1.25pc{}}(h0\bar{h}\bar{l})\cr}]$
			Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]$
			Trigonal antiprism	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to 1.1pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}20) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0) {\hbox to 1.1pc{}}(2\bar{1}\bar{1}0)\cr}]$
3	b	..2	Trigonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to 1.1pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10)\cr}]$
			Trigon through origin (j)	$[{\hbox to 6pc{}}\hbox{or } \matrix{(\bar{1}010) {\hbox to 1.15pc{}}(1\bar{1}00) {\hbox to 1.05pc{}}(01\bar{1}0)\cr}]$
2	a	3..	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) {\hbox to 1.1pc{}}(000\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&1&&2\cr}]$
32	$[D_{3}]$
RHOMBOHEDRAL AXES
6	c	1	Trigonal trapezohedron	$[{\hbox to 7pc{}}\matrix{(hkl) {\hbox to .85pc{}}(lhk) {\hbox to .85pc{}}(klh)\cr}]$
			Twisted trigonal antiprism (f)	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hk(\overline{h\!+\!k})) {\hbox to 1.45pc{}}((\overline{h\!+\!k})hk) {\hbox to 1.25pc{}}(k(\overline{h\!+\!k})h)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}(h\!+\!k)) {\hbox to 1.45pc{}}(\bar{h}(h\!+\!k)\bar{k}) {\hbox to 1.25pc{}}((h\!+\!k)\bar{k}\bar{h})\cr}]$
			Trigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hk(2k\!-\!h)) {\hbox to 1pc{}}((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]$
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}(h\!-\!2k)) {\hbox to 1pc{}}(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]$
			Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hhl) &(lhh) &(hlh)\cr}]$
			Trigonal antiprism	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}) {\hbox to .65pc{}}(\bar{2}11) {\hbox to .65pc{}}(1\bar{2}1)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}2) {\hbox to .66pc{}}(\bar{1}2\bar{1}) {\hbox to .65pc{}}(2\bar{1}\bar{1})\cr}]$
3	b	.2	Trigonal prism	$[{\hbox to 7pc{}}\matrix{(01\bar{1}) {\hbox to .65pc{}}(\bar{1}01) {\hbox to .65pc{}}(1\bar{1}0)\cr}]$
			Trigon through origin (d)	$[{\hbox to 6pc{}}\hbox{or } \matrix{(0\bar{1}1) {\hbox to .7pc{}}(10\bar{1}) {\hbox to .65pc{}}(\bar{1}10)\cr}]$
2	a	3.	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(111) {\hbox to .65pc{}}(\bar{1}\bar{1}\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3m&&2&&1\cr}]$
3m1	$[C_{3v}]$
HEXAGONAL AXES
6	c	1	Ditrigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hkil) {\hbox to 1.5pc{}}(ihkl) {\hbox to 1.45pc{}}(kihl)\cr}]$
			Truncated trigon (e)	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}l) {\hbox to 1.45pc{}}(\bar{h}\bar{i}\bar{k}l) {\hbox to 1.45pc{}}(\bar{i}\bar{k}\bar{h}l)\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to 1.35pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.3pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.25pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
			Hexagonal pyramid	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]$
			Hexagon	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to 1.1pc{}}(\bar{2}110) {\hbox to 1.05pc{}}(1\bar{2}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}20) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0) {\hbox to 1.05pc{}}(2\bar{1}\bar{1}0)\cr}]$
3	b	.m.	Trigonal pyramid	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) {\hbox to 1.25pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]$
			Trigon (d)
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to 1.1pc{}}(\bar{1}100) {\hbox to 1.05pc{}}(0\bar{1}10)\cr}]$
			Trigon through origin	$[{\hbox to 6pc{}}\hbox{or } \matrix{(\bar{1}010) {\hbox to 1.1pc{}}(1\bar{1}00) {\hbox to 1.1pc{}}(01\bar{1}0)\cr}]$
1	a	3m.	Pedion or monohedron	$[{\hbox to 7.15pc{}}(0001) \hbox{ or } (000\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&1&&m\cr}]$
31m	$[C_{3v}]$
HEXAGONAL AXES
6	c	1	Ditrigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hkil) &(ihkl) &(kihl)\cr}]$
			Truncated trigon (d)	$[{\hbox to 7pc{}}\matrix{(khil) &(hikl) &(ikhl)\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(khi0) {\hbox to .65pc{}}(hik0) {\hbox to .65pc{}}(ikh0)\cr}]$
			Hexagonal pyramid	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) {\hbox to .6pc{}}(\bar{h}h0l) {\hbox to .65pc{}}(0\bar{h}hl)\cr}]$
			Hexagon	$[{\hbox to 7pc{}}\matrix{(0h\bar{h}l) {\hbox to .6pc{}}(h\bar{h}0l) {\hbox to .65pc{}}(\bar{h}0hl)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to .45pc{}}(\bar{1}100) {\hbox to .45pc{}}(0\bar{1}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(01\bar{1}0) {\hbox to .45pc{}}(1\bar{1}00) {\hbox to .45pc{}}(\bar{1}010)\cr}]$
3	b	..m	Trigonal pyramid	$[{\hbox to 7.1pc{}}(hh\overline{2h}l){\hbox to .2pc{}}(\overline{2h}hhl){\hbox to .25pc{}} (h\overline{2h}hl)]$
			Trigon (c)
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to .45pc{}}(\bar{2}110) {\hbox to .45pc{}}(1\bar{2}10)\cr}]$
			Trigon through origin	$[{\hbox to 6pc{}}\hbox{or }\matrix{(\bar{1}\bar{1}20) {\hbox to .45pc{}}(2\bar{1}\bar{1}0) {\hbox to .45pc{}}(\bar{1}2\bar{1}0)\cr}]$
1	a	3.m	Pedion or monohedron	$[{\hbox to 7.1pc{}}(0001) \hbox{ or } (000\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox {Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&m&&1\cr}]$
3m	$[C_{3v}]$
RHOMBOHEDRAL AXES
6	c	1	Ditrigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hkl) &(lhk) &(klh)\cr}]$
			Truncated trigon (c)	$[{\hbox to 7pc{}}\matrix{(khl) &(hlk) &(lkh)\cr}]$
			Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hk(\overline{h\!+\!k})) {\hbox to 1.4pc{}}((\overline{h \!+\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]$
			Truncated trigon through origin	$[{\hbox to 7pc{}}\matrix{(kh(\overline{h\!+\!k})) {\hbox to 1.4pc{}}(h(\overline{h \!+\! k})k) &((\overline{h \!+\! k})kh)\cr}]$
			Hexagonal pyramid	$[{\hbox to 7pc{}}\matrix{(hk(2k\!-\!h)) {\hbox to .95pc{}}((2k\! -\! h)hk) {\hbox to .45pc{}}(k(2k \!- \!h)h)\cr}]$
			Hexagon	$[{\hbox to 7pc{}}\matrix{(kh(2k\!-\!h)) {\hbox to 1pc{}}(h(2k\! -\! h)k) {\hbox to .4pc{}}((2k\! - \!h)kh)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(01\bar{1}) {\hbox to .65pc{}}(\bar{1}01) {\hbox to .6pc{}}(1\bar{1}0)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(10\bar{1}) {\hbox to .65pc{}}(0\bar{1}1) {\hbox to .6pc{}}(\bar{1}10)\cr}]$
3	b	.m	Trigonal pyramid	$[{\hbox to 7pc{}}\matrix{(hhl) &(lhh) &(hlh)\cr}]$
			Trigon (b)
			Trigonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}) {\hbox to .65pc{}}(\bar{2}11) {\hbox to .6pc{}}(1\bar{2}1)\cr}]$
			Trigon through origin	$[{\hbox to 6pc{}}\hbox{or }\matrix{(\bar{1}\bar{1}2) {\hbox to .65pc{}}(2\bar{1}\bar{1}) {\hbox to .65pc{}}(\bar{1}2\bar{1})\cr}]$
1	a	3m	Pedion or monohedron	$[{\hbox to 7.2pc{}}(111) \hbox{ or }(\bar{1}\bar{1}\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3m&&1&&m\cr}]$
$[\openup4pt\matrix{\bar{3}m1\hfill\cr \bar{3} {\displaystyle{2 \over m}} 1\hfill\cr}]$	$[D_{3d}]$
HEXAGONAL AXES
12	d	1	$[\matrix{\hbox{Ditrigonal scalenohedron or}\hfill\cr \hbox{hexagonal scalenohedron}\hfill\cr Trigonal\ antiprism\ sliced\ off\ by\hfill\cr pinacoid\ (j)\hfill\cr}]$	$[{\hbox to 7pc{}}\matrix{(hkil)\quad &(ihkl) &\quad(kihl)\cr (khi\bar{l})\quad &(hik\bar{l}) &\quad(ikh\bar{l})\cr (\bar{h}\bar{k}\bar{i}\bar{l})\quad &(\bar{i}\bar{h}\bar{k}\bar{l}) &\quad(\bar{k}\bar{i}\bar{h}\bar{l})\cr (\bar{k}\bar{h}\bar{i}l)\quad &(\bar{h}\bar{i}\bar{k}l) &\quad(\bar{i}\bar{k}\bar{h}l)\cr}]$

