International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 724-726

Table 3.2.1.3 

Th. Hahna and H. Klappera

Table 3.2.1.3| 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. The 47 crystal forms are shown in Fig. 3.2.1.1[link]. (Note that the numbering of the forms in column 1 does not correspond to the numbering used in Fig. 3.2.1.1[link].)

No.Crystal formPoint formNumber of faces or pointsEigensymmetryGenerating point groups with oriented face (site) symmetries between parentheses
1 Pedion or monohedron Single point 1 [\infty m] [\matrix{{\bf 1(1)}\semi\ {\bf 2(2)}\semi\ {\bi m}({\bi m})\semi\ {\bf 3(3)}\semi\ {\bf 4(4)}\semi\hfill\cr {\bf 6(6)}\semi\ {\bi m}{\bi m}{\bf 2}({\bi m}{\bi m}{\bf 2})\semi\ {\bf 4}{\bi m}{\bi m}({\bf 4}{\bi m}{\bi m})\semi\hfill\cr {\bf 3}{\bi m}({\bf 3}{\bi m})\semi\ {\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},{\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{..}})\semi \left\{\matrix{\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}\right.\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})\semi\ \left\{\matrix{{\bf 321}({\bf 3\hbox{..}})\cr {\bf 312}({\bf 3\hbox{..}})\cr {\bf 32}\phantom 2({\bf 3\hbox{.}})\cr}\right.\semi \left\{\matrix{\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})\cr \overline{{\bf 3}}{\bi m}({\bf 3}{\bi m})\cr}\right.\semi\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 \left\{\matrix{\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}\right.\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)}\semi \ {\bi m}{\bf (1)}\semi \ {\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)}\semi \ 222(1)][\semi \ mm 2(1)\semi \ {\bi m}{\bi m}{\bi m}({\bi m\hbox{..}}, {\bi \hbox{.}m\hbox{.}}, {\bi \hbox{..}m})]
7 Rhombic dipyramid Rectangular prism 8 mmm [{\bi m}{\bi m}{\bi m}{\bf (1)}]
8 Tetragonal pyramid Square 4 4mm [{\bf 4(1)}\semi \ {\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)}\semi \ \left\{\matrix{\overline{{\bf 4}}{\bf 2}{\bi m}({\bi \hbox{..}m})\cr \overline{{\bf 4}}{\bi m}{\bf 2}({\bi \hbox{.}m\hbox{.}})\cr}\right.]
10 Tetragonal prism Square through origin 4 [\displaystyle{4 \over m}mm] [4(1)\semi \ \overline{4}(1)\semi \ \displaystyle{{\bf 4} \over {\bi m}}({\bi m\hbox{..}})\semi \ {\bf 422(\hbox{..}2},{\bf \hbox{.}2\hbox{.})}\semi \ 4mm(..m, .m.)\semi ]
[\left\{\matrix{\overline{{\bf 4}}{\bf 2}{\bi m}{\bf (\hbox{.}2\hbox{.})}\ \ {\rm and} \ \ \overline{4}2m(..m)\cr \overline{{\bf 4}}{\bi m}{\bf 2(\hbox{..}2)} \ \ {\rm and} \ \ \overline{4}m2(.m.)\cr}\right.\semi]
[{\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] [\left\{\matrix{\overline{{\bf 4}}{\bf 2}{\bi m}{\bf (1)}\cr \overline{{\bf 4}}{\bi m}{\bf 2(1)}\cr}\right.]
14 Tetragonal dipyramid Tetragonal prism 8 [\displaystyle{4 \over m}mm] [{\displaystyle{{\bf 4} \over {\bi m}}}{\bf (1)}\semi \ 422{(1)}]; [\left\{\matrix{\overline{4}2m(1)\cr \overline{{4}}m{2(1)}\cr}\right.\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\ \left\{\matrix{\overline{4}2m(1)\cr \overline{{4}}m{2(1)}\cr}\right.\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)}\semi \left\{\matrix{{\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}\right.]
18 Trigonal prism Trigon through origin 3 [\overline{6}2m] [\matrix{3(1)\semi \left\{\matrix{{\bf 321}({\bf \hbox{.}2\hbox{.}})\cr {\bf 312}({\bf \hbox{..}2})\cr {\bf 32} \ \ ({\bf \hbox{.}2})\cr}\right.\semi\ \left\{\matrix{3m1(.m.)\cr 31m(..m) \cr 3m (.m)\cr}\right.\semi\hfill\cr \overline{{\bf 6}}({\bi m\hbox{..}})\semi \left\{\matrix{\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}\right.\hfill\cr}]
19 Trigonal trapezohedron Twisted trigonal antiprism 6 32 [\left\{\matrix{{\bf 321(1)}\cr {\bf 312(1)}\cr {\bf 32} {\bf (1)}\cr}\right.]
20 Ditrigonal pyramid Truncated trigon 6 3m [\left\{\matrix{{\bf 3}{\bi m}{\bf 1(1)}\cr {\bf 31}{\bi m}{\bf (1)}\cr {\bf 3}{\bi m} {\bf (1)}\cr}\right.]
