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. 742-771

Section 3.2.3. Tables of the crystallographic point-group types

H. Klapper,a Th. Hahna and M. I. Aroyoc

3.2.3. Tables of the crystallographic point-group types

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The crystallographic point-group types are listed in Tables 3.2.3.1[link] and 3.2.3.2[link] for two-dimensional and for three-dimensional space, respectively. No listings are presented for the noncrystallographic point-group types (i.e. having axes of orders other than 1, 2, 3, 4 and 6), but their symbols can be found in Tables 3.2.1.5[link], 3.2.1.6[link] and 3.2.3.3[link] (cf. Section 3.2.1.4[link] for a review of noncrystallographic point groups). The two icosahedral point groups 235 and [m\bar3\bar5] are treated in detail in Section 3.2.1.4.2[link], while their crystallographic data are shown in Table 3.2.3.3[link].

Table 3.2.3.1| top | pdf |
The ten two-dimensional crystallographic point groups

The point groups are listed in blocks according to crystal system and are specified by their Hermann–Mauguin symbols. For each point group, the stereographic projections show (on the left) the general position and (on the right) the symmetry elements. The list of Wyckoff positions includes: Columns 1 to 4: multiplicity, Wyckoff letter, oriented site-symmetry symbol, coordinate doublets; Under the stereographic projections: edge forms (in roman type) and point forms (in italics); if there are two entries, the second entry refers to a limiting (noncharacteristic) form; Last column: Miller indices of equivalent edges [for hexagonal groups, Bravais–Miller indices (hki) are used].

OBLIQUE SYSTEM
1       [Scheme scheme1]  
1 a 1 [x,y] Single edge (hk)
        Single point  
2       [Scheme scheme2]  
2 a 1 [x,y\quad {\bar x},\bar y] Pair of parallel edges [\matrix{(hk) &(\bar{h}\bar{k})\cr}]
        Line segment through origin  
1 o 2 0, 0 Point in origin  
RECTANGULAR SYSTEM
m       [Scheme scheme3]  
2 b 1 [x,y\quad \bar x,y ] Pair of edges [\matrix{(hk) &(\bar{h}k)\cr}]
        Line segment  
        Pair of parallel edges [\matrix{(10) &(\bar{1}0)\cr}]
        Line segment through origin  
1 a .m. [0,y] Single edge (01) or [(0\bar{1})]
        Single point  
2mm       [Scheme scheme4]  
4 c 1 [x,y\quad \bar x,\bar y\quad \bar x,y\quad x,\bar y] Rhomb [\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{h}k) &(h\bar{k})\cr}]
        Rectangle  
2 b .m. [0,y\quad 0,\bar y] Pair of parallel edges [\matrix{(01) &(0\bar{1})\cr}{\hbox to 4.6pc{}}]
        Line segment through origin  
2 a ..m [x,0\quad \bar x,0] Pair of parallel edges [\matrix{(10) &(\bar{1}0)\cr}{\hbox to 4.6pc{}}]
        Line segment through origin  
1 o 2mm [0, 0] Point in origin  
SQUARE SYSTEM
4       [Scheme scheme5]  
4 a 1 [x,y\quad \bar x,\bar y\quad \bar y,x\quad y,\bar x ] Square [\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{k}h) &(k\bar{h})\cr}]
        Square  
1 o 4.. [0,0] Point in origin  
4mm       [Scheme scheme6]  
8 c 1 [\matrix{x,y& \bar x,\bar y &\bar y,x &y,\bar x\cr} ] Ditetragon [\matrix{(hk) &(\bar{h}{\hbox to .5pt{}}\bar{k}) &(\bar{k}h) &(k\bar{h})\cr}]
      [\matrix{\bar x,y& x,\bar y & y,x &\bar y,\bar x\cr}] Truncated square [\matrix{(\bar{h}k) &(h\bar{k}) &(kh) &(\bar{k}{\hbox to .5pt{}}\bar{h})\cr}]
4 b ..m [x,x\quad \bar x,\bar x\quad \bar x,x\quad x,\bar x] Square [\matrix{(11) &(\bar{1}\bar{1}) &(\bar{1}1) &(1\bar{1})\cr}]
        Square  
4 a .m. [x,0\quad \bar x,0\quad 0,x \quad 0,\bar x ] Square [\matrix{(10) &(\bar{1}0) &(01) &(0\bar{1})\cr}]
        Square  
1 o 4mm [0,0] Point in origin  
HEXAGONAL SYSTEM
3       [Scheme scheme7]  
3 a 1 [x,y\quad \bar y,x-y \quad \bar x+y,\bar x ] Trigon [\matrix{(hki) &(ihk) &(kih)\cr}]
        Trigon  
1 o 3.. [0,0] Point in origin  
3m1       [Scheme scheme8]  