			Dihexagonal prism	$[{\hbox to 7pc{}}\matrix{(hki0)\quad {\hbox to .65pc{}}(ihk0) \quad{\hbox to .7pc{}}(kih0)\cr}]$
			Truncated hexagon through origin	$[{\hbox to 7pc{}}\matrix{(khi0)\quad {\hbox to .65pc{}}(hik0)\quad {\hbox to .7pc{}}(ikh0)\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{i}0)\quad {\hbox to .65pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to .65pc{}}\quad(\bar{k}\bar{i}\bar{h}0)\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}0)\quad {\hbox to .65pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to .65pc{}}\quad(\bar{i}\bar{k}\bar{h}0)\cr}]$
			Hexagonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l)\quad {\hbox to .25pc{}}(\overline{2h}hhl)\quad {\hbox to .2pc{}}(h\overline{2h}hl)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}{\hbox to 1pt{}}\bar{l})\quad {\hbox to .25pc{}}(h\overline{2h}h\bar{l})\quad {\hbox to .2pc{}}(\overline{2h}hh\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}2h\bar{l})\quad {\hbox to .25pc{}}(2h\bar{h}\bar{h}\bar{l})\quad {\hbox to .2pc{}}(\bar{h}2h\bar{h}\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}2hl)\quad {\hbox to .25pc{}}(\bar{h}2h\bar{h}l)\quad {\hbox to .2pc{}}(2h\bar{h}\bar{h}l)\cr}]$
6	c	.m.	Rhombohedron	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l)\quad {\hbox to .65pc{}}(\bar{h}h0l)\quad {\hbox to .65pc{}}(0\bar{h}hl)\cr}]$
			Trigonal antiprism (i)	$[{\hbox to 7pc{}}\matrix{(0h\bar{h}\bar{l})\quad {\hbox to .65pc{}}(h\bar{h}0\bar{l})\quad {\hbox to .65pc{}}(\bar{h}0h\bar{l})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0)\quad {\hbox to .45pc{}}(\bar{1}100)\quad {\hbox to .5pc{}}(0\bar{1}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(01\bar{1}0)\quad {\hbox to .45pc{}}(1\bar{1}00)\quad {\hbox to .5pc{}}(\bar{1}010)\cr}]$
6	b	.2.	Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0)\quad {\hbox to .45pc{}}(\bar{2}110)\quad {\hbox to .5pc{}}(1\bar{2}10)\cr}]$
			Hexagon through origin (g)	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}20)\quad {\hbox to .45pc{}}(\bar{1}2\bar{1}0)\quad {\hbox to .5pc{}}(2\bar{1}\bar{1}0)\cr}]$
2	a	3m.	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001)\quad {\hbox to .45pc{}}(000\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox {Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2&&2mm\cr}]$
$[\openup4pt\matrix{\bar{3}1m\cr \bar{3}1{\displaystyle{2 \over m}}\cr}]$	$[D_{3d}]$
HEXAGONAL AXES
12	d	1	Ditrigonal scalenohedron or hexagonal scalenohedron	$[{\hbox to 7pc{}}\matrix{(hkil) {\hbox to 1.5pc{}}(ihkl) {\hbox to 1.35pc{}}(kihl)\cr (\bar{k}\bar{h}\bar{i}\bar{l}) {\hbox to 1.5pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.35pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]$
			Trigonal antiprism sliced off by pinacoid (l)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{i}\bar{l}) {\hbox to 1.5pc{}}(\bar{i}\bar{h}\bar{k}\bar{l}) {\hbox to 1.4pc{}}(\bar{k}\bar{i}\bar{h}\bar{l})\cr (khil) {\hbox to 1.5pc{}}(hikl) {\hbox to 1.4pc{}}(ikhl)\cr}]$
			Dihexagonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) {\hbox to 1.3pc{}}(ihk0) {\hbox to 1.3pc{}}(kih0)\cr}]$
			Truncated hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.3pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.3pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{i}0) {\hbox to 1.3pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to 1.25pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
				$[{\hbox to 7pc{}}\matrix{(khi0) {\hbox to 1.3pc{}}(hik0) {\hbox to 1.25pc{}}(ikh0)\cr}]$
			Hexagonal dipyramid	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) {\hbox to 1.3pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(0\bar{h}h\bar{l}) {\hbox to 1.3pc{}}(\bar{h}h0\bar{l}) {\hbox to 1.25pc{}}(h0\bar{h}\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}0h\bar{l}) {\hbox to 1.3pc{}}(h\bar{h}0\bar{l}) {\hbox to 1.25pc{}}(0h\bar{h}\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(0h\bar{h}l) {\hbox to 1.3pc{}}(h\bar{h}0l) {\hbox to 1.25pc{}}(\bar{h}0hl)\cr}]$
6	c	..m	Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]$
			Trigonal antiprism (k)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to 1.1pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}20) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0) {\hbox to 1.05pc{}}(2\bar{1}\bar{1}0)\cr}]$
6	b	..2	Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to 1.05pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10)\cr}]$
			Hexagon through origin (i)	$[{\hbox to 7pc{}}\matrix{(\bar{1}010) {\hbox to 1.1pc{}}(1\bar{1}00) {\hbox to 1.1pc{}}(01\bar{1}0)\cr}]$
2	a	3.m	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) {\hbox to 1.1pc{}}(000\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2\cr}]$
$[\openup4pt\matrix{\bar{3}m\hfill\cr \bar{3}{\displaystyle{2 \over m}}\hfill\cr}]$	$[D_{3d}]$
RHOMBOHEDRAL AXES
12	d	1	Ditrigonal scalenohedron or hexagonal scalenohedron	$[{\hbox to 7pc{}}\matrix{(hkl) &(lhk) &(klh)\cr(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]$
			Trigonal antiprism sliced off by pinacoid (i)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}\bar{l}) &(\bar{l}\bar{h}\bar{k}) &(\bar{k}\bar{l}\bar{h})\cr (khl) &(hlk) &(lkh)\cr}]$
			Dihexagonal prism	$[{\hbox to 7pc{}}\matrix{(hk(\overline{h\!+\!k})) {\hbox to 1.45pc{}}((\overline{h\!+\!k})hk) {\hbox to 1.25pc{}}(k(\overline{h\!+\!k})h)\cr}]$
			Truncated hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}(h\!+\!k)) {\hbox to 1.45pc{}}(\bar{h}(h\!+\!k)\bar{k}) {\hbox to 1.25pc{}}((h\!+\!k)\bar{k}\bar{h})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}(h\!+\!k)) {\hbox to 1.45pc{}}((h\!+\!k)\bar{h}\bar{k}) {\hbox to 1.25pc{}}(\bar{k}(h\!+\!k)\bar{h})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(kh(\overline{h\!+\!k})) {\hbox to 1.45pc{}}(h(\overline{h\!+\!k})k) {\hbox to 1.25pc{}}((\overline{h\!+\!k})kh)\cr}]$
			Hexagonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hk(2k\!-\!h)) {\hbox to 1pc{}}((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(\bar{k}\bar{h}(h\!-\!2k)) {\hbox to 1pc{}}(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{k}(h\!-\!2k)) {\hbox to 1pc{}}((h\!-\!2k)\bar{h}\bar{k}) &(\bar{k}(h\!-\!2k)\bar{h})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(kh(2k\!-\!h)) {\hbox to 1pc{}}(h(2k\!-\!h)k) &((2k\!-\!h)kh)\cr}]$
6	c	.m	Rhombohedron	$[{\hbox to 7pc{}}\matrix{(hhl) &(lhh) &(hlh)\cr}]$
			Trigonal antiprism (h)	$[{\hbox to 7pc{}}\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}) {\hbox to .65pc{}}(\bar{2}11) {\hbox to .65pc{}}(1\bar{2}1)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(\bar{1}\bar{1}2) {\hbox to .65pc{}}(\bar{1}2\bar{1}) {\hbox to .65pc{}}(2\bar{1}\bar{1})\cr}]$
6	b	.2	Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(01\bar{1}) {\hbox to .65pc{}}(\bar{1}01) {\hbox to .65pc{}}(1\bar{1}0)\cr}]$
			Hexagon through origin (f)	$[{\hbox to 7pc{}}\matrix{(0\bar{1}1) {\hbox to .65pc{}}(10\bar{1}) {\hbox to .65pc{}}(\bar{1}10)\cr}]$
2	a	3m	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(111) {\hbox to .65pc{}}(\bar{1}\bar{1}\bar{1})\cr}]$
			Line segment through origin (c)
Symmetry of special projections
$[\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 6mm&&2&&2mm\cr}]$
HEXAGONAL SYSTEM
6	$[C_{6}]$
6	b	1	Hexagonal pyramid	$[{\hbox to 5pc{}}\matrix{(hkil) &(ihkl) &(kihl) &(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]$
			Hexagon (d)
			Hexagonal prism	$[{\hbox to 5pc{}}\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0) {\hbox to .65pc{}}(\bar{h}\bar{k}\bar{i}0) {\hbox to .65pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
			Hexagon through origin
1	a	6..	Pedion or monohedron	$[{\hbox to 5pc{}}(0001) \hbox{ or }(000\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&m&&m}]$
$[\bar{6}]$	$[C_{3h}]$
6	c	1	Trigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hkil) &(ihkl) &(kihl)\cr}]$
			Trigonal prism (l)	$[{\hbox to 7pc{}}\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]$
3	b	m..	Trigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) &{\hbox to -2pt{}}(ihk0) &{\hbox to -2pt{}}(kih0)\cr}]$
			Trigon through origin (j)
2	a	3..	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) &{\hbox to -4.5pt{}}(000\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3&&m&&m}]$
	$[C_{6h}]$
12	c	1	Hexagonal dipyramid	$[{\hbox to 5pc{}}\matrix{(hkil) &(ihkl) &(kihl) &(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]$
			Hexagonal prism (l)	$[{\hbox to 5pc{}}\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l}) &(\bar{h}\bar{k}\bar{i}\bar{l}) &(\bar{i}\bar{h}\bar{k}\bar{l}) &(\bar{k}\bar{i}\bar{h}\bar{l})\cr}]$
6	b	m..	Hexagonal prism	$[{\hbox to 5pc{}}\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0) {\hbox to .65pc{}}(\bar{h}\bar{k}\bar{i}0) {\hbox to .65pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
			Hexagon through origin (j)
2	a	6..	Pinacoid or parallelohedron	$[{\hbox to 5pc{}}\matrix{(0001) {\hbox to .45pc{}}(000\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&2mm&&2mm\cr}]$
622	$[D_{6}]$
12	d	1	Hexagonal trapezohedron	$[{\hbox to 3.5pc{}}\matrix{(hkil) {\hbox to 1.2pc{}}(ihkl) {\hbox to 1.25pc{}}(kihl) {\hbox to 1.25pc{}}(\bar{h}\bar{k}\bar{i}l) {\hbox to 1.15pc{}}(\bar{i}\bar{h}\bar{k}l) {\hbox to 1.2pc{}}(\bar{k}\bar{i}\bar{h}l)\cr}]$
			Twisted hexagonal antiprism (n)	$[{\hbox to 3.5pc{}}\matrix{(khi\bar{l}) {\hbox to 1.2pc{}}(hik\bar{l}) {\hbox to 1.2pc{}}(ikh\bar{l}) {\hbox to 1.25pc{}}(\bar{k}\bar{h}\bar{i}\bar{l}) {\hbox to 1.15pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.25pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]$
			Dihexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(hki0) {\hbox to 1pc{}}(ihk0) {\hbox to 1.1pc{}}(kih0) {\hbox to 1.05pc{}}(\bar{h}\bar{k}\bar{i}0) {\hbox to 1pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to 1.05pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
			Truncated hexagon through origin	$[{\hbox to 3.5pc{}}\matrix{(khi0) {\hbox to 1pc{}}(hik0) {\hbox to 1.1pc{}}(ikh0) {\hbox to 1.05pc{}}(\bar{k}\bar{h}\bar{i}0) {\hbox to 1pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.05pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
			Hexagonal dipyramid	$[{\hbox to 3.5pc{}}\matrix{(h0\bar{h}l) {\hbox to .95pc{}}(\bar{h}h0l) {\hbox to 1.05pc{}}(0\bar{h}hl) {\hbox to 1.05pc{}}(\bar{h}0hl) {\hbox to .9pc{}}(h\bar{h}0l) {\hbox to 1.05pc{}}(0h\bar{h}l)\cr}]$
			Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(0h\bar{h}\bar{l}) {\hbox to .95pc{}}(h\bar{h}0\bar{l}) {\hbox to 1.05pc{}}(\bar{h}0h\bar{l}) {\hbox to 1.05pc{}}(0\bar{h}h\bar{l}) {\hbox to .95pc{}}(\bar{h}h0\bar{l}) {\hbox to 1pc{}}(h0\bar{h}\bar{l})\cr}]$
			Hexagonal dipyramid	$[{\hbox to 3.5pc{}}\matrix{(hh\overline{2h}l) {\hbox to .55pc{}}(\overline{2h}hhl) {\hbox to .6pc{}}(h\overline{2h}hl) {\hbox to .6pc{}}(\bar{h}\bar{h}2hl) {\hbox to .55pc{}}(2h\bar{h}\bar{h}l) {\hbox to .65pc{}}(\bar{h}2h\bar{h}l)\cr}]$
			Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(hh\overline{2h}\bar{l}) {\hbox to .55pc{}}(h\overline{2h}h\bar{l}) {\hbox to .6pc{}}(\overline{2h}hh\bar{l}) {\hbox to .6pc{}}(\bar{h}\bar{h}2h\bar{l}) {\hbox to .55pc{}}(\bar{h}2h\bar{h}\bar{l}) {\hbox to .65pc{}}(2h\bar{h}\bar{h}\bar{l})\cr}]$
6	c	..2	Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) &(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]$
			Hexagon through origin (l)
6	b	.2.	Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) &(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]$
			Hexagon through origin (j)
2	a	6..	Pinacoid or parallelohedron	$[{\hbox to 3.5pc{}}\matrix{(0001) &(000\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2mm\cr}]$
6mm	$[C_{6v}]$
12	d	1	Dihexagonal pyramid	$[{\hbox to 3.5pc{}}\matrix{(hkil) {\hbox to 1.2pc{}}(ihkl) {\hbox to 1.45pc{}}(kihl) {\hbox to 1.45pc{}}(\bar{h}\bar{k}\bar{i}l) {\hbox to 1.4pc{}}(\bar{i}\bar{h}\bar{k}l) {\hbox to 1.5pc{}}(\bar{k}\bar{i}\bar{h}l)\cr}]$
			Truncated hexagon (f)	$[{\hbox to 3.5pc{}}\matrix{(khil) {\hbox to 1.2pc{}}(hikl) {\hbox to 1.45pc{}}(ikhl) {\hbox to 1.45pc{}}(\bar{k}\bar{h}\bar{i}l) {\hbox to 1.4pc{}}(\bar{h}\bar{i}\bar{k}l) {\hbox to 1.5pc{}}(\bar{i}\bar{k}\bar{h}l)\cr}]$
			Dihexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(hki0) {\hbox to 1.05pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0) {\hbox to 1.3pc{}}(\bar{h}\bar{k}\bar{i}0) {\hbox to 1.25pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to 1.3pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
			Truncated hexagon through origin	$[{\hbox to 3.5pc{}}\matrix{(khi0) {\hbox to 1.05pc{}}(hik0) {\hbox to 1.2pc{}}(ikh0) {\hbox to 1.35pc{}}(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.2pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.35pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
6	c	.m.	Hexagonal pyramid	$[{\hbox to 3.5pc{}}\matrix{(h0\bar{h}l) {\hbox to 1pc{}}(\bar{h}h0l) {\hbox to 1.2pc{}}(0\bar{h}hl) {\hbox to 1.3pc{}}(\bar{h}0hl) {\hbox to 1.15pc{}}(h\bar{h}0l) {\hbox to 1.3pc{}}(0h\bar{h}l)\cr}]$
			Hexagon (e)
			Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(10\bar{1}0) {\hbox to 1pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10) {\hbox to 1.05pc{}}(\bar{1}010) &(1\bar{1}00) {\hbox to 1.1pc{}}(01\bar{1}0)\cr}]$
			Hexagon through origin
6	b	..m	Hexagonal pyramid	$[{\hbox to 3.5pc{}}\matrix{(hh\overline{2h}l) {\hbox to .55pc{}}(\overline{2h}hhl) &(h\overline{2h}hl) &(\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr}]$
			Hexagon (d)
			Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(11\bar{2}0) {\hbox to 1pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10) {\hbox to 1.05pc{}}(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0)\cr}]$
			Hexagon through origin
1	a	6mm	Pedion or monohedron	$[{\hbox to 3.6pc{}}(0001) \hbox{ or } (000\bar{1})]$
			Single point (a)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&m&&m\cr}]$
$[\bar{6}m2]$	$[D_{3h}]$
12	e	1	Ditrigonal dipyramid	$[{\hbox to 8pc{}}\matrix{(hkil) {\hbox to 1.45pc{}}(ihkl) {\hbox to 1.5pc{}}(kihl)\cr}]$
			Edge-truncated trigonal prism (o)	$[{\hbox to 8pc{}}\matrix{(hki\bar{l}) {\hbox to 1.45pc{}}(ihk\bar{l}) {\hbox to 1.45pc{}}(kih\bar{l})\cr}]$
				$[{\hbox to 8pc{}}\matrix{(\bar{k}\bar{h}\bar{i}l) {\hbox to 1.45pc{}}(\bar{h}\bar{i}\bar{k}l) {\hbox to 1.4pc{}}(\bar{i}\bar{k}\bar{h}l)\cr}]$
				$[{\hbox to 8pc{}}\matrix{(\bar{k}\bar{h}\bar{i}\bar{l}) {\hbox to 1.45pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.4pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]$
			Hexagonal dipyramid	$[{\hbox to 8pc{}}\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]$
			Hexagonal prism	$[{\hbox to 8pc{}}\matrix{(hh\overline{2h}\hskip 1pt\bar{l}) &(\overline{2h}hh\bar{l}) &(h\overline{2h}h\bar{l})\cr}]$
				$[{\hbox to 8pc{}}\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]$
				$[{\hbox to 8pc{}}\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]$
6	d	m..	Ditrigonal prism	$[{\hbox to 8pc{}}\matrix{(hki0) {\hbox to 1.3pc{}}(ihk0) {\hbox to 1.3pc{}}(kih0)\cr}]$
			Truncated trigon through origin (l)	$[{\hbox to 8pc{}}\matrix{(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.3pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.25pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
			Hexagonal prism	$[{\hbox to 8pc{}}\matrix{(11\bar{2}0) {\hbox to 1.05pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]$
			Hexagon through origin	$[{\hbox to 8pc{}}\matrix{(\bar{1}\bar{1}20) {\hbox to 1.05pc{}}(\bar{1}2\bar{1}0) {\hbox to 1.1pc{}}(2\bar{1}\bar{1}0)\cr}]$
6	c	.m.	Trigonal dipyramid	$[{\hbox to 8pc{}}\matrix{(h0\bar{h}l) {\hbox to 1.25pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]$
			Trigonal prism (n)	$[{\hbox to 8pc{}}\matrix{(h0\bar{h}\bar{l}) {\hbox to 1.25pc{}}(\bar{h}h0\bar{l}) {\hbox to 1.25pc{}}(0\bar{h}h\bar{l})\cr}]$
3	b	mm2	Trigonal prism	$[{\hbox to 8pc{}}\matrix{(10\bar{1}0) {\hbox to 1.1pc{}}(\bar{1}100) {\hbox to 1.05pc{}}(0\bar{1}10)\cr}]$
			Trigon through origin (j)	$[{\hbox to 6.75pc{}}\hbox{ or }\matrix{(\bar{1}010) {\hbox to 1.1pc{}}(1\bar{1}00) {\hbox to 1.05pc{}}(01\bar{1}0)\cr}]$
2	a	3m.	Pinacoid or parallelohedron	$[{\hbox to 8pc{}}\matrix{(0001) {\hbox to 1.1pc{}}(000\bar{1})\cr}]$
			Line segment through origin (g)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&m&&2mm\cr}]$
$[\bar{6}2m]$	$[D_{3h}]$
12	e	1	Ditrigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hkil) {\hbox to 1.1pc{}}(ihkl) {\hbox to 1pc{}}(kihl)\cr}]$
			Edge-truncated trigonal prism (l)	$[{\hbox to 7pc{}}\matrix{(hki\bar{l}) {\hbox to 1.1pc{}}(ihk\bar{l}) {\hbox to .95pc{}}(kih\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(khi\bar{l}) {\hbox to 1.1pc{}}(hik\bar{l}) {\hbox to .95pc{}}(ikh\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(khil) {\hbox to 1.1pc{}}(hikl) {\hbox to 1.pc{}}(ikhl)\cr}]$
			Hexagonal dipyramid	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(h0\bar{h}\bar{l}) &(\bar{h}h0\bar{l}) &(0\bar{h}h\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l})\cr}]$
				$[{\hbox to 7pc{}}\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]$
6	d	m..	Ditrigonal prism	$[{\hbox to 7pc{}}\matrix{(hki0) &(ihk0) {\hbox to .9pc{}}(kih0)\cr}]$
			Truncated trigon through origin (j)	$[{\hbox to 7pc{}}\matrix{(khi0) &(hik0) {\hbox to .9pc{}}(ikh0)\cr}]$
			Hexagonal prism	$[{\hbox to 7pc{}}\matrix{(10\bar{1}0) {\hbox to .65pc{}}(\bar{1}100) {\hbox to .7pc{}}(0\bar{1}10)\cr}]$
			Hexagon through origin	$[{\hbox to 7pc{}}\matrix{(01\bar{1}0) {\hbox to .7pc{}}(1\bar{1}00) {\hbox to .65pc{}}(\bar{1}010)\cr}]$
6	c	..m	Trigonal dipyramid	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}l) {\hbox to .4pc{}}(\overline{2h}hhl) {\hbox to .4pc{}}(h\overline{2h}hl)\cr}]$
			Trigonal prism (i)	$[{\hbox to 7pc{}}\matrix{(hh\overline{2h}\hskip1pt\bar{l}) {\hbox to .4pc{}}(\overline{2h}hh\bar{l}) {\hbox to .35pc{}}(h\overline{2h}h\bar{l})\cr}]$
3	b	m2m	Trigonal prism	$[{\hbox to 7pc{}}\matrix{(11\bar{2}0) {\hbox to .7pc{}}(\bar{2}110) {\hbox to .65pc{}}(1\bar{2}10)\cr}]$
			Trigon through origin (f)	$[{\hbox to 6.15pc{}}\hbox{or }(\bar{1}\bar{1}20){\hbox to .7pc{}} (2\bar{1}\bar{1}0){\hbox to .65pc{}} (\bar{1}2\bar{1}0)]$
2	a	3.m	Pinacoid or parallelohedron	$[{\hbox to 7pc{}}\matrix{(0001) {\hbox to .65pc{}}(000\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&2mm&&m\cr}]$
$[\openup4pt\!\matrix{6/mmm\hfill\cr \displaystyle{6 \over m}{2 \over m}{2 \over m}\hfill\cr}]$	$[D_{6h}]$