21 Rhombohedron Trigonal antiprism 6 [\overline{3}m] [\overline{{\bf 3}}{\bf (1)}\semi \left\{\matrix{321(1)\cr 312(1)\cr 32{} (1)\cr}\right.\semi\ \left\{\matrix{\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}\right.]
22 Ditrigonal prism Truncated trigon through origin 6 [\overline{6}2m] [\matrix{\left\{\matrix{321(1)\cr 312(1)\cr 32  (1)\cr}\right.\semi\ \left\{\matrix{3m1(1)\cr 31m(1)\cr 3m (1)\cr}\right.\semi\hfill\cr\noalign{\vskip3pt}\left\{\matrix{\overline{{\bf 6}}{\bi m}{\bf 2}({\bi m\hbox{..}})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bi m\hbox{..}})\cr}\right.\hfill\cr}]
23 Hexagonal pyramid Hexagon 6 6mm [\left\{\matrix{3m1(1)\cr 31m (1)\cr 3m  (1)\cr}\right.\semi\ {\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] [\left\{\matrix{321(1)\cr 312(1) \cr 32 (1)\cr}\right.\semi\ \ \overline{{\bf 6}}{\bf (1)}\semi \ \left\{\matrix{\overline{{\bf 6}}{\bi m}{\bf 2}({\bi \hbox{.}m\hbox{.}})\cr \overline{{\bf 6}}{\bf 2}{\bi m}({\bi \hbox{..}m})\cr}\right.]
25 Hexagonal prism Hexagon through origin 6 [\displaystyle{6 \over m}mm] [\overline{3}(1)\semi \ \left\{\matrix{321(1)\cr 312(1)\cr 32 (1)\cr}\right.\semi\left\{\matrix{3m1 (1)\cr 31m (1)\cr 3m (1)\cr}\right.\semi]
[\left\{\matrix{\overline{{\bf 3}}{\bi m}{\bf 1}({\bf\hbox{.} 2\hbox{.}})\ \ \hbox{and}\ \ \overline{{3}}m1(.m.)\hfill\cr \overline{{\bf 3}}{\bf 1}{\bi m}({\bf \hbox{..}2}) \ \ \hbox{and}\ \ \overline{3}1m(..m)\hfill\cr \overline{{\bf 3}}{\bi m}({\bf \hbox{.}2}) \ \ \hbox{and}\ \ \overline{3}m(.m)\hfill\cr}\right.\semi]
[\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\ \left\{\matrix{\overline{6}m2(m..)\cr \overline{6}2m(m..)\cr}\right.\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] [\left\{\matrix{\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}\right.]
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] [\left\{\matrix{\overline{{\bf 6}}{\bi m}{\bf 2} {\bf (1)}\cr \overline{{\bf 6}}{\bf 2}{\bi m} {\bf (1)}\cr}\right.]
30 Dihexagonal prism Truncated hexagon 12 [\displaystyle{6 \over m}mm] [\matrix{\left\{\matrix{\overline{3}m1 (1)\cr \overline{3}1m (1) \cr \overline{3}m  (1)\cr}\right.\semi\ 622 (1)\semi \ 6mm (1)\semi \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] [\left\{\matrix{\overline{3}m1 (1)\cr \overline{3}1m (1)\cr \overline{3}m (1)\cr}\right.\semi\ \displaystyle{6 \over m}(1)\semi \ 622(1)];
[\left\{\matrix{\overline{6}m2 (1)\cr \overline{6}2m (1)\cr}\right.\semi\ \ \displaystyle{{\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 {.}})\semi \ \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 {..}})\semi \ {\bi m}\overline{{\bf 3}}({\bf 2}{\bi m}{\bi m\hbox {..}})\semi \hfill\cr {\bf 432}({\bf 4\hbox {..}})\semi \ \overline{{\bf 4}}{\bf 3}{\bi m}({\bf 2\hbox {.}}{\bi m}{\bi m})\semi \ {\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 {.}})\semi \ {\bf 432}({\bf \hbox {.}3\hbox {.}})\semi \ {\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)\semi \ {\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)\semi \ \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)\semi \ \overline{{\bf 4}}{\bf 3}{\bi m}({\bi \hbox {..}m})]
40 Rhomb-dodecahedron Cuboctahedron 12 [m\overline{3}m] [\matrix{23(1)\semi \ m\overline{3}(m..)\semi \ {\bf 432}({\bf \hbox{..}2})\semi \hfill\cr \overline{4}3m(..m)\semi \ {\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)\semi \ 432 (1)\semi \ {\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)\semi \ 432(1)\semi \ {\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)\semi \ \overline{4}3m (1)\semi \ {\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 3.2.3.2[link].
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{.}})\ \ \hbox{and}\ \ \overline{4}2m(..m)]' and as `[\overline{{\bf 3}}{\bi m}{\bf 1}({\bf \hbox{.}2\hbox{.}})\ \ \hbox{and}\ \ \overline{3}m1(.m.)]'.