6 b 1 [\matrix{x,y & \bar y,x-y &\bar x+y,\bar x\cr}] Ditrigon [{\hbox to 9pt{}}\matrix{(hki) &{\hbox to 2pt{}}(ihk) &{\hbox to 3pt{}}(kih)\hfill\cr}]
      [\matrix{\bar y,\bar x & \bar x+y,y & x,x-y\cr}] Truncated trigon [{\hbox to 9pt{}}\matrix{(\bar{k}\bar{h}\bar{i\hskip-2pt\phantom l}) &{\hbox to 2pt{}}(\bar{\phantom l\hskip-2.5pt i}\bar{k}\bar{h}) &{\hbox to 3pt{}}(\bar{h}\bar{\hskip-2.5pt\phantom li}\bar{k})\cr}]
        Hexagon [{\hbox to 11pt{}}\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]
        Hexagon [{\hbox to 11pt{}}\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]
3 a .m. [x,\bar x\quad x,2x\quad 2\bar x,\bar x ] Trigon [{\hbox to 11.5pt{}}\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]
        Trigon or [\matrix{(\bar{1}01) &(1\bar{1}0) &(01\bar{1})\cr}]
1 o 3m. [0,0] Point in origin  
31m       [Scheme scheme9]  
6 b 1 [\matrix{x,y& \bar y,x-y & \bar x+y,\bar x\cr}] Ditrigon [{\hbox to 8pt{}}\matrix{(hki) {\hbox to 13pt{}}(ihk) {\hbox to 12pt{}}(kih)\cr}]
      [\matrix{y,x & x-y,\bar y & \bar x,\bar x+y\cr}] Truncated trigon [{\hbox to 8pt{}}\matrix{(khi) {\hbox to 13pt{}}(ikh) {\hbox to 12pt{}}(hik)\cr}]
        Hexagon [{\hbox to 11pt{}}\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]
        Hexagon [{\hbox to 11pt{}}\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]
3 a ..m [x,0 \quad 0,x \quad \bar x,\bar x] Trigon [{\hbox to 11pt{}}\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]
        Trigon or [\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]
1 o 3.m [0,0] Point in origin  
6       [Scheme scheme10]  
6 a 1 [x,y \quad \bar y,x-y \quad \bar x+y,\bar x ] Hexagon [\matrix{(hki) &(ihk) &(kih)\cr}]
      [\bar x,\bar y\quad y,\bar x+y \quad x-y,x] Hexagon [\matrix{(\bar{h}\bar{k}\bar{\phantom l\hskip-2.5pti}) &(\bar{\phantom l\hskip-2.5pti}\bar{h}\bar{k}) &(\bar{k}\bar{\phantom l\hskip-2.5pti}\bar{h})\cr}]
1 o 6.. [0,0] Point in origin  
6mm       [Scheme scheme11]  
12 c 1 [\matrix{x,y & \bar y,x-y &\bar x+y,\bar x}] Dihexagon [\matrix{(hki)& (ihk)& (kih)}]
      [\matrix{\bar x,\bar y & y,\bar x+y & x-y,x}] Truncated hexagon [\matrix{(\bar{h}\bar{k}\bar{i})&(\bar{i}\bar{h}\bar{k}) &(\bar{k}\bar{i}\bar{h})}]
      [\matrix{\bar y,\bar x & \bar x+y,y & x,x-y}]   [\matrix{(\bar{k}\bar{h}\bar{i})&(\bar{i}\bar{k}\bar{h}) &(\bar{h}\bar{i}\bar{k})}]
      [\matrix{y,x &x-y,\bar y &\bar x,\bar x+y}]   [\matrix{(khi) &(ikh) &(hik)}]
6 b .m. [\matrix{x,\bar x & x,2x & 2\bar x,\bar x }] Hexagon [\matrix{(10\bar{1}) &(\bar{1}10) &(0\bar{1}1)\cr}]
      [\matrix{\bar x,x & \bar x,2\bar x & 2x,x}] Hexagon [\matrix{(\bar{1}01) &(1\bar{1}0) &(01\bar{1})\cr}]
6 a ..m [\matrix{x,0 & 0,x & \bar x,\bar x }] Hexagon [\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]
      [\matrix{\bar x,0 & 0,\bar x & x,x}] Hexagon [\matrix{(\bar{1}\bar{1}2) &(2\bar{1}\bar{1}) &(\bar{1}2\bar{1})\cr}]
1 o 6mm [0,0] Point in origin  

Table 3.2.3.2| top | pdf |
The 32 three-dimensional crystallographic point groups

The point groups are listed in blocks according to crystal system and are specified by their short and (if different) full Hermann–Mauguin symbols and their Schoenflies symbols. For each point group, the stereographic projections show (on the left) the general position and (on the right) the symmetry elements. The list of Wyckoff positions includes: Columns 1 to 4: multiplicity, Wyckoff letter, oriented site-symmetry symbol, coordinate triplets; Under the stereographic projections: face forms (in roman type) and point forms (in italics); if there is more than one entry, subsequent entries refer to limiting (noncharacteristic) forms; Last column: Miller indices of equivalent faces [for trigonal and hexagonal groups, Bravais–Miller indices (hkil) are used if referred to hexagonal axes].

TRICLINIC SYSTEM
1 [C_{1}] [Scheme scheme12]  
1 a 1 [x,y,z] Pedion or monohedron (hkl)
        Single point  
        Symmetry of special projections  
