Table 10.1.2.3

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. 791-793

Table 10.1.2.3

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

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 $[\overline{{\bf 3}}{\bi m}{\bf 1}({\bf \hbox{.}2\hbox{.}})\ \& \ \overline{3}m1(.m.)]$ .