24	g	1	Dihexagonal dipyramid	$[{\hbox to 3.5pc{}}\matrix{(hkil) {\hbox to 1.2pc{}}(ihkl) {\hbox to 1.25pc{}}(kihl) {\hbox to 1.2pc{}}(\bar{h}\bar{k}\bar{i}l) {\hbox to 1.2pc{}}(\bar{i}\bar{h}\bar{k}l) {\hbox to 1.2pc{}}(\bar{k}\bar{i}\bar{h}l)\cr}]$
			Edge-truncated hexagonal prism (r)	$[{\hbox to 3.5pc{}}\matrix{(khi\bar{l}) {\hbox to 1.2pc{}}(hik\bar{l}) {\hbox to 1.2pc{}}(ikh\bar{l}) {\hbox to 1.25pc{}}(\bar{k}\bar{h}\bar{i}\bar{l}) {\hbox to 1.2pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.15pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]$
				$[{\hbox to 3.5pc{}}\matrix{(\bar{h}\bar{k}\bar{i}\bar{l}) {\hbox to 1.15pc{}}(\bar{i}\bar{h}\bar{k}\bar{l}) {\hbox to 1.2pc{}}(\bar{k}\bar{i}\bar{h}\bar{l}) {\hbox to 1.2pc{}}(hki\bar{l}) {\hbox to 1.25pc{}}(ihk\bar{l}) {\hbox to 1.15pc{}}(kih\bar{l})\cr}]$
				$[{\hbox to 3.5pc{}}\matrix{(\bar{k}\bar{h}\bar{i}l) {\hbox to 1.15pc{}}(\bar{h}\bar{i}\bar{k}l) {\hbox to 1.2pc{}}(\bar{i}\bar{k}\bar{h}l) {\hbox to 1.2pc{}}(khil) {\hbox to 1.25pc{}}(hikl) {\hbox to 1.2pc{}}(ikhl)\cr}]$
12	f	m..	Dihexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(hki0) {\hbox to 1.05pc{}}(ihk0) {\hbox to 1pc{}}(kih0) {\hbox to 1.1pc{}}(\bar{h}\bar{k}\bar{i}0) {\hbox to 1pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to 1pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]$
			Truncated hexagon through origin (p)	$[{\hbox to 3.5pc{}}\matrix{(khi0) {\hbox to 1.05pc{}}(hik0) {\hbox to 1pc{}}(ikh0) {\hbox to 1.1pc{}}(\bar{k}\bar{h}\bar{i}0) {\hbox to 1pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]$
12	e	.m.	Hexagonal dipyramid	$[{\hbox to 3.5pc{}}\matrix{(h0\bar{h}l) {\hbox to 1pc{}}(\bar{h}h0l) {\hbox to 1pc{}}(0\bar{h}hl) {\hbox to 1pc{}}(\bar{h}0hl) {\hbox to 1pc{}}(h\bar{h}0l) {\hbox to 1pc{}}(0h\bar{h}l)\cr}]$
			Hexagonal prism (o)	$[{\hbox to 3.5pc{}}\matrix{(0h\bar{h}\bar{l}) {\hbox to 1pc{}}(h\bar{h}0\bar{l}) {\hbox to 1pc{}}(\bar{h}0h\bar{l}) {\hbox to 1pc{}}(0\bar{h}h\bar{l}) {\hbox to 1pc{}}(\bar{h}h0\bar{l}) {\hbox to 1pc{}}(h0\bar{h}\bar{l})\cr}]$
12	d	..m	Hexagonal dipyramid	$[{\hbox to 3.5pc{}}\matrix{(hh\overline{2h}l) {\hbox to .55pc{}}(\overline{2h}hhl) {\hbox to .6pc{}}(h\overline{2h}hl) {\hbox to .6pc{}}(\bar{h}\bar{h}2hl) {\hbox to .6pc{}}(2h\bar{h}\bar{h}l) {\hbox to .55pc{}}(\bar{h}2h\bar{h}l)\cr}]$
			Hexagonal prism (n)	$[{\hbox to 3.5pc{}}\matrix{(hh\overline{2h}\bar{l}) {\hbox to .55pc{}}(h\overline{2h}h\bar{l}) {\hbox to .6pc{}}(\overline{2h}hh\bar{l}) {\hbox to .6pc{}}(\bar{h}\bar{h}2h\bar{l}) {\hbox to .6pc{}}(\bar{h}2h\bar{h}\bar{l}) {\hbox to .55pc{}}(2h\bar{h}\bar{h}\bar{l})\cr}]$
6	c	mm2	Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) &(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]$
			Hexagon through origin (l)
6	b	m2m	Hexagonal prism	$[{\hbox to 3.5pc{}}\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) &(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]$
			Hexagon through origin (j)
2	a	6mm	Pinacoid or parallelohedron	$[{\hbox to 3.5pc{}}\matrix{(0001) &(000\bar{1})\cr}]$
			Line segment through origin (e)
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2mm\cr}]$
CUBIC SYSTEM
23	T
12	c	1	$[\matrix{\hbox{Pentagon\hbox{-}tritetrahedron or tetartoid}\hfill\cr \hbox{or tetrahedral pentagon\hbox{-}dodecahedron}\hfill\cr Snub\ tetrahedron\ (= pentagon\hbox{-}tritetra\hbox{-}\hfill\cr hedron + two\ tetrahedra)\ (\;j)\hfill\cr}]$	$[{\hbox to 7pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k})\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h)\cr}]$
			$[\left\{\matrix{\hbox{Trigon-tritetrahedron}\hfill\cr \hbox{or tristetrahedron (for }\|h\| \lt \|l\|\hbox{)}\hfill\cr Tetrahedron\ truncated\ by\ tetrahedron\hfill\cr (for\ \|x\| \lt \|z\|)\hfill\cr \cr \hbox{Tetragon-tritetrahedron or deltohedron}\hfill\cr \hbox{or deltoid-dodecahedron (for}\ \|h\| \gt \|l\|\hbox{)}\hfill\cr Cube\ \&\ two\ tetrahedra\ (for\ \|x\| \gt \|z\|)\hfill\cr}\right\}]$	$[{\hbox to 7pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\hfill\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr}]$
			$[\matrix{\hbox{Pentagon-dodecahedron}\hfill\cr \hbox{or dihexahedron or pyritohedron}\hfill\cr Irregular\ icosahedron\hfill\cr (=pentagon\hbox{-}dodecahedron + octahedron)\hfill\cr}]$	$[{\hbox to 7pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr(l0k)&(l0\bar{k})&({\bar l}0k)&({\bar l}0\bar{k})\cr(kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0)\cr}]$
			$[\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}]$	$[{\hbox to 7pc{}}\openup-1pt\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]$
6	b	2..	$[\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\ (\;f)\hfill\cr}]$	$[{\hbox to 7pc{}}\openup-1pt\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]$
4	a	.3.	$[\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\ (e)\hfill\cr}]$	$[{\hbox to 6.1pc{}}\openup-1pt\matrix{{}}(111)&(\bar{1}\bar{1}1)&(\bar{1}1\bar{1})&(1\bar{1}\bar{1})\hfill\cr \hbox{or }(\bar{1}\bar{1}\bar{1})&(11\bar{1})&(1\bar{1}1)&(\bar{1}11)\hfill\cr}\hfill]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 2mm&&3&&m\cr}]$
$[\openup 6pt\matrix{m{\bar 3}\cr \displaystyle{2 \over m}\bar{3}}]$	$[T_{h}]$
24	d	1	$[\matrix{\hbox{Didodecahedron or diploid}\hfill\cr\hbox{or dyakisdodecahedron}\hfill\cr Cube\ \&\ octahedron\ \&\hfill\cr pentagon\hbox{-}dodecahedron\ (l)\hfill\cr}]$	$[{\hbox to 6.5pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr(lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k})\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h)\cr\noalign{\vskip10pt}(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl)\cr(\bar{l}\bar{h}\bar{k}) &(\bar{l}hk) &(lh\bar{k}) &(l\bar{h}k)\cr (\bar{k}\bar{l}\bar{h}) &(k\bar{l}h) &(\bar{k}lh) &(kl\bar{h})\cr}]$
			$[\openup-1pt\left\{\matrix{\hbox{Tetragon-trioctahedron or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for}\ \|h\| \lt \|l\|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\ rhomb\hbox{-}\hfill\cr dodecahedron\hfill\cr (for\ \|x\| \lt \|z\|)\hfill\cr \cr \hbox{Trigon-trioctahedron or trisoctahedron}\hfill\cr \hbox{(for }\|h\| \gt \|l\|\hbox{)}\hfill\cr Cube\ truncated\ by\ octahedron\hfill\cr (for\ \|x\| \gt \|z\|)\hfill\cr}\right\}]$	$[{\hbox to 6.5pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr\noalign{\vskip8pt} (\bar{h}\bar{h}\bar{l}) &(hh\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (\bar{l}\bar{h}\bar{h}) &(\bar{l}hh) &(lh\bar{h}) &(l\bar{h}h)\cr (\bar{h}\bar{l}\bar{h}) &(h\bar{l}h) &(\bar{h}lh) &(hl\bar{h})\cr}]$
12	c	m..	$[\matrix{\hbox{Pentagon-dodecahedron}\hfill\cr \hbox{or dihexahedron or pyritohedron}\hfill\cr Irregular\ icosahedron\hfill\cr (\!= pentagon\hbox{-}dodecahedron + octahedron)\; (\;j)\hfill\cr}]$	$[{\hbox to 6.5pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k})\cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0)\cr}]$
			$[\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}]$	$[{\hbox to 6.5pc{}}\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr(101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr(110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]$
8	b	.3.	$[\!\matrix{\hbox{Octahedron}\hfill\cr Cube\ (i)\hfill\cr}\hfill]$	$[{\hbox to 6.5pc{}}\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\cr(\bar{1}\bar{1}\bar{1}) &(11\bar{1}) &(1\bar{1}1) &(\bar{1}11)\cr}]$
6	a	2mm..	$[\!\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\ (e)\hfill\cr}]$	$[{\hbox to 6.5pc{}}\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr(001) &(00\bar{1})\cr}]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 2mm&&6&&2mm\cr}]$
432	O
24	d	1	$[\matrix{\hbox{Pentagon-trioctahedron}\hfill\cr \hbox{or gyroid}\hfill\cr \hbox{or pentagon-icositetrahedron}\hfill\cr Snub\ cube (=cube\ +\hfill\cr octahedron + pentagon\hbox{-}\hfill\cr trioctahedron)\ (k)\hfill\cr}]$	$[{\hbox to 2.5pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &&(kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr(lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) &&(\bar{l}kh) &(\bar{l}\bar{k}\bar{h}) &(lk\bar{h}) &(l\bar{k}h)\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) &&(h\bar{l}k) &(\bar{h}\bar{l}\bar{k}) &(\bar{h}lk) &(hl\bar{k})\cr}]$
			$[\left\{\matrix{\hbox{Tetragon-trioctahedron}\hfill\cr \hbox{or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for }\|h\| \;\lt\; \|l\|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\hfill\cr rhomb\hbox{-}dodecahedron\hfill\cr (for\ \|x\| \;\lt\; \|z\|)\hfill\cr\cr \hbox{Trigon-trioctahedron}\hfill\cr \hbox{or trisoctahedron}\hfill\cr \hbox{(for }\|h\| \;\gt\; \|l\|)\hfill\cr Cube\ truncated\ by\ octahedron\hfill\cr (for\ \|x\| \;\gt\; \|z\|)\hfill\cr}\right\}]$	$[{\hbox to 2.5pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) &&(hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h}) &&(\bar{l}hh) &(\bar{l}\bar{h}\bar{h}) &(lh\bar{h}) &(l\bar{h}h)\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h) &&(h\bar{l}h) &(\bar{h}\bar{l}\bar{h}) &(\bar{h}lh) &(hl\bar{h})\cr}]$
			$[\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}]$	$[{\hbox to 2.5pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) &&(k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) &&(\bar{l}k0) &(\bar{l}\bar{k}0) &(lk0) &(l\bar{k}0)\cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) &&(0\bar{l}k) &(0\bar{l}\bar{k}) &(0lk) &(0l\bar{k})\cr}]$
12	c	..2	$[\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\ (i)\hfill\cr}]$	$[{\hbox to 2.5pc{}}\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]$
8	b	.3.	$[\matrix{\hbox{Octahedron}\hfill\cr Cube\ (g)\hfill\cr}]$	$[{\hbox to 2.5pc{}}\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\cr (\bar{1}\bar{1}\bar{1}) &(11\bar{1}) &(1\bar{1}1) &(\bar{1}11)\cr}]$
6	a	4..	$[\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\ (e)\hfill\cr}]$	$[{\hbox to 2.5pc{}}\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&3m&&2mm\cr}]$
$[\bar{4}3m]$	$[T_{d}]$
24	d	1	$[\matrix{\hbox{Hexatetrahedron}\hfill\cr \hbox{or hexakistetrahedron}\hfill\cr Cube\ truncated\ by\hfill\cr two\ tetrahedra\ (\;j)\hfill\cr}]$	$[{\hbox to 1pc{}}\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &&(khl) &(\bar{k}\bar{h}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) &&(lkh) &(l\bar{k}\bar{h}) &(\bar{l}k\bar{h}) &(\bar{l}\bar{k}h)\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) &&(hlk) &(\bar{h}l\bar{k}) &(\bar{h}\bar{l}k) &(h\bar{l}\bar{k})\cr}]$
			$[\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}]$	$[{\hbox to 1pc{}}\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) &&(k0l) &(\bar{k}0l) &(k0\bar{l}) &(\bar{k}0\bar{l})\cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) &&(lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)\cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) &&(0lk) &(0l\bar{k}) &(0\bar{l}k) &(0\bar{l}\bar{k})}]$
12	c	..m	$[\left\{\matrix{\hbox{Trigon-tritetrahedron}\hfill\cr \hbox{or tristetrahedron}\hfill\cr \hbox{(for }\|h\| \lt \|l\|\hbox{)}\hfill\cr Tetrahedron\ truncated\hfill\cr by\ tetrahedron\ (i)\hfill\cr (for\ \|x\| \lt \|z\|)\hfill\cr \cr\hbox{Tetragon-tritetrahedron}\hfill\cr \hbox{or deltohedron}\hfill\cr \hbox{or deltoid-dodecahedron}\hfill\cr \hbox{(for }\|h\|\gt \|l\|\hbox{)}\hfill\cr Cube\ \&\ two\ tetrahedra\ (i)\hfill\cr (for\ \|x\| \gt \|z\|)\hfill\cr}\right\}]$	$[{\hbox to 1pc{}}\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr}]$
			$[\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}]$	$[{\hbox to 1pc{}}\matrix{(110) &(\bar{1}\bar{1}0) &(\bar{1}10) &(1\bar{1}0)\cr (011) &(0\bar{1}\bar{1}) &(0\bar{1}1) &(01\bar{1})\cr (101) &(\bar{1}0\bar{1}) &(10\bar{1}) &(\bar{1}01)\cr}]$
6	b	2.mm	$[\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\ (\;f)\hfill\cr}]$	$[{\hbox to 1pc{}}\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]$
4	a	.3m	$[\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\ (e)\hfill\cr}]$	$[\matrix{}(111) {\hbox to .65pc{}}(\bar{1}\bar{1}1) {\hbox to .65pc{}}(\bar{1}1\bar{1}) {\hbox to .7pc{}}(1\bar{1}\bar{1})\cr \hbox{or }(\bar{1}\bar{1}\bar{1}) {\hbox to .65pc{}}(11\bar{1}) {\hbox to .65pc{}}(1\bar{1}1) {\hbox to .75pc{}}(\bar{1}11)\cr}]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&3m&&m\cr}]$
$[\openup 6pt\matrix{m\bar{3}m\cr\displaystyle{4 \over m}\bar{3}{2 \over m}\cr}]$	$[O_{h}]$
48	f	l	$[\matrix{\hbox{Hexaoctahedron}\hfill\cr \hbox{or hexakisoctahedron}\hfill\cr Cube\ truncated\ by\hfill\cr octahedron\ and\ by\ rhomb\hbox{-}\hfill\cr dodecahedron\ (n)\hfill\cr}]$	$[\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &&(kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) &&(\bar{l}kh) &(\bar{l}\bar{k}\bar{h}) &(lk\bar{h}) &(l\bar{k}h)\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) &&(h\bar{l}k) &(\bar{h}\bar{l}\bar{k}) &(\bar{h}lk) &(hl\bar{k})\cr\noalign{\vskip6pt} (\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl) &&(\bar{k}\bar{h}l) &(khl) &(\bar{k}h\bar{l}) &(k\bar{h}\bar{l})\cr (\bar{l}\bar{h}\bar{k}) &(\bar{l}hk) &(lh\bar{k}) &(l\bar{h}k) &&(l\bar{k}\bar{h}) &(lkh) &(\bar{l}\bar{k}h) &(\bar{l}k\bar{h})\cr (\bar{k}\bar{l}\bar{h}) &(k\bar{l}h) &(\bar{k}lh) &(kl\bar{h}) &&(\bar{h}l\bar{k}) &(hlk) &(h\bar{l}\bar{k}) &(\bar{h}\bar{l}k)\cr}]$
24	e	..m	$[\left\{\matrix{\hbox{Tetragon-trioctahedron}\hfill\cr \hbox{or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for }\|h\| \lt \|l\|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\ rhomb\hbox{-}\hfill\cr dodecahedron\ (m)\hfill\cr (for\ \|x\| \lt \|z\|)\hfill\cr\cr \hbox{Trigon-trioctahedron}\hfill\cr \hbox{or trisoctahedron}\hfill\cr \hbox{(for }\|h\| \gt \|l\|)\hfill\cr Cube\ truncated\ by\hfill\cr octahedron\ (m)\hfill\cr (for\ \|x\| \gt \|z\|)\hfill\cr}\right\}]$	$[\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) &&(hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h}) &&(\bar{l}hh) &(\bar{l}\bar{h}\bar{h}) &(lh\bar{h}) &(l\bar{h}h)\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h) &&(h\bar{l}h) &(\bar{h}\bar{l}\bar{h}) &(\bar{h}lh) &(hl\bar{h})\cr}]$
24	d	m..	$[\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\hfill\cr by\ cube\ (k)\hfill\cr}]$	$[\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) &&(k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) &&(\bar{l}k0) &(\bar{l}\bar{k}0) &(lk0) &(l\bar{k}0)\cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) &&(0\bar{l}k) &(0\bar{l}\bar{k}) &(0lk) &(0l\bar{k})\cr}]$
12	c	m.m2	$[\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\ (i)\hfill\cr}]$	$[\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]$
8	b	.3m	$[\matrix{\hbox{Octahedron}\hfill\cr Cube\ (g)\hfill\cr}]$	$[\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1}) {\hbox to 1.5pc{}}(11\bar{1}) &(\bar{1}\bar{1}\bar{1}) &(1\bar{1}1) &(\bar{1}11)\cr}]$

