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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 10.1, pp. 770-790

Table 10.1.2.2 

Th. Hahna* and H. Klappera‡

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

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}]     [Scheme scheme12]
1 a 1 Pedion or monohedron (hkl)
      Single point (a)  
Symmetry of special projections  
Along any direction  
1  
[\bar{1}] [C_{i}]     [Scheme scheme13]
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}]     [Scheme scheme14] [Scheme scheme15]
        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}]     [Scheme scheme16] [Scheme scheme17]
        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}]    
[2/m] [C_{2h}]     [Scheme scheme18] [Scheme scheme19]
        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}]     [Scheme scheme20]
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}]     [Scheme scheme21]
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}]     [Scheme scheme22]
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}]     [Scheme scheme23]
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}]     [Scheme scheme24]
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}]  
[4/m] [C_{4h}]     [Scheme scheme25]
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}]     [Scheme scheme26]
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}]     [Scheme scheme27]
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}]     [Scheme scheme28]
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}]     [Scheme scheme29]
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}]   [Scheme scheme30]
         
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}]     [Scheme scheme31]
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}]     [Scheme scheme32]
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}]     [Scheme scheme33]
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}]     [Scheme scheme34]
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}]     [Scheme scheme35]
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}]     [Scheme scheme36]
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}]     [Scheme scheme37]
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}]     [Scheme scheme38]
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}]     [Scheme scheme39]
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}]     [Scheme scheme40]
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}]     [Scheme scheme41]
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}]     [Scheme scheme42]
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}]     [Scheme scheme43]
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}]     [Scheme scheme44]
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}]     [Scheme scheme45]
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}]  
[6/m] [C_{6h}]     [Scheme scheme46]
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}]     [Scheme scheme47]
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}]     [Scheme scheme48]
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}]     [Scheme scheme49]
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}]     [Scheme scheme50]
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}]     [Scheme scheme51]
         
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     [Scheme scheme52]
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}]     [Scheme scheme53]
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     [Scheme scheme54]
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}]     [Scheme scheme55]
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}]     [Scheme scheme56]
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}]