        Along any direction  
        1  
[\bar{1}] [C_{i}] [Scheme scheme13]  
2 a 1 [x,y,z\quad \bar x,\bar y,\bar z] Pinacoid or parallelohedron [(hkl) \quad(\bar{h}\bar{k}\bar{l})]
        Line segment through origin  
1 o [\bar 1] [0,0,0] Point in origin  
        Symmetry of special projections  
        Along any direction  
        2  
MONOCLINIC SYSTEM
2 [C_{2}] [Scheme scheme14]  
UNIQUE AXIS b  
2 b 1 [x,y,z\quad \bar x,y,\bar z] Sphenoid or dihedron [(hkl) \quad(\bar{h}k\bar{l})]
        Line segment  
        Pinacoid or parallelohedron [(h0l) \quad(\bar{h}0\bar{l})]
        Line segment through origin  
1 a 2 [0,y,0] Pedion or monohedron [(010) \hbox{ or } (0\bar{1}0)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&2&&m\cr}]  
2 [C_{2}] [Scheme scheme15]  
UNIQUE AXIS c  
2 b 1 [x,y,z \quad \bar x,\bar y,z] Sphenoid or dihedron [(hkl) \quad(\bar{h}\bar{k}l)]
        Line segment  
        Pinacoid or parallelohedron [(hk0) \quad(\bar{h}\bar{k}0)]
        Line segment through origin  
1 a 2 [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&m&&2}]  
m [C_{s}] [Scheme scheme16]  
UNIQUE AXIS b  
2 b 1 [x,y,z\quad x,\bar y,z] Dome or dihedron [(hkl) \quad(h\bar{k}l)]
        Line segment  
        Pinacoid or parallelohedron [(010) \quad(0\bar{1}0)]
        Line segment through origin  
1 a m [x,0,z ] Pedion or monohedron [(h0l)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&1&&m\cr}]  
m [C_{s}] [Scheme scheme17]  
UNIQUE AXIS c  
2 b 1 [x,y,z\quad x,y,\bar z] Dome or dihedron [(hkl) \quad(hk\bar{l})]
        Line segment  
        Pinacoid or parallelohedron [(001) \quad(00\bar{1})]
        Line segment through origin  
1 a m [x,y,0] Pedion or monohedron [(hk0)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&m&&1\cr}]  
2/m [C_{2h}] [Scheme scheme18]  
UNIQUE AXIS b  
4 c 1 [x,y,z\quad \bar x,y,\bar z\quad \bar x,\bar y,\bar z\quad x,\bar y,z] Rhombic prism [(hkl) \quad(\bar{h}k\bar{l}) \quad(\bar{h}\bar{k}\bar{l}) \quad(h\bar{k}l)]
        Rectangle through origin  
2 b m [x,0,z\quad \bar x,0,\bar z] Pinacoid or parallelohedron [(h0l) \quad(\bar{h}0\bar{l})]
        Line segment through origin  
2 a 2 [0,y,0\quad 0,\bar y,0] Pinacoid or parallelohedron [(010) \quad(0\bar{1}0)]
        Line segment through origin  
1 o 2/m [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along } [100]&&\hbox{Along } [010]&&\hbox{Along }[001]\cr 2mm&&2&&2mm\cr }]  
2/m [C_{2h}] [Scheme scheme19]  
UNIQUE AXIS c  
4 c 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,\bar y,\bar z\quad x,y,\bar z] Rhombic prism [(hkl) \quad(\bar{h}\bar{k}l) \quad(\bar{h}\bar{k}\bar{l}) \quad(hk\bar{l})]
        Rectangle through origin  
2 b m [x,y,0\quad \bar x,\bar y,0] Pinacoid or parallelohedron [(hk0) \quad(\bar{h}\bar{k}0)]
        Line segment through origin  
2 a 2 [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [(001) \quad(00\bar{1})]
        Line segment through origin  
1 o 2/m [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along } [100]&&\hbox{Along } [010]&&\hbox{Along }[001]\cr 2mm&&2mm&&2}]  
ORTHORHOMBIC SYSTEM
222 [D_{2}] [Scheme scheme20]  
4 d 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z] Rhombic disphenoid or rhombic tetrahedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]
        Rhombic tetrahedron  
        Rhombic prism [\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 [\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]
        Rectangle through origin  
        Rhombic prism [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]
        Rectangle through origin  
2 c ..2 [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(001) {\hbox to 7.5pt{}}(00\bar{1})\cr}]
        Line segment through origin  