6	a	4m.m	$[\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\ (e)\hfill\cr}]$	$[\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&6mm&&2mm\cr}]$

In Tables 10.1.2.1 and 10.1.2.2 , some point groups are described in two or three versions, in order to bring out the relations to the corresponding space groups (cf. Section 2.2.3 ):

(a) The three monoclinic point groups 2, m and are given with two settings, one with `unique axis b' and one with `unique axis c'.
(b) The two point groups $[\overline{4}2m]$ and $[\overline{6}m2]$ are described for two orientations with respect to the crystal axes, as $[\overline{4}2m]$ and $[\overline{4}m2]$ and as $[\overline{6}m2]$ and $[\overline{6}2m]$ .
(c) The five trigonal point groups 3, $[\overline{3}]$ , 32, 3m and $[\overline{3}m]$ are treated with two axial systems, `hexagonal axes' and `rhombohedral axes'.
(d) The hexagonal-axes description of the three trigonal point groups 32, 3m and $[\overline{3}m]$ is given for two orientations, as 321 and 312, as 3m1 and 31m, and as $[\overline{3}m1]$ and $[\overline{3}1m]$ ; this applies also to the two-dimensional point group 3m.

The presentation of the point groups is similar to that of the space groups in Part 7 . 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 two-dimensional groups. For an explanation of the symbols see Section 2.2.4 and Chapter 12.1 .

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 2.1.2.1 and illustrated in Figs. 2.2.6.1 to 2.2.6.10 .

In the right-hand projection, the graphical symbols of the symmetry elements are the same as those used in the space-group diagrams; they are listed in Chapter 1.4 . Note that the symbol of a symmetry centre , a small circle, is also used for a face-pole in the left-hand 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 left-hand 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.

(i) As general face poles, where they represent general crystal faces which form a polyhedron, the `general crystal form ' (face form) $[\{hkl\}]$ of the point group (see below). In two dimensions, edges, edge poles, edge forms and polygons take the place of faces, face poles, crystal forms (face forms) and polyhedra in three dimensions.

Face poles marked as dots lie above the projection plane and represent faces which intersect the positive c axis² (positive Miller index l), those marked as circles lie below the projection plane (negative Miller index l). In two dimensions, edge poles always lie on the pole circle.
(ii) As general points (centres of atoms) that span a polyhedron or polygon, the `general crystallographic point form' x, y, z. This interpretation is of interest in the study of coordination polyhedra, atomic groups and molecular shapes. The polyhedron or polygon of a point form is dual to the polyhedron of the corresponding crystal form.³

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 under Crystal and point forms; the explanation of the listed data is to be found under 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 three-dimensional 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 2.2.14 . The relations between the axes of the three-dimensional point group and those of its two-dimensional projections can easily be derived with the help of the stereographic projection. No projection symmetries are listed for the two-dimensional 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 $[\overline{1}]$ all faces have face symmetry 1, whereas projections along any direction have symmetry 2; in point group 422, the faces (001) and $[(00\overline{1})]$ have face symmetry 4, whereas the projection along [001] has symmetry 4mm.

10.1.2.2. Crystal and point forms

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For a point group $[{\cal P}]$ a crystal form is a set of all symmetrically equivalent faces; a point form is a set of all symmetrically equivalent points. Crystal and point forms in point groups correspond to `crystallographic orbits ' in space groups; cf. Section 8.3.2 .

Two kinds of crystal and point forms with respect to $[{\cal P}]$ can be distinguished. They are defined as follows:

(i) General form: A face is called `general' if only the identity operation transforms the face onto itself. Each complete set of symmetrically equivalent `general faces' is a general crystal form. The multiplicity of a general form , i.e. the number of its faces, is the order of $[{\cal P}]$ . In the stereographic projection, the poles of general faces do not lie on symmetry elements of $[{\cal P}]$ .

A point is called `general' if its site symmetry, i.e. the group of symmetry operations that transforms this point onto itself, is 1. A general point form is a complete set of symmetrically equivalent `general points'.
(ii) Special form: A face is called `special' if it is transformed onto itself by at least one symmetry operation of $[{\cal P}]$ , in addition to the identity. Each complete set of symmetrically equivalent `special faces' is called a special crystal form. The face symmetry of a special face is the group of symmetry operations that transforms this face onto itself; it is a subgroup of $[{\cal P}]$ . The multiplicity of a special crystal form is the multiplicity of the general form divided by the order of the face-symmetry group. In the stereographic projection, the poles of special faces lie on symmetry elements of $[{\cal P}]$ . The Miller indices of a special crystal form obey restrictions like $[\{hk0\}]$ , $[\{hhl\}, \{100\}]$ .

A point is called `special ' if its site symmetry is higher than 1. A special point form is a complete set of symmetrically equivalent `special points'. The multiplicity of a special point form is the multiplicity of the general form divided by the order of the site-symmetry group. It is thus the same as that of the corresponding special crystal form. The coordinates of the points of a special point form obey restrictions, like x, y, 0; x, x, z; x, 0, 0. The point 0, 0, 0 is not considered to be a point form.

In two dimensions, point groups 1, 2, 3, 4 and 6 and, in three dimensions, point groups 1 and $[\overline{1}]$ have no special crystal and point forms.

General and special crystal and point forms can be represented by their sets of equivalent Miller indices $[\{hkl\}]$ 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 2.2.11 and 8.3.2 . 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 $[{\cal P}]$ for which the face poles (points) are positioned on the same set of conjugate symmetry elements of $[{\cal P}]$ ; 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 so-called `characteristic triangle' of the stereographic projection.

Examples

(1) All tetragonal pyramids $[\{hkl\}]$ and tetragonal prisms $[\{hk0\}]$ in point group 4 have face symmetry 1 and belong to the same general `Wyckoff position' 4b, with Wyckoff letter b.
(2) All tetragonal pyramids and tetragonal prisms in point group 4mm belong to two special `Wyckoff positions', depending on the orientation of their face-symmetry groups m with respect to the crystal axes: For the `oriented face symmetry' .m., the forms $[\{h0l\}]$ and $[\{100\}]$ belong to Wyckoff position 4c; for the oriented face symmetry ..m, the forms $[\{hhl\}]$ and $[\{110\}]$ belong to Wyckoff position 4b. The face symmetries .m. and ..m are not conjugate in point group 4mm. For the analogous `oriented site symmetries' in space groups, see Section 2.2.12 .

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:

(i) the point group $[{\cal P}]$ by which a form is generated (generating point group), i.e. the point group in which it occurs;
(ii) the full symmetry (inherent symmetry ) of a form (considered as a polyhedron by itself), here called eigensymmetry $[{\cal C}]$ . The eigensymmetry point group $[{\cal C}]$ is either the generating point group itself or a supergroup of it.

Examples

(1) Each tetragonal pyramid $[\{hkl\}\ (l \neq 0)]$ of Wyckoff position 4b in point group 4 has generating symmetry 4 and eigensymmetry 4mm; each tetragonal prism $[\{hk0\}]$ of the same Wyckoff position has generating symmetry 4 again, but eigensymmetry .
(2) A cube $[\{100\}]$ may have generating symmetry 23, $[m\overline{3}]$ , 432, $[\overline{4}3m]$ or $[m\overline{3}m]$ , but its eigensymmetry is always $[m\overline{3}m]$ .

The eigensymmetries and the generating symmetries of the 47 crystal forms (point forms) are listed in Table 10.1.2.3 . 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 10.2.2 and 10.2.3 .

Table 10.1.2.3| top | pdf |
The 47 crystallographic face and point forms, their names, eigensymmetries, and their occurrence in the crystallographic point groups (generating point groups)

The oriented face (site) symmetries of the forms are given in parentheses after the Hermann–Mauguin symbol (column 6); a symbol such as [mm2(.m.,m..)] indicates that the form occurs in point group mm2 twice, with face (site) symmetries .m. and m... Basic (general and special) forms are printed in bold face, limiting (general and special) forms in normal type. The various settings of point groups 32, 3m, $[\overline{3}m, \overline{4}2m]$ and $[\overline{6}m2]$ are connected by braces.

