Crystal and point forms

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. 764-766

Section 10.1.2.2. Crystal and point forms

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

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