2 b .2. [0,y,0\quad 0,\bar y,0] Pinacoid or parallelohedron [\matrix{(010) {\hbox to 7.5pt{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 a 2.. [x,0,0\quad \bar x,0,0] Pinacoid or parallelohedron [\matrix{(100) {\hbox to 7.5pt{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 o 222 [0,0,0] Point in origin  
        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 [x,y,z\quad \bar x,\bar y,z\quad x,\bar y,z\quad \bar x,y,z] Rhombic pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(h\bar{k}l) &(\bar{h}kl)\cr}]
        Rectangle  
        Rhombic prism [\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.. [0,y,z\quad 0,\bar y,z] Dome or dihedron [\matrix{(0kl) &(0\bar{k}l)\cr}]
        Line segment  
        Pinacoid or parallelohedron [\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 b .m. [x,0,z\quad \bar x,0,z] Dome or dihedron [\matrix{(h0l) &(\bar{h}0l)\cr}]
        Line segment  
        Pinacoid or parallelohedron [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 a mm2 [0,0,z ] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr m&&m&&2mm\cr}]  
[\openup 6pt\matrix{m\ m\ m\cr{\displaystyle{2 \over m}\ {2 \over m}\ {2 \over m}}\cr}] [D_{2h}] [Scheme scheme22]  
8 g 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z] Rhombic dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]
      [\bar x,\bar y,\bar z\quad x,y,\bar z\quad x,\bar y,z\quad \bar x,y,z] Rectangular prism [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl)\cr}]
4 f ..m [x,y,0\quad \bar x,\bar y,0\quad \bar x,y,0\quad x,\bar y,0] Rhombic prism [\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  
4 e .m. [x,0,z\quad \bar x,0,z\quad \bar x,0,\bar z\quad x,0,\bar z] Rhombic prism [\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]
        Rectangle through origin  
4 d m.. [0,y,z\quad 0,\bar y,z\quad 0,y,\bar z\quad 0,\bar y,\bar z] Rhombic prism [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]
        Rectangle through origin  
2 c mm2 [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
2 b m2m [0,y,0\quad 0,\bar y,0 ] Pinacoid or parallelohedron [\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 a 2mm [x,0,0\quad \bar x,0,0] Pinacoid or parallelohedron [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 o mmm [0,0,0] Point in origin  
        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 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\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.. [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        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 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
        Tetragonal tetrahedron  
        Tetragonal prism [\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.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 4..] [0,0,0] Point in origin  
        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 [x,y,z \quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,\bar y,\bar z\quad x,y,\bar z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal prism [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
4 b m.. [x,y,0\quad \bar x,\bar y,0\quad \bar y,x,0\quad y,\bar x,0] Tetragonal prism [\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  
2 a 4.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 4/m.. [0,0,0] Point in origin  
        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 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal trapezohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,y,\bar z\quad x,\bar y,\bar z\quad y,x,\bar z\quad \bar y,\bar x,\bar z] Twisted tetragonal antiprism [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]
        Ditetragonal prism [\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 [\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 [\matrix{(h0l) {\hbox to .8pc{}}(\bar{h}0l) {\hbox to .8pc{}}(0hl) {\hbox to .8pc{}}(0\bar{h}l)\cr}]