No.	Crystal form	Point form	Number of faces or points	Eigensymmetry	Generating point groups with oriented face (site) symmetries between parentheses
1	Pedion or monohedron	Single point	1	$[\infty m]$	$[\matrix{{\bf 1(1)};\ {\bf 2(2)};\ {\bi m}({\bi m});\ {\bf 3(3)};\ {\bf 4(4)};\hfill\cr {\bf 6(6)};\ {\bi m}{\bi m}{\bf 2}({\bi m}{\bi m}{\bf 2});\ {\bf 4}{\bi m}{\bi m}({\bf 4}{\bi m}{\bi m});\hfill\cr {\bf 3}{\bi m}({\bf 3}{\bi m});\ {\bf 6}{\bi m}{\bi m}({\bf 6}{\bi m}{\bi m})\hfill\cr}]$
2	Pinacoid or parallelohedron	Line segment through origin	2	$[\displaystyle{\infty \over m}m]$	$[\matrix{\overline{{\bf 1}} {\bf (1)}\semi\ 2(1)\semi\ m(1)\semi\ {\displaystyle{{\bf 2} \over {\bi m}}}({\bf 2}{\hbox{\bf.}}{\bi m})\semi\ {\bf 222}({\bf 2\hbox{..}}, {\bf \hbox{.}2\hbox{.}}, {\bf \hbox{..}2})\semi\hfill\cr mm2(.m., m..)\semi\ {\bi m}{\bi m}{\bi m}({\bf 2}{\bi m}{\bi m}, {\bi m}{\bf 2}{\bi m}, {\bi m}{\bi m}{\bf 2})\semi\hfill\cr \overline{{\bf 4}}({\bf 2\hbox{..}})\semi\ {\displaystyle{{\bf 4} \over {\bi m}}} ({\bf 4\hbox{..}})\semi\ {\bf 422}({\bf 4\hbox{..}}), \cases{\overline{{\bf 4}}{\bf 2}{\bi m}({\bf 2\hbox{.}}{\bi m}{\bi m})\cr \overline{{\bf 4}}{\bi m}{\bf 2}({\bf 2}{\bi m}{\bi m\hbox{.}})\cr}\semi\hfill\cr {\displaystyle{{\bf 4} \over {\bi m}}}{\bi m}{\bi m}({\bf 4}{\bi m}{\bi m})\semi\ \overline{{\bf 3}}({\bf 3\hbox{..}})\semi\ \cases{{\bf 321}({\bf 3\hbox{..}})\cr {\bf 312}({\bf 3\hbox{..}})\semi\cr {\bf 32}\phantom 2({\bf 3\hbox{.}})\cr} \cases{\overline{{\bf 3}}{\bi m}{\bf 1}({\bf 3}{\bi m\hbox{.}})\cr \overline{{\bf 3}}{\bf 1}{\bi m}({\bf 3\hbox{.}}{\bi m})\semi\cr \overline{{\bf 3}}{\bi m}{\bf 1}({\bf 3}{\bi m})\cr}\cr \overline{{\bf 6}} ({\bf 3\hbox{..}})\semi\ {\displaystyle{{\bf 6} \over {\bi m}}} ({\bf 6\hbox{..}})\semi\ {\bf 622}({\bf 6\hbox{..}})\semi\hfill\cr \cases{\overline{{\bf 6}}{\bi m}{\bf 2}({\bf 3}{\bi m\hbox{.}})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bf 3\hbox{.}}{\bi m})\cr}\semi\ {\displaystyle{{\bf 6} \over {\bi m}}}{\bi mm}({\bf 6}{\bi mm})\hfill\cr}]$
3	Sphenoid, dome, or dihedron	Line segment	2	mm2	$[{\bf 2(1)};\ {\bi m}{\bf (1)};\ {\bi m}{\bi m}{\bf 2}({\bi \hbox{.}m\hbox{.}}, {\bi m\hbox{..}})]$
4	Rhombic disphenoid or rhombic tetrahedron	Rhombic tetrahedron	4	222	$[{\bf 222(1)}]$
5	Rhombic pyramid	Rectangle	4	mm2	$[{\bi m}{\bi m}{\bf 2(1)}]$
6	Rhombic prism	Rectangle through origin	4	mmm	$[{\bf 2}/{\bi m}{\bf (1)};\ 222(1)]$ ^† $[;\ mm 2(1);\ {\bi m}{\bi m}{\bi m}({\bi m\hbox{..}}, {\bi \hbox{.}m\hbox{.}}, {\bi \hbox{..}m})]$
7	Rhombic dipyramid	Quad	8	mmm	$[{\bi m}{\bi m}{\bi m}{\bf (1)}]$
8	Tetragonal pyramid	Square	4	4mm	$[{\bf 4(1)};\ {\bf 4}{\bi m}{\bi m}({\bi \hbox{..}m}, {\bi \hbox{.}m\hbox{.}})]$
9	Tetragonal disphenoid or tetragonal tetrahedron	Tetragonal tetrahedron	4	$[\overline{4}2m]$	$[\overline{{\bf 4}}{\bf (1)};\ \cases{\overline{{\bf 4}}{\bf 2}{\bi m}({\bi \hbox{..}m})\cr \overline{{\bf 4}}{\bi m}{\bf 2}({\bi \hbox{.}m\hbox{.}})\cr}]$
10	Tetragonal prism	Square through origin	4	$[\displaystyle{4 \over m}mm]$	$[4(1);\ \overline{4}(1);\ {{\bf 4} \over {\bi m}}({\bi m\hbox{..}});\ {\bf 422(\hbox{..}2},{\bf \hbox{.}2\hbox{.})};\ 4mm(..m, .m.);]$
					^‡ $[\left\{\matrix{\overline{{\bf 4}}{\bf 2}{\bi m}{\bf (\hbox{.}2\hbox{.})}\ \& \ \overline{4}2m(..m)\cr \overline{{\bf 4}}{\bi m}{\bf 2(\hbox{..}2)} \ \& \ \overline{4}m2(.m.)\cr}\right.;]$
					$[{\displaystyle{\bf{4}\over \bi{m}}}{\bi mm}({\bi m\hbox{.}m}{\bf 2}, {\bi m}{\bf 2}{\bi m\hbox{.}})]$
11	Tetragonal trapezohedron	Twisted tetragonal antiprism	8	422	$[{\bf 422(1)}]$
12	Ditetragonal pyramid	Truncated square	8	4mm	$[{\bf 4}{\bi m}{\bi m}{\bf (1)}]$
13	Tetragonal scalenohedron	Tetragonal tetrahedron cut off by pinacoid	8	$[\overline{4}2m]$	$[\cases{\overline{{\bf 4}}{\bf 2}{\bi m}{\bf (1)}\cr \overline{{\bf 4}}{\bi m}{\bf 2(1)}\cr}]$
14	Tetragonal dipyramid	Tetragonal prism	8	$[\displaystyle{4 \over m}mm]$	$[{\displaystyle{{\bf 4} \over {\bi m}}}{\bf (1)};\ 422{(1)}]$ ^†;^‡ $[\cases{\overline{4}2m(1)\cr \overline{{4}}m{2(1)}\cr}\semi\ {\displaystyle{{\bf 4} \over {\bi m}}}{\bi m}{\bi m}({\bi \hbox{.}m}, {\bi \hbox{.}m\hbox{.}})]$
15	Ditetragonal prism	Truncated square through origin	8	$[\displaystyle{4 \over m}mm]$	$[422(1)\semi\ 4mm(1)\semi\ \cases{\overline{4}2m(1)\cr \overline{{4}}m{2(1)}\cr}\semi\ {\displaystyle{{\bf 4} \over {\bi m}}}{\bi m}{\bi m}({\bi m\hbox{..}})]$
16	Ditetragonal dipyramid	Edge-truncated tetragonal prism	16	$[\displaystyle{4 \over m}mm]$	$[{\displaystyle{{\bf 4} \over {\bi m}}}{\bi m}{\bi m}{\bf (1)}]$
17	Trigonal pyramid	Trigon	3	3m	$[{\bf 3(1)}; \cases{{\bf 3}{\bi m}{\bf 1}({\bi \hbox{.}m\hbox{.}})\cr {\bf 31}{\bi m}({\bi \hbox{..}m})\cr {\bf 3}{\bi m}{}({\bi \hbox{.}m})\cr}]$
18	Trigonal prism	Trigon through origin	3	$[\overline{6}2m]$	$[\matrix{3(1)\semi \cases{{\bf 321}({\bf \hbox{.}2\hbox{.}})\cr {\bf 312}({\bf \hbox{..}2})\ \semi\ \cr {\bf 32} \ \ ({\bf \hbox{.}2})\cr}\ \cases{3m1(.m.)\cr 31m(..m);\cr 3m (.m)\cr}\hfill\cr \overline{{\bf 6}}({\bi m\hbox{..}})\semi \cases{\overline{{\bf 6}}{\bi m}{\bf 2}({\bi m}{\bi m}{\bf 2})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bi m}{\bf 2}{\bi m})\cr}\hfill\cr}]$
19	Trigonal trapezohedron	Twisted trigonal antiprism	6	32	$[\cases{{\bf 321(1)}\cr {\bf 312(1)}\cr {\bf 32} {\bf (1)}\cr}]$
20	Ditrigonal pyramid	Truncated trigon	6	3m	$[{\bf 3}{\bi m}{\bf (1)}]$
21	Rhombohedron	Trigonal antiprism	6	$[\overline{3}m]$	$[\overline{{\bf 3}}{\bf (1)}\semi \cases{321(1)\cr 312(1)\semi\cr 32{} (1)\cr}\ \cases{\overline{{\bf 3}}{\bi m}{\bf 1}({\bi \hbox{.}m\hbox{.}})\cr \overline{{\bf 3}}{\bf 1}{\bi m}({\bi \hbox{..}m})\cr \overline{{\bf 3}}{\bi m} \ ({\bi \hbox{.}m})\cr}]$
22	Ditrigonal prism	Truncated trigon through origin	6	$[\overline{6}2m]$	$[\matrix{\cases{321(1)\cr 312(1)\semi\cr 32 (1)\cr}\ \cases{3m1(1)\cr 31m(1)\semi\cr 3m (1)\cr}\hfill\cr\noalign{\vskip3pt}\cases{\overline{{\bf 6}}{\bi m}{\bf 2}({\bi m\hbox{..}})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bi m\hbox{..}})\cr}\hfill\cr}]$
23	Hexagonal pyramid	Hexagon	6	6mm	$[\cases{3m1(1)\cr 31m (1)\semi\cr 3m (1)\cr}\ {\bf 6(1)}\semi\ {\bf 6}{\bi m}{\bi m}({\bi \hbox{..}m},{\bi \hbox{.}m\hbox{.}})]$
24	Trigonal dipyramid	Trigonal prism	6	$[\overline{6}2m]$	$[\cases{321(1)\cr 312(1);\cr 32 (1)\cr}\ \ \overline{{\bf 6}}{\bf (1)};\ \cases{\overline{{\bf 6}}{\bi m}{\bf 2}({\bi \hbox{.}m\hbox{.}})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bi \hbox{..}m})\cr}]$
25	Hexagonal prism	Hexagon through origin	6	$[\displaystyle{6 \over m}mm]$	$[\overline{3}(1)\semi \ \cases{321(1)\cr 312(1)\semi\cr 32 \phantom{1}(1)\cr}\cases{3m1 (1)\cr 31m (1)\cr 3m \phantom{1}(1)\cr}]$
					^‡ $[\cases{\overline{{\bf 3}}{\bi m}{\bf 1}({\bf\hbox{.} 2\hbox{.}})\ \& \ \overline{{3}}m1(.m.)\hfill\cr \overline{{\bf 3}}{\bf 1}{\bi m}({\bf \hbox{..}2}) \ \& \ \overline{3}1m(..m)\semi\hfill\cr \overline{{\bf 3}}{\bi m}({\bf \hbox{.}2}) \ \& \ \overline{3}m(.m)\hfill\cr}]$
					$[\displaylines{6(1)\semi\ {{\bf 6} \over {\bi m}}({\bi m\hbox{..}})\semi\ {\bf 622}({\bf \hbox{.}2\hbox{.}},{\bf \hbox{..}2})\semi\hfill\cr 6mm(..m,.m.)\semi\ \cases{\overline{6}m2(m..)\cr \overline{6}2m(m..)\cr}\semi\hfill\cr{\displaystyle{{\bf 6} \over {\bi m}}} {\bi m}{\bi m} ({\bi m}{\bf 2}{\bi m}, {\bi m}{\bi m}{\bf 2})\hfill\cr}]$
26	Ditrigonal scalenohedron or hexagonal scalenohedron	Trigonal antiprism sliced off by pinacoid	12	$[\overline{3}m]$	$[\cases{\overline{{\bf 3}}{\bi m}{\bf 1} {\bf (1)}\cr \overline{{\bf 3}}{\bf 1}{\bi m} {\bf (1)}\cr \overline{{\bf 3}}{\bi m} {\bf (1)}\cr}]$
27	Hexagonal trapezohedron	Twisted hexagonal antiprism	12	622	$[{\bf 622(1)}]$
28	Dihexagonal pyramid	Truncated hexagon	12	6mm	$[{\bf 6}{\bi m}{\bi m} {\bf (1)}]$
29	Ditrigonal dipyramid	Edge-truncated trigonal prism	12	$[\overline{6}2m]$	$[\cases{\overline{{\bf 6}}{\bi m}{\bf 2} {\bf (1)}\cr \overline{{\bf 6}}{\bf 2}{\bi m} {\bf (1)}\cr}]$
30	Dihexagonal prism	Truncated hexagon	12	$[\displaystyle{6 \over m}mm]$	$[\matrix{\cases{\overline{3}m1 (1)\cr \overline{3}1m (1);\cr \overline{3}m (1)\cr}\ 622 (1);\ 6mm (1);\hfill\cr\cr{\displaystyle{{\bf 6} \over {\bi m}}}{\bi m}{\bi m}({\bi m\hbox {..}})\hfill\cr}]$
31	Hexagonal dipyramid	Hexagonal prism	12	$[\displaystyle{6 \over m}mm]$	$[\cases{\overline{3}m1 (1)\cr \overline{3}1m (1)\semi\cr \overline{3}m \phantom{1}(1)\cr}\ {6 \over m}(1)\semi \ 622(1)]$ ^†;
31	Hexagonal dipyramid	Hexagonal prism	12	$[\displaystyle{6 \over m}mm]$	$[\cases{\overline{6}m2 (1)\cr \overline{6}2m (1)\cr}\semi\ \ {{\bf 6} \over {\bi m}}{\bi m}{\bi m}({\bi \hbox {..}m},{\bi \hbox {.}m\hbox {.}})]$
32	Dihexagonal dipyramid	Edge-truncated hexagonal prism	24	$[\displaystyle{6 \over m}mm]$	$[{\displaystyle{{\bf 6} \over {\bi m}}}{\bi m}{\bi m} {\bf (1)}]$
33	Tetrahedron	Tetrahedron	4	$[\overline{4}3m]$	$[{\bf 23}({\bf \hbox {.}3\hbox {.}});\ \overline{{\bf 4}}{\bf 3}{\bi m}({\bf\hbox {.} 3}{\bi m})]$
34	Cube or hexahedron	Octahedron	6	$[m\overline{3}m]$	$[\!\matrix{{\bf 23}({\bf 2\hbox {..}});\ {\bi m}\overline{{\bf 3}}({\bf 2}{\bi m}{\bi m\hbox {..}});\hfill\cr {\bf 432}({\bf 4\hbox {..}});\ \overline{{\bf 4}}{\bf 3}{\bi m}({\bf 2\hbox {.}}{\bi m}{\bi m});\ {\bi m}\overline{{\bf 3}}{\bi m}({\bf 4}{\bi m\hbox {.}m})\cr}]$
35	Octahedron	Cube	8	$[m\overline{3}m]$	$[{\bi m}\overline{{\bf 3}}({\bf \hbox {.}3\hbox {.}});\ {\bf 432}({\bf \hbox {.}3\hbox {.}});\ {\bi m}{\overline{{\bf 3}}}{\bi m}({\bf\hbox{.}}{{\bf 3}}{\bi m})]$
36	Pentagon-tritetrahedron or tetartoid or tetrahedral pentagon-dodecahedron	Snub tetrahedron (= pentagon-tritetrahedron + two tetrahedra)	12	23	$[{\bf 23}{\bf (1)}]$
37	Pentagon-dodecahedron or dihexahedron or pyritohedron	Irregular icosahedron (= pentagon-dodecahedron + octahedron)	12	$[m\overline{3}]$	$[23(1);\ {\bi m}\overline{{\bf 3}}({\bi m\hbox{..}})]$
38	Tetragon-tritetrahedron or deltohedron or deltoid-dodecahedron	Cube and two tetrahedra	12	$[\overline{4}3m]$	$[23 (1); \ \overline{{\bf 4}}{\bf 3}{\bi m}(\bf\hbox{..}{\bi m})]$
39	Trigon-tritetrahedron or tristetrahedron	Tetrahedron truncated by tetrahedron	12	$[\overline{4}3m]$	$[23(1); \ \overline{{\bf 4}}{\bf 3}{\bi m}({\bi \hbox {..}m})]$
40	Rhomb-dodecahedron	Cuboctahedron	12	$[m\overline{3}m]$	$[\matrix{23(1); \ m\overline{3}(m..);\ {\bf 432}({\bf \hbox{..}2});\hfill\cr \overline{4}3m(..m);\ {\bi m}\overline{{\bf 3}}{\bi m}({\bi m}{\bi \hbox{.}m}{\bf 2})\hfill\cr}]$
41	Didodecahedron or diploid or dyakisdodecahedron	Cube & octahedron & pentagon-dodecahedron	24	$[m\overline{3}]$	$[{\bi m}\overline{{\bf 3}}{\bf (1)}]$
42	Trigon-trioctahedron or trisoctahedron	Cube truncated by octahedron	24	$[m\overline{3}m]$	$[m\overline{3} (1); \ 432 (1); \ {\bi m}\overline{{\bf 3}}{\bi m}({\bi \hbox{..}m})]$
43	Tetragon-trioctahedron or trapezohedron or deltoid-icositetrahedron	Cube & octahedron & rhomb-dodecahedron	24	$[m\overline{3}m]$	$[m\overline{3} (1); \ 432(1); \ {\bi m}\overline{{\bf 3}}{\bi m}({\bi \hbox{..}m})]$
44	Pentagon-trioctahedron or gyroid	Cube + octahedron + pentagon-trioctahedron	24	432	$[{\bf 432} {\bf (1)}]$
45	Hexatetrahedron or hexakistetrahedron	Cube truncated by two tetrahedra	24	$[\overline{4}3m]$	$[\overline{{\bf 4}}{\bf 3}{\bi m} {\bf (1)}]$
46	Tetrahexahedron or tetrakishexahedron	Octahedron truncated by cube	24	$[m\overline{3}m]$	$[432 (1); \ \overline{4}3m (1);\ {\bi m}\overline{{\bf 3}}{\bi m}({\bi m\hbox{..}})]$
47	Hexaoctahedron or hexakisoctahedron	Cube truncated by octahedron and by rhomb-dodecahedron	48	$[m\overline{3}m]$	$[{\bi m}\overline{{\bf 3}}{\bi m} {\bf (1)}]$

^†These limiting forms occur in three or two non-equivalent orientations (different types of limiting forms); cf. Table 10.1.2.2

.
^‡In point groups $[\overline{4}2m]$ and $[\overline{3}m]$ , 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 $[{\bf \overline{4}2}{\bi m}({\bf\hbox{.} 2\hbox{.}})\ \& \ \overline{4}2m(..m)]$ and as $[\overline{{\bf 3}}{\bi m}{\bf 1}({\bf \hbox{.}2\hbox{.}})\ \& \ \overline{3}m1(.m.)]$ .

With the help of the two groups $[{\cal P}]$ and $[{\cal C}]$ , each crystal or point form occurring in a particular point group can be assigned to one of the following two categories:

(i) characteristic form, if eigensymmetry $[{\cal C}]$ and generating symmetry $[{\cal P}]$ are the same;
(ii) noncharacteristic form, if $[{\cal C}]$ is a proper supergroup of $[{\cal P}]$ .

The importance of this classification will be apparent from the following examples.

Examples

(1) A pedion and a pinacoid are noncharacteristic forms in all crystallographic point groups in which they occur:
(2) all other crystal or point forms occur as characteristic forms in their eigensymmetry group $[{\cal C}]$ ;
(3) a tetragonal pyramid is noncharacteristic in point group 4 and characteristic in 4mm;
(4) a hexagonal prism can occur in nine point groups (12 Wyckoff positions) as a noncharacteristic form; in , it occurs in two Wyckoff positions as a characteristic form.

The general forms of the 13 point groups with no, or only one, symmetry direction (`monoaxial groups') $[1, 2, 3, 4, 6, \overline{1}, m, \overline{3}, \overline{4}]$ , $[\overline{6} =]$ [3/m, 2/m, 4/m, 6/m] 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 10.2.2 ).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 10.1.2.3 ). 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 topology⁶ 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

(1) In point group 4, the (noncharacteristic) crystal forms $[{\{hkl\}\ (l \neq 0)}]$ (tetragonal pyramids) of eigensymmetry 4mm are basic forms of the general Wyckoff position 4b, whereas the forms $[\{hk0\}]$ (tetragonal prisms) of higher eigensymmetry are `limiting general forms'. The face poles of forms $[\{hk0\}]$ lie on the horizontal mirror plane of the supergroup .
(2) In point group 4mm, the (characteristic) special crystal forms $[\{h0l\}]$ with eigensymmetry 4mm are `basic forms' of the special Wyckoff position 4c, whereas $[\{100\}]$ with eigensymmetry is a `limiting special form'. The face poles of $[\{100\}]$ are located on the intersections of the vertical mirror planes of the point group 4mm with the horizontal mirror plane of the supergroup , i.e. on twofold axes of .

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 [6/mmm] 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 `realizationsFace form' of one type of limiting form in point groups 23, $[m\overline{3}]$ and 432 is explained below in Section 10.1.2.4 , Notes on crystal and point forms, item (viii).

Examples

(1) In point group 32, the limiting general crystal forms are of four types :

(i) ditrigonal prisms, eigensymmetry $[\overline{6}2m]$ (face poles on horizontal mirror plane of holohedry );
(ii) trigonal dipyramids, eigensymmetry $[\overline{6}2m]$ (face poles on one kind of vertical mirror plane);
(iii) rhombohedra, eigensymmetry $[\overline{3}m]$ (face poles on second kind of vertical mirror plane);
(iv) hexagonal prisms, eigensymmetry (face poles on horizontal twofold axes).

Types (i) and (ii) have the same eigensymmetry but different topologies; types (i) and (iv) have the same topology but different eigensymmetries; type (iii) differs from the other three types in both eigensymmetry and topology.

(2) In point group 222, the face poles of the three types of general limiting forms, rhombic prisms, are located on the three (non-equivalent) symmetry planes of the holohedry mmm. Geometrically, the axes of the prisms are directed along the three non-equivalent orthorhombic symmetry directions. The three types of limiting forms have the same eigensymmetry and the same topology but different orientations.

Similar cases occur in point groups 422 and 622 (cf. the first footnote to Table 10.1.2.3¹⁰).

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 pentagon-dodecahedron to a regular pentagon-dodecahedron. 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 `ditetragonal-pyramidal class'; some of these names are listed in Table 10.1.2.4 .

Table 10.1.2.4| top | pdf |
Names and symbols of the 32 crystal classes

System used in this volume	Point group		Schoenflies symbol	Class names
	International symbol			Class names
	Short	Full		Groth (1921)	Friedel (1926)
Triclinic	1	1	$[C_{1}]$	Pedial (asymmetric)	Hemihedry
Triclinic	$[\overline{1}]$	$[\overline{1}]$	$[C_{i}(S_{2})]$	Pinacoidal	Holohedry
Monoclinic	2	2	$[C_{2}]$	Sphenoidal	Holoaxial hemihedry
	m	m	$[C_{s}(C_{1h})]$	Domatic	Antihemihedry
		$[\displaystyle{2 \over m}]$	$[C_{2h}]$	Prismatic	Holohedry
Orthorhombic	222	222	$[D_{2} (V)]$	Disphenoidal	Holoaxial hemihedry
	mm2	mm2	$[C_{2v}]$	Pyramidal	Antihemihedry
	mmm	$[\displaystyle{2 \over m} {2 \over m} {2 \over m}]$	$[D_{2h} (V_{h})]$	Dipyramidal	Holohedry
Tetragonal	4	4	$[C_{4}]$	Pyramidal	Tetartohedry with 4-axis
	$[\overline{4}]$	$[\overline{4}]$	$[S_{4}]$	Disphenoidal	Sphenohedral tetartohedry
		$[\displaystyle{4 \over m}]$	$[C_{4h}]$	Dipyramidal	Parahemihedry
	422	422	$[D_{4}]$	Trapezohedral	Holoaxial hemihedry
	4mm	4mm	$[C_{4v}]$	Ditetragonal-pyramidal	Antihemihedry with 4-axis
	$[\overline{4}2m]$	$[\overline{4}2m]$	$[D_{2d} (V_{d})]$	Scalenohedral	Sphenohedral antihemihedry
	4/mmm	$[\displaystyle{4 \over m} {2 \over m} {2 \over m}]$	$[D_{4h}]$	Ditetragonal-dipyramidal	Holohedry
					Hexagonal	Rhombohedral
Trigonal	3	3	$[C_{3}]$	Pyramidal	Ogdohedry	Tetartohedry
	$[\overline{3}]$	$[\overline{3}]$	$[C_{3i}(S_{6})]$	Rhombohedral	Paratetartohedry	Parahemihedry
	32	32	$[D_{3}]$	Trapezohedral	Holoaxial tetartohedry with 3-axis	Holoaxial hemihedry
	3m	3m	$[C_{3v}]$	Ditrigonal-pyramidal	Hemimorphic antitetartohedry	Antihemihedry
	$[\overline{3}m]$	$[\overline{3} \displaystyle{2 \over m}]$	$[D_{3d}]$	Ditrigonal-scalenohedral	Parahemihedry with 3-axis	Holohedry
Hexagonal	6	6	$[C_{6}]$	Pyramidal	Tetartohedry with 6-axis
	$[\overline{6}]$	$[\overline{6}]$	$[C_{3h}]$	Trigonal-dipyramidal	Trigonohedral antitetartohedry
		$[\displaystyle{6 \over m}]$	$[C_{6h}]$	Dipyramidal	Parahemihedry with 6-axis
	622	622	$[D_{6}]$	Trapezohedral	Holoaxial hemihedry
	6mm	6mm	$[C_{6v}]$	Dihexagonal-pyramidal	Antihemihedry with 6-axis
	$[\overline{6}2m]$	$[\overline{6}2m]$	$[D_{3h}]$	Ditrigonal-dipyramidal	Trigonohedral antihemihedry
		$[\displaystyle{6 \over m} {2 \over m} {2 \over m}]$	$[D_{6h}]$	Dihexagonal-dipyramidal	Holohedry
Cubic	23	23	T	$[\!\matrix{\hbox{Tetrahedral-pentagondodecahedral}\hfill\cr\quad (=\hbox{tetartoidal})\hfill\cr}]$	Tetartohedry
	$[m\overline{3}]$	$[\displaystyle{2 \over m} \overline{3}]$	$[T_{h}]$	$[\!\matrix{\hbox{Disdodecahedral}\hfill\cr\quad(=\hbox{diploidal})\hfill\cr}]$	Parahemihedry
	432	432	O	$[\!\matrix{\hbox{Pentagon-icositetrahedral}\hfill\cr\quad(=\hbox{gyroidal})\hfill\cr}]$	Holoaxial hemihedry
	$[\overline{4}3m]$	$[\overline{4}3m]$	$[T_{d}]$	$[\!\matrix{\hbox{Hexakistetrahedral}\hfill\cr \quad(=\hbox{hextetrahedral})\hfill\cr}]$	Antihemihedry
	$[m\overline{3}m]$	$[\displaystyle{4 \over m} \overline{3} {2 \over m}]$	$[O_{h}]$	$[\!\matrix{\hbox{Hexakisoctahedral}\hfill\cr\quad(=\hbox{hexoctahedral})\hfill\cr}]$	Holohedry

10.1.2.3. Description of crystal and point forms

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The main part of each point-group table 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 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 10.1.2.4 , 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. Section 2.2.11 ).

Column 3: Face symmetry or site symmetry, given in the form of an `oriented point-group symbol', analogous to the oriented site-symmetry symbols of space groups (cf. Section 2.2.12 ). The face symmetry is also the symmetry of etch pits, striations and other face markings. For the two-dimensional point groups, this column contains the edge symmetry, which can be either 1 or m.

Column 4: 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 10.1.2.3 , which may be useful for determinative purposes, as explained in Sections 10.2.2 and 10.2.3 . 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 10.1.2.3 . Note that the same point form, `line segment', corresponds to both sphenoid and dome. The letter in parentheses after each name of a point form is explained below.

Column 5: Miller indices (hkl) for the symmetrically equivalent 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 [h + k + i = 0] .

Coordinates x, y, z for the symmetrically equivalent points of a point form are not listed explicitly because they can be obtained from data in this volume as follows: after the name of the point form, a letter is given in parentheses. This is the Wyckoff letter of the corresponding position in the symmorphic P space group that belongs to the point group under consideration. The coordinate triplets of this (general or special) position apply to the point form of the point group.

The triplets of Miller indices (hkl) and point coordinates x, y, z 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 2.2.10 and 8.3.5 . For all point groups, except those referred to a hexagonal coordinate system, the correspondence between the (hkl) and the x, y, z triplets is immediately obvious.⁸

The sets of symmetrically equivalent crystal faces also represent the sets of equivalent reciprocal-lattice points, as well as the sets of equivalent X-ray (neutron) reflections.

Examples

(1) In point group $[\overline{4}]$ , the general crystal form 4b is listed as $[(hkl)\ (\overline{h}{\hbox to 1pt{}}\overline{k}l)\ (k\overline{h}{\hbox to 1pt{}}\overline{l})\ (\overline{k}h\overline{l})]$ : the corresponding general position 4h of the symmorphic space group $[P\overline{4}]$ reads ; $[\overline{x},\overline{y}, z]$ ; $[y,\overline{x},\overline{z}]$ ; $[\overline{y},x,\overline{z}]$ .
(2) In point group 3, the general crystal form 3b is listed as (hkil) (ihkl) (kihl) with ; the corresponding general position 3d of the symmorphic space group P3 reads ; $[\overline{y}, x - y, z]$ ; $[- x + y,\overline{x},z]$ .
(3) The Miller indices of the cubic point groups are arranged in one, two or four blocks of $[(3 \times 4)]$ entries. The first block (upper left) belongs to point group 23. The second block (upper right) belongs to the diagonal twofold axes in 432 and $[m\overline{3}m]$ or to the diagonal mirror plane in $[\overline{4}3m]$ . In point groups $[m\overline{3}]$ and $[m\overline{3}m]$ , the lower one or two blocks are derived from the upper blocks by application of the inversion.