        Tetragonal prism [\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 [\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 [\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. [x,0,0\quad \bar x,0,0\quad 0,x,0\quad 0,\bar x,0] Tetragonal prism [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .6pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Square through origin  
4 b ..2 [x,x,0\quad \bar x,\bar x,0\quad \bar x,x,0\quad x,\bar x,0] Tetragonal prism [\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  
2 a 4.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 422 [0,0,0] Point in origin  
        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 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Ditetragonal pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [x,\bar y,z\quad \bar x,y,z\quad \bar y,\bar x,z\quad y,x,z] Truncated square [\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]
        Ditetragonal prism [\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 [\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. [x,0,z\quad \bar x,0,z\quad 0,x,z\quad 0,\bar x,z] Tetragonal pyramid [\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\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 [x,x,z\quad \bar x,\bar x,z\quad \bar x,x,z\quad x,\bar x,z] Tetragonal pyramid [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\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 [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        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 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal scalenohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [\bar x,y,\bar z\quad x,\bar y,\bar z\quad \bar y,\bar x,z\quad y,x,z] Tetragonal tetrahedron cut off by pinacoid [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(\bar{k}\bar{h}l) &(khl)\cr}]
        Ditetragonal prism [\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 [\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 [\matrix{(h0l) &(\bar{h}0l) {\hbox to 0.75pc{}}(0\bar{h}\bar{l}) {\hbox to 0.85pc{}}(0h\bar{l})\cr}]
        Tetragonal prism [\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 [x,x,z\quad \bar x,\bar x,z\quad x,\bar x,\bar z\quad \bar x,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\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  
        Tetragonal prism [\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. [x,0,0\quad \bar x,0,0\quad 0,\bar x,0\quad 0,x,0] Tetragonal prism [\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  
2 a 2.mm [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to 0.65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 42m] [0,0,0] Point in origin  
        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 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal scalenohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [x,\bar y,z\quad \bar x,y,z\quad y,x,\bar z\quad \bar y,\bar x,\bar z] Tetragonal tetrahedron cut off by pinacoid [\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 [\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 [\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 [\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 [\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. [x,0,z\quad \bar x,0,z\quad 0,\bar x,\bar z\quad 0,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\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  
        Tetragonal prism [\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 [x,x,0\quad \bar x,\bar x,0\quad x,\bar x,0\quad \bar x,x,0] Tetragonal prism [\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  
2 a 2mm. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) &{\hbox to -2pt{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 4 m2] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&m&&2mm\cr}]  
[\openup6pt\matrix{4/mmm\hfill\cr{\displaystyle{4 \over m}{2 \over m}{2 \over m}}\hfill}] [D_{4h}] [Scheme scheme30]  