10.1.2.4. Notes on crystal and point forms

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(i) As mentioned in Section 10.1.2.2 , each set of Miller indices of a given point group represents infinitely many face forms with the same name. Exceptions occur for the following cases.

Some special crystal forms occur with only one representative. Examples are the pinacoid $[\{001\}]$ , the hexagonal prism $[\{10\overline{1}0\}]$ and the cube $[\{100\}]$ . The Miller indices of these forms consist of fixed numbers and signs and contain no variables.

In a few noncentrosymmetric point groups, a special crystal form is realized by two representatives: they are related by a centre of symmetry that is not part of the point-group symmetry. These cases are

(a) the two pedions (001) and $[(00\overline{1})]$ ;
(b) the two trigonal prisms $[\{10\overline{1}0\}]$ and $[\{\overline{1}010\}]$ ; similarly for two dimensions;
(c) the two trigonal prisms $[\{11\overline{2}0\}]$ and $[\{\overline{1}{\hbox to 1pt{}}\overline{1}20\}]$ ; similarly for two dimensions;
(d) the positive and negative tetrahedra $[\{111\}]$ and $[\{\overline{1}{\hbox to 1pt{}}\overline{1}{\hbox to 1pt{}}\overline{1}\}]$ .

In the point-group tables, both representatives of these forms are listed, separated by `or', for instance `(001) or $[(00\overline{1})]$ '.

(ii) In crystallography, the terms tetragonal, trigonal, hexagonal, as well as tetragon, trigon and hexagon, imply that the cross sections of the corresponding polyhedra, or the polygons, are regular tetragons (squares), trigons or hexagons. Similarly, ditetragonal, ditrigonal, dihexagonal, as well as ditetragon, ditrigon and dihexagon, refer to semi-regular cross sections or polygons.
(iii) Crystal forms can be `open' or `closed'. A crystal form is `closed' if its faces form a closed polyhedron; the minimum number of faces for a closed form is 4. Closed forms are disphenoids, dipyramids, rhombohedra, trapezohedra, scalenohedra and all cubic forms; open forms are pedions, pinacoids, sphenoids (domes), pyramids and prisms.

A point form is always closed. It should be noted, however, that a point form dual to a closed face form is a three-dimensional polyhedron, whereas the dual of an open face form is a two- or one-dimensional polygon, which, in general, is located `off the origin' but may be centred at the origin (here called `through the origin').
(iv) Crystal forms are well known; they are described and illustrated in many textbooks. Crystal forms are `isohedral' polyhedra that have all faces equivalent but may have more than one kind of vertex; they include regular polyhedra. The in-sphere of the polyhedron touches all the faces.

Crystallographic point forms are less known; they are described in a few places only, notably by A. Niggli (1963), by Fischer et al. (1973), and by Burzlaff & Zimmermann (1977). The latter publication contains drawings of the polyhedra of all point forms. Point forms are `isogonal' polyhedra (polygons) that have all vertices equivalent but may have more than one kind of face;⁹ again, they include regular polyhedra. The circumsphere of the polyhedron passes through all the vertices.

In most cases, the names of the point-form polyhedra can easily be derived from the corresponding crystal forms: the duals of n-gonal pyramids are regular n-gons off the origin, those of n-gonal prisms are regular n-gons through the origin. The duals of di-n-gonal pyramids and prisms are truncated (regular) n-gons, whereas the duals of n-gonal dipyramids are n-gonal prisms.

In a few cases, however, the relations are not so evident. This applies mainly to some cubic point forms [see item (v) below]. A further example is the rhombohedron, whose dual is a trigonal antiprism (in general, the duals of n-gonal streptohedra are n-gonal antiprisms).¹⁰ The duals of n-gonal trapezohedra are polyhedra intermediate between n-gonal prisms and n-gonal antiprisms; they are called here `twisted n-gonal antiprisms'. Finally, the duals of di-n-gonal scalenohedra are n-gonal antiprisms `sliced off' perpendicular to the prism axis by the pinacoid $[\{001\}]$ .¹¹

(v) Some cubic point forms have to be described by `combinations' of `isohedral' polyhedra because no common names exist for `isogonal' polyhedra. The maximal number of polyhedra required is three. The shape of the combination that describes the point form depends on the relative sizes of the polyhedra involved, i.e. on the relative values of their central distances. Moreover, in some cases even the topology of the point form may change.

Example

`Cube truncated by octahedron' and `octahedron truncated by cube'. Both forms have 24 vertices, 14 faces and 36 edges but the faces of the first combination are octagons and trigons, those of the second are hexagons and tetragons. These combinations represent different special point forms x, x, z and 0, y, z. One form can change into the other only via the (semi-regular) cuboctahedron 0, y, y, which has 12 vertices, 14 faces and 24 edges.

The unambiguous description of the cubic point forms by combinations of `isohedral' polyhedra requires restrictions on the relative sizes of the polyhedra of a combination. The permissible range of the size ratios is limited on the one hand by vanishing, on the other hand by splitting of vertices of the combination. Three cases have to be distinguished:

(a) The relative sizes of the polyhedra of the combination can vary independently. This occurs whenever three edges meet in one vertex. In Table 10.1.2.2 , the names of these point forms contain the term `truncated'.

Examples

(1) `Octahedron truncated by cube' (24 vertices, dual to tetrahexahedron).
(2) `Cube truncated by two tetrahedra' (24 vertices, dual to hexatetrahedron), implying independent variation of the relative sizes of the two truncating tetrahedra.

(b) The relative sizes of the polyhedra are interdependent. This occurs for combinations of three polyhedra whenever four edges meet in one vertex. The names of these point forms contain the symbol `&'.

Example

`Cube & two tetrahedra' (12 vertices, dual to tetragon-tritetrahedron); here the interdependence results from the requirement that in the combination a cube edge is reduced to a vertex in which faces of the two tetrahedra meet. The location of this vertex on the cube edge is free. A higher symmetrical `limiting' case of this combination is the `cuboctahedron', where the two tetrahedra have the same sizes and thus form an octahedron.
(c) The relative sizes of the polyhedra are fixed. This occurs for combinations of three polyhedra if five edges meet in one vertex. These point forms are designated by special names (snub tetrahedron, snub cube, irregular icosahedron), or their names contain the symbol `+'.

The cuboctahedron appears here too, as a limiting form of the snub tetrahedron (dual to pentagon-tritetrahedron) and of the irregular icosahedron (dual to pentagon-dodecahedron) for the special coordinates 0, y, y.

(vi) Limiting crystal forms result from general or special crystal forms for special values of certain geometrical parameters of the form.

Examples

(1) A pyramid degenerates into a prism if its apex angle becomes 0, i.e. if the apex moves towards infinity.
(2) In point group 32, the general form is a trigonal trapezohedron $[\{hkl\}]$ ; this form can be considered as two opposite trigonal pyramids, rotated with respect to each other by an angle χ. The trapezohedron changes into the limiting forms `trigonal dipyramid' $[\{hhl\}]$ for $[\chi = 0^{\circ}]$ and `rhombohedron' $[\{h0l\}]$ for $[\chi = 60^{\circ}]$ .

(vii) One and the same type of polyhedron can occur as a general, special or limiting form.

Examples

(1) A tetragonal dipyramid is a general form in point group , a special form in point group and a limiting general form in point groups 422 and $[\overline{4}2m]$ .
(2) A tetragonal prism appears in point group $[\overline{4}2m]$ both as a basic special form (4b) and as a limiting special form (4c).

(viii) A peculiarity occurs for the cubic point groups. Here the crystal forms $[\{hhl\}]$ are realized as two topologically different kinds of polyhedra with the same face symmetry, multiplicity and, in addition, the same eigensymmetry. The realization of one or other of these forms depends upon whether the Miller indices obey the conditions $[|h| \gt |l|]$ or $[|h| \lt |l|]$ , i.e. whether, in the stereographic projection, a face pole is located between the directions [110] and [111] or between the directions [111] and [001]. These two kinds of polyhedra have to be considered as two realizations of one type of crystal form because their face poles are located on the same set of conjugate symmetry elements. Similar considerations apply to the point forms x, x, z.

In the point groups $[m\overline{3}m]$ and $[\overline{4}3m]$ , the two kinds of polyhedra represent two realizations of one special `Wyckoff position'; hence, they have the same Wyckoff letter. In the groups 23, $[m\overline{3}]$ and 432, they represent two realizations of the same type of limiting general forms. In the tables of the cubic point groups, the two entries are always connected by braces.

The same kind of peculiarity occurs for the two icosahedral point groups, as mentioned in Section 10.1.4 and listed in Table 10.1.4.3 .

10.1.2.5. Names and symbols of the crystal classes

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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 point-group symbols. As examples, two sets (both translated into English) that are frequently found in the literature are given in Table 10.1.2.4. 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'.

10.1.3. Subgroups and supergroups of the crystallographic point groups

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In this section, the sub- and supergroup relations between the crystallographic point groups are presented in the form of a `family tree'.¹² Figs. 10.1.3.1 and 10.1.3.2 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 $[{\cal P}]$ of order $[k_{P}]$ the lines to groups of lower order connect $[{\cal P}]$ with all its maximal subgroups $[{\cal H}]$ with orders $[k_{H}]$ ; the index [i] of each subgroup is given by the ratio of the orders $[k_{P}/k_{H}]$ . The lines to groups of higher order connect $[{\cal P}]$ with all its minimal supergroups $[{\cal S}]$ with orders $[k_{S}]$ ; the index [i] of each supergroup is given by the ratio $[k_{S}/k_{P}]$ . 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.

Figure 10.1.3.1| top | pdf |

Maximal subgroups and minimal supergroups of the two-dimensional 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.

Figure 10.1.3.2| top | pdf |

Maximal subgroups and minimal supergroups of the three-dimensional 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 $[{\cal H}_{1}, {\cal H}_{2}, \ldots, {\cal H}_{n}]$ of a group $[{\cal P}]$ are conjugate subgroups if $[{\cal H}_{1}, {\cal H}_{2}, \ldots, {\cal H}_{n}]$ are symmetrically equivalent in $[{\cal P}]$ , i.e. if for every pair $[{\cal H}_{i}, {\cal H}_{j}]$ at least one symmetry operation $[\hbox{\sf W}]$ of $[{\cal P}]$ exists which maps $[{\cal H}_{i}]$ onto $[{\cal H}_{j}: \hbox{\sf W}^{-1} {\cal H}_{i}\hbox{\sf W} = {\cal H}_{j}]$ ; cf. Section 8.3.6 .

Examples

(1) Point group 3m has three different mirror planes which are equivalent due to the threefold axis. In each of the three maximal subgroups of type m, one of these mirror planes is retained. Hence, the three subgroups m are conjugate in 3m. This set of conjugate subgroups is represented by one dashed line in Figs. 10.1.3.1 and 10.1.3.2 .
(2) Similarly, group 432 has three maximal conjugate subgroups of type 422 and four maximal conjugate subgroups of type 32.

The subgroup $[{\cal H}]$ of a group $[{\cal P}]$ is a normal (or invariant) subgroup if no subgroup $[{\cal H}']$ of $[{\cal P}]$ exists that is conjugate to $[{\cal H}]$ in $[{\cal P}]$ . Note that this does not imply that $[{\cal H}]$ is also a normal subgroup of any supergroup of $[{\cal P}]$ . Subgroups of index [2] are always normal and maximal. (The role of normal subgroups for the structure of space groups is discussed in Section 8.1.6 .)

Examples

(1) Fig. 10.1.3.2 shows two solid lines between point groups 422 and 222, indicating that 422 has two maximal normal subgroups 222 of index [2]. The symmetry elements of one subgroup are rotated by 45° around the c axis with respect to those of the other subgroup. Thus, in one subgroup the symmetry elements of the two secondary, in the other those of the two tertiary tetragonal symmetry directions (cf. Table 2.2.4.1 ) are retained, whereas the primary twofold axis is the same for both subgroups. There exists no symmetry operation of 422 that maps one subgroup onto the other. This is illustrated by the stereograms below. The two normal subgroups can be indicated by the `oriented symbols' 222. and 2.22.
(2) Similarly, group 432 has one maximal normal subgroup, 23.

Figs. 10.1.3.1 and 10.1.3.2 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 basis of the translationengleiche or t subgroups and supergroups of space groups; this is set out in Section 2.2.15 . Tables of the sub- and supergroups of the plane groups and space groups are contained in Parts 6 and 7 . A general discussion of sub- and supergroups of crystallographic groups, together with further explanations and examples, is given in Section 8.3.3 .

10.1.4. Noncrystallographic point groups

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10.1.4.1. Description of general point groups

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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 $[\geq]$ 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 $[{1 \over 2}n]$ -fold rotation point with )
$[{1 \over 2}n]$	$[{1 \over 2}n]$	Regular $[{1 \over 2}n]$ -gon	Regular $[{1 \over 2}n]$ -gon	1, 3
$[{1 \over 2}nm]$	n	Semiregular di- $[{1 \over 2}n]$ -gon	Truncated $[{1 \over 2}n]$ -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 $[\geq]$ 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	$[C_{n}]$	n	n-gonal pyramid	Regular n-gon	4
$[\overline{n}]$	$[S_{n}]$	n	$[{1 \over 2}n]$ -gonal streptohedron	$[{1 \over 2}n]$ -gonal antiprism	$[\overline{4}]$
	$[C_{nh}]$	2n	n-gonal dipyramid	n-gonal prism
n22	$[D_{n}]$	2n	n-gonal trapezohedron	Twisted n-gonal antiprism	422
nmm	$[C_{nv}]$	2n	Di-n-gonal pyramid	Truncated n-gon	4mm
$[\overline{n}2m]$	$[D_{{1 \over 2}nd}]$	2n	n-gonal scalenohedron	$[{1 \over 2}n]$ -gonal antiprism sliced off by pinacoid	$[\overline{4}2m]$
$[n/mmm, \ \displaystyle{n \over m}{2 \over m}{2 \over m}]$	$[D_{nh}]$	4n	Di-n-gonal dipyramid	Edge-truncated n-gonal prism
-gonal system (single n-fold symmetry axis with )
n	$[C_{n}]$	n	n-gonal pyramid	Regular n-gon	1, 3
$[\overline{n} = n \times \overline{1}]$	$[C_{ni}]$	2n	n-gonal streptohedron	n-gonal antiprism	$[\overline{1},\ \overline{3} = 3 \times \overline{1}]$
n2	$[D_{n}]$	2n	n-gonal trapezohedron	Twisted n-gonal antiprism	32
nm	$[C_{nv}]$	2n	Di-n-gonal pyramid	Truncated n-gon	3m
$[\overline{n}m, \ \overline{n} \displaystyle{2 \over m}]$	$[D_{nd}]$	4n	Di-n-gonal scalenohedron	n-gonal antiprism sliced off by pinacoid	$[\overline{3}m]$
-gonal system (single n-fold symmetry axis with )
n	$[C_{n}]$	n	n-gonal pyramid	Regular n-gon	2, 6
$[\overline{n} = {\textstyle{1 \over 2}}n/m]$	$[C_{{1 \over 2}nh}]$	n	$[{\textstyle{1 \over 2}}n]$ -gonal dipyramid	$[{\textstyle{1 \over 2}}n]$ -gonal prism	$[\overline2 \equiv m, \ \overline{6} \equiv 3/m]$
	$[C_{nh}]$	2n	n-gonal dipyramid	n-gonal prism	$[2/m, \ 6/m]$
n22	$[D_{n}]$	2n	n-gonal trapezohedron	Twisted n-gonal antiprism	222, 622
nmm	$[C_{nv}]$	2n	Di-n-gonal pyramid	Truncated n-gon	mm2, 6mm
$[\overline{n}2m = {\textstyle{1 \over 2}}n/m2m]$	$[D_{{1 \over 2}nh}]$	2n	Di- $[{\textstyle{1 \over 2}}n]$ -gonal dipyramid	Truncated $[{\textstyle{1 \over 2}}n]$ -gonal prism	$[\overline{6}2m]$
$[n/mmm, \ \displaystyle{n \over m}{2 \over m}{2 \over m}]$	$[D_{nh}]$	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
$[m\overline{3}, \displaystyle{2 \over m}\overline{3}]$	$[T_{h}]$	24	Didodecahedron	Cube & octahedron & pentagon-dodecahedron	$[m\overline{3}]$
432	O	24	Pentagon-trioctahedron	Snub cube	432
$[\overline{4}3m]$	$[T_{d}]$	24	Hexatetrahedron	Cube truncated by two tetrahedra	$[\overline{4}3m]$
$[m\overline{3}m, \ \displaystyle{4 \over m}\overline{3}{2 \over m}]$	$[O_{h}]$	48	Hexaoctahedron	Cube truncated by octahedron and by rhomb-dodecahedron	$[m\overline{3}m]$
Icosahedral system^† (for details see Table 10.1.4.3)
235	I	60	Pentagon-hexecontahedron	Snub pentagon-dodecahedron	–
$[m\overline{3}\overline{5}, \ {\displaystyle{2 \over m}}\overline{3}\overline{5}]$	$[I_{h}]$	120	Hecatonicosahedron	Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron	–
Cylindrical system ^‡
∞	$[C_{\infty}]$	∞	Rotating cone	Rotating circle	–
$[\infty/m \equiv \overline{\infty}]$	$[C_{\infty h} \equiv S_{\infty} \equiv C_{\infty i}]$	∞	Rotating double cone	Rotating finite cylinder	–
∞2	$[D_{\infty}]$	∞	`Anti-rotating' double cone	`Anti-rotating' finite cylinder	–
∞m	$[C_{\infty v}]$	∞	Stationary cone	Stationary circle	–
$[\infty/mm \equiv \overline{\infty}m, \ {\displaystyle{\infty \over m}{2 \over m}} \equiv \overline{\infty} {\displaystyle{2 \over m}}]$	$[D_{\infty h} \equiv D_{\infty d}]$	∞	Stationary double cone	Stationary finite cylinder	–
Spherical system ^§
$[2 \infty]$	K	∞	Rotating sphere	Rotating sphere	–
$[m \overline{\infty},\ {\displaystyle{2 \over m}} \overline{\infty}]$	$[K_{h}]$	∞	Stationary sphere	Stationary sphere	–

^†The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and $[\bar{5}\bar{3}m]$ (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 $[\infty 2]$ , enantiomorphism occurs, i.e. forms with opposite senses of rotation. Class $[\infty/m \equiv \overline{\infty}]$ 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 $[\infty 2]$ 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 $[2 \infty]$ 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 $[m \overline{\infty}]$ contain infinitely many mirror planes through the centres of the spheres. Group $[2 \infty]$ is sometimes symbolized by $[\infty \infty]$ ; group $[m \overline{\infty}]$ by $[\overline{\infty}\; \overline{\infty}]$ or $[\infty \infty m]$ . 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
			$[\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\ icosahedron\hfill\cr (\hbox{poles between axes 2 and 3})\hfill\cr \cr\hbox{Deltoid\hbox{-}hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \&\ \hfill\cr pentagon\hbox{-}dodecahedron\ \&\ icosahedron\hfill\cr (\hbox{poles between axes 3 and 5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\hfill\cr pentagon\hbox{-}dodecahedron\hfill\cr (\hbox{poles between axes 5 and 2)}\hfill\cr}\right.]$	$[\matrix{(0kl)\hbox{ with } \|l\| \lt 0.382 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \lt 0.382 \|y\|\hfill\cr \cr\cr (0kl)\hbox{ with }0.382 \|k\| \lt \|l\| \lt 1.618 \|k\|\hfill\cr 0,y,z\ with\ 0.382 \|y\| \lt \|z\| \lt 1.618 \|y\|\hfill\cr\cr\cr\cr (0kl)\hbox{ with }\|l\| \gt 1.618 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \gt 1.618 \|y\|\hfill\cr\cr\cr}]$
30	c	2..	$[\!\matrix{\hbox{Rhomb-triacontahedron}\hfill\cr Icosadodecahedron\ (\!= pentagon\hbox{-}\hfill\cr dodecahedron\ \&\ icosahedron)\hfill\cr}]$	$[\!\matrix{(100)\hfill\cr x,0,0\hfill\cr\cr}]$

20	b	.3.	$[\!\matrix{\hbox{Regular icosahedron}\hfill\cr Regular\ pentagon\hbox{-}dodecahedron\hfill\cr}]$	$[\!\matrix{(111)\hfill\cr x,\;x,\;x\hfill\cr}]$
12	a	..5	$[\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr {Regular\ icosahedron}\hfill\cr}]$	$[\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with } \tau = {\textstyle{1 \over 2}}(\sqrt{5} + 1) = 1.618]$
Symmetry of special projections
$[\matrix{\hbox{Along } [001]&&&\hbox{Along } [111]&&&\hbox{Along } [1\tau 0]\cr 2mm&&&3m&&&5m}]$
$[\matrix{m\bar{3}\bar{5}\hfill\cr\cr\displaystyle{2 \over m}\bar{3}\bar{5}\hfill\cr}]$	$[I_{h}]$
120	e	l	Hecatonicosahedron or hexaicosahedron	(hkl)
			Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron	x, y, z
60	d	m..	$[\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\ icosahedron\hfill\cr \hbox{(poles between axes 2 and }\overline{3})\hfill\cr \cr \hbox{Deltoid-hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \& \ pentagon\hbox{-}dodecahedron\ \& \hfill\cr icosahedron\hfill\cr \hbox{(poles between axes }\overline{3} \hbox{ and } \overline{5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\ pentagon\hbox{-}dodecahedron\hfill\cr \hbox{(poles between axes }\overline{5} \hbox{ and } 2)\hfill\cr}\right.]$	$[\matrix{(0kl) \hbox{ with } \|l\| \;\lt\; 0.382 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \;\lt\; 0.382 \|y\|\hfill\cr \cr\cr (0kl) \hbox{ with } 0.382 \|k\| \;\lt\; \|l\| \;\lt\; 1.618 \|k\|\hfill\cr 0,y,z\ with\ 0.382 \|y\| \;\lt\; \|z\| \;\lt\; 1.618 \|y\|\hfill\cr \cr\cr\cr (0kl) \hbox{ with } \|l\| \;\gt\; 1.618 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \;\gt\; 1.618 \|y\|\hfill\cr\cr}]$
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)	$[\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr Regular\ icosahedron\hfill\cr}]$	$[\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with }\tau = {1 \over 2}(\sqrt{5} + 1) = 1.618]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&6mm&&10mm\cr}]$

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, [(2N + 1)] -gonal and [(4N + 2)] -gonal system, each general class contains infinitely many point groups, which differ in their principal n-fold symmetry axis, with $[n = 4, 8, 12,\ldots]$ for the 4N-gonal system, $[n = 1, 3, 5,\ldots]$ for the -gonal system and $[n = 2,6,10,\ldots]$ 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. Section 2.2.4 and Chapter 12.1 ). This extends also to the infinite groups where symbols like $[\infty m]$ or $[C_{\infty v}]$ 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 [(4N + 2)] -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 [(2N + 1)] -gonal and the [(4N + 2)] -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 $[\overline{n}]$ or rotoreflection axes $[\tilde{n}]$ . The equivalence between these two descriptions is apparent from the following examples: $[\matrix{&n = 4N\hfill &:&\overline{4} = \tilde{4}\hfill &\overline{8} = \tilde{8} \hfill \ldots &\overline{n} = \tilde{n}\hfill\cr &n = 2N + 1 &:&\overline{1} = \tilde{2}\hfill &\overline{3} = \tilde{6} = 3 \times \overline{1}\hfill \ldots &\overline{n} = \widetilde{2n} = n \times \overline{1}\hfill\cr &n = 4N + 2 &:&\overline{2} = \tilde{1} = m &\overline{6} = \tilde{3} = 3/m \hfill \ldots &\overline{n} = \!\widetilde{\;{1 \over 2}n} = {1 \over 2}n/m.\cr}]$ 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

10.1.4.2. The two icosahedral groups

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The two point groups 235 and $[m\bar{3}\bar{5}]$ 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, $[B_{12}]$ 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 $[63.43^{\circ}]$ . Each fivefold axis is surrounded by five threefold and five twofold axes, with angular distances of $[37.38^{\circ}]$ between a fivefold and a threefold axis and of $[31.72^{\circ}]$ between a fivefold and a twofold axis. The angles between neighbouring threefold axes are $[41.81^{\circ}]$ , between neighbouring twofold axes $[36^{\circ}]$ . The smallest angle between a threefold and a twofold axis is $[20.90^{\circ}]$ .

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 $[\overline{5}m]$ (for $[m\bar{3}\bar{5}]$ ) 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 $[\overline{3}m]$ (for $[m\bar{3}{\hbox to .5pt{}}\bar{5}]$ ) with index [10]. There occur, furthermore, five maximal conjugate cubic subgroups of types 23 (for 235) and $[m\overline{3}]$ (for $[m\bar{3}\bar{5}]$ ) 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 $[m\bar{3}\bar{5}]$ instead of the customary symbols 532 and $[\bar{5}\bar{3}m]$ .

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' $[m\bar{3}\bar{5}]$ , 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:

(i) For the face forms the cubic point groups 23 and $[m\overline{3}]$ (Table 10.1.2.2 ), and for the point forms the cubic space groups P23 (195) and $[Pm\overline{3}]$ (200) have to be considered. For each `initial' triplet (hkl), the set of Miller indices of the (general or special) crystal form with the same face symmetry in 23 (for group 235) or $[m\overline{3}]$ (for $[m\bar{3}\bar{5}]$ ) is taken. For each `initial' triplet x, y, z, the coordinate triplets of the (general or special) position with the same site symmetry in P23 or $[Pm\overline{3}]$ are taken; this procedure is similar to that described in Section 10.1.2.3 for the crystallographic point forms.

(ii) To obtain the complete set of icosahedral Miller indices and point coordinates, the `cubic' (hkl) triplets (as rows) and x, y, z triplets (as columns) have to be multiplied with the identity matrix and with

(a) the matrices $[Y, Y^{2}, Y^{3}]$ and $[Y^{4}]$ for the Miller indices;
(b) the matrices $[Y^{-1}, Y^{-2}, Y^{-3}]$ and $[Y^{-4}]$ for the point coordinates.

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 $[\eqalign{Y &= Y^{-4} = \pmatrix{\hfill {1 \over 2} &\hfill {\phantom-}g &\hfill {\phantom-}G\cr \hfill g &\hfill G &-{1 \over 2}\cr -G &\hfill{1 \over 2} &\hfill g\cr},\ \ Y^{2} = Y^{-3} = \pmatrix{\hfill-g &\hfill G &\hfill {1 \over 2}\cr \hfill G &\hfill{\phantom-}{1 \over 2} &\hfill-g\cr \hfill-{1 \over 2} &\hfill g &\hfill-G\cr},\cr Y^{3} &= Y^{-2} = \pmatrix{\hfill-g &\hfill G &\hfill-{1 \over 2}\cr \hfill{\phantom-} G &\hfill {1 \over 2} &\hfill g\cr \hfill {1 \over 2} &\hfill-g &\hfill-G\cr},{\hbox to 6pt{}} Y^{4} = Y^{-1} = \pmatrix{\hfill{\phantom-}{1 \over 2} &\hfill g &\hfill-G\cr \hfill g &\hfill G &\hfill {1 \over 2}\cr \hfill G &\hfill-{1 \over 2} &\hfill g\cr},}]$ with²⁰ $[\eqalign{G &= {\sqrt{5} + 1 \over 4} = {\tau \over 2} = \cos 36^{\circ} = 0.80902 \simeq {72 \over 89}\cr g &= {\sqrt{5} - 1 \over 4} = {\tau - 1 \over 2} = \cos 72^{\circ} = 0.30902 \simeq {17 \over 55}.}]$ These matrices correspond to counter-clockwise rotations of 72, 144, 216 and $[288^{\circ}]$ around a fivefold axis parallel to $[[1\tau 0]]$ .

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 $[B_{12}]$ 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.

10.1.4.3. Sub- and supergroups of the general point groups

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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 [ $[\infty]$ ] 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 [ [(4N + 2)] -gonal system, including the [(2N + 1)] -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 [ $[\infty]$ ]. The order of the rotation point may be [n = 4N, n = 4N + 2] or [n = 2N + 1] . In the first case, the subgroups belong to the 4N-gonal system, in the latter two cases, they belong to the [(4N + 2)] -gonal system. [In the diagram of the -gonal system, the [(2N + 1)] -gonal groups appear with the symbols $[{1 \over 2}nm]$ and $[{1 \over 2}n]$ .] The subgroups of the circular groups are not maximal because for any given value of N there exists a group with [N' = 2N] 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 [(4N + 2)] -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 [N = 4] in Fig. 10.1.4.2 .

Fig. 10.1.4.3 for three dimensions illustrates that the two spherical groups $[2/m \overline{\infty}]$ and $[2 \infty]$ 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 [ $[\infty]$ ].

Each of the two icosahedral groups 235 and $[2/m\bar{3}\bar{5}]$ 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 [ $[\infty]$ ]. Depending upon whether the main symmetry axis has the multiplicity 4N, [4N + 2] or [2N + 1] , the subgroups belong to the 4N-gonal, [(4N + 2)] -gonal or [(2N + 1)] -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 [(4N + 2)] -gonal and the [(2N + 1)] -gonal systems are combined, with the consequence that the five classes of the -gonal system now appear with the symbols $[\overline{{1 \over 2}n}{2 \over m}, {1 \over 2}n2, {1 \over 2}nm,\ \overline{{1 \over 2}n}]$ and $[{1 \over 2}n]$ . 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).

Acknowledgements

The authors are indebted to A. Niggli (Zürich) for valuable suggestions on Section 10.1.4 , in particular for providing a sketch of Fig. 10.1.4.3 . We thank H. Wondratschek (Karlsruhe) for stimulating and constructive discussions. We are grateful to R. A. Becker (Aachen) for the careful preparation of the diagrams.

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