16 g 1 [x,y,z \quad \bar x,\bar y,z \quad \bar y,x,z \quad y,\bar x,z] Ditetragonal dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,y,\bar z \quad x,\bar y,\bar z \quad y,x,\bar z \quad \bar y,\bar x,\bar z] Edge-truncated tetragonal prism [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]
      [\bar x,\bar y,\bar z \quad x,y,\bar z \quad y,\bar x,\bar z \quad \bar y,x,\bar z]   [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [x,\bar y,z \quad \bar x,y,z \quad \bar y,\bar x,z \quad y,x,z]   [\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]
8 f .m. [x,0,z\quad \bar x,0,z\quad 0,x,z\quad 0,\bar x,z] Tetragonal dipyramid [\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]
      [\bar x,0,\bar z\quad x,0,\bar z\quad 0,x,\bar z\quad 0,\bar x,\bar z] Tetragonal prism [\matrix{(\bar{h}0\bar{l}) &(h0\bar{l}) &(0h\bar{l}) &(0\bar{h}\bar{l})\cr}]
8 e ..m [x,x,z\quad \bar x,\bar x,z\quad \bar x,x,z\quad x,\bar x,z] Tetragonal dipyramid [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]
      [\bar x,x,\bar z\quad x,\bar x,\bar z\quad x,x,\bar z\quad \bar x,\bar x,\bar z] Tetragonal prism [\matrix{(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) &(hh\bar{l}) &(\bar{h}\bar{h}\bar{l})\cr}]
8 d m.. [x,y,0\quad \bar x,\bar y,0\quad \bar y,x,0\quad y,\bar x,0] Ditetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
      [\bar x,y,0\quad x,\bar y,0\quad y,x,0\quad \bar y,\bar x,0] Truncated square through origin [\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. [x,0,0\quad \bar x,0,0\quad 0,x,0\quad 0,\bar x,0] Tetragonal prism [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .7pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Square through origin  
4 b m.m2 [x,x,0\quad \bar x,\bar x,0\quad \bar x,x,0\quad x,\bar x,0] Tetragonal prism [\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  
2 a 4mm [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 4/mmm [0,0,0] Point in origin  
        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 [x,y,z \quad \bar y,x-y,z \quad \bar x+y,\bar x,z] Trigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
        Trigon  
        Trigonal prism [\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]
        Trigon through origin  
1 a 3.. [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
      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 [x,y,z\quad z,x,y\quad y,z,x] Trigonal pyramid [\matrix{(hkl) &(lhk) &(klh)\cr}]
        Trigon  
        Trigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Trigon through origin  
1 a 3. [x,x,x] Pedion or monohedron [(111) \hbox{ or } (\bar{1}\bar{1}\bar{1})]
        Single point  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Rhombohedron [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z] Trigonal antiprism [\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 [\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]
        Hexagon through origin [\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.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to .4pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3..] [0,0,0] Point in origin  
        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 [x,y,z\quad z,x,y\quad y,z,x] Rhombohedron [\matrix{(hkl) &(lhk) &(klh)\cr}]
      [\bar x,\bar y,\bar z\quad \bar z,\bar x,\bar y\quad \bar y,\bar z,\bar x] Trigonal antiprism [\matrix{(\bar{h}\bar{k}\bar{l}) &(\bar{l}\bar{h}\bar{k}) &(\bar{k}\bar{l}\bar{h})\cr}]
        Hexagonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\! +\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]
        Hexagon through origin [\matrix{(\bar{h}\bar{k}(h\!+\!k)) &((h\! +\! k)\bar{h}\bar{k}) &(\bar{k}(h\! +\! k)\bar{h})\cr}]
2 a 3. [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3.] [0,0,0] Point in origin  
        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}]  
312 [D_{3}] [Scheme scheme36]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Trigonal trapezohedron [\matrix{(hkil) {\hbox to 1.5pc{}}(ihkl) {\hbox to 1.45pc{}}(kihl)\cr}]
      [\bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z] Twisted trigonal antiprism [\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 [\matrix{(hki0) {\hbox to 1.35pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]
        Truncated trigon through origin [\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 [\matrix{(h0\bar{h}l) {\hbox to 1.3pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]
        Trigonal prism [\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 [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Trigonal antiprism [\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 [\matrix{(11\bar{2}0) {\hbox to 1.1pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]
        Hexagon through origin [\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 [x,\bar x,0\quad x,2x,0\quad 2\bar x,\bar x,0] Trigonal prism [(10\bar{1}0)\quad(\bar{1}100)\quad(0\bar{1}10)]
        Trigon through origin or [(\bar{1}010)\quad(1\bar{1}00)\quad(01\bar{1}0)]
2 a 3.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to 1.1pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o 3.2 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&1&&2\cr}]  
321 [D_{3}] [Scheme scheme35]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Trigonal trapezohedron [\matrix{(hkil) {\hbox to 1.45pc{}}(ihkl) {\hbox to 1.5pc{}}(kihl)\cr}]
      [y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z] Twisted trigonal antiprism [\matrix{(khi\bar{l}) {\hbox to 1.45pc{}}(hik\bar{l}) {\hbox to 1.45pc{}}(ikh\bar{l})\cr}]
        Ditrigonal prism [\matrix{(hki0) {\hbox to 1.3pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]
        Truncated trigon through origin [\matrix{(khi0) {\hbox to 1.3pc{}}(hik0) {\hbox to 1.3pc{}}(ikh0)\cr}]
        Trigonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Trigonal prism [\matrix{(hh\overline{2h}{\hbox to 1pt{}}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l})\cr}]
        Rhombohedron [\matrix{(h0\bar{h}l) {\hbox to 1.25pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]
        Trigonal antiprism [\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 [\matrix{(10\bar{1}0) {\hbox to 1.05pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) {\hbox to 1.05pc{}}(1\bar{1}00) {\hbox to 1.1pc{}}(\bar{1}010)\cr}]
3 b .2. [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Trigonal prism [(11\bar{2}0) \quad(\bar{2}110) \quad(1\bar{2}10)]
        Trigon through origin or [(\bar{1}\bar{1}20)\quad(2\bar{1}\bar{1}0)\quad(\bar{1}2\bar{1}0)]
2 a 3.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to 1.05pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o 32. [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&2&&1\cr}]  
32 [D_{3}] [Scheme scheme37]  
RHOMBOHEDRAL AXES  
6 c 1 [x,y,z \quad z,x,y\quad y,z,x] Trigonal trapezohedron [\matrix{(hkl) {\hbox to .85pc{}}(lhk) {\hbox to .85pc{}}(klh)\cr}]
      [\bar z,\bar y,\bar x\quad \bar y,\bar x,\bar z\quad \bar x,\bar z,\bar y] Twisted trigonal antiprism [\matrix{(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]
        Ditrigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Truncated trigon through origin [\matrix{(\bar{k}\bar{h}(h\!+\!k)) &(\bar{h}(h\!+\!k)\bar{k}) &((h\!+\!k)\bar{k}\bar{h})\cr}]
        Trigonal dipyramid [\matrix{(hk(2k\!-\!h)) &((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]
        Trigonal prism [\matrix{(\bar{k}\bar{h}(h\!-\!2k)) &(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]
        Rhombohedron [\matrix{(hhl) &(lhh) &(hlh)\cr}]
        Trigonal antiprism [\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]
        Hexagonal prism [\matrix{(11\bar{2})&(\bar{2}11) &(1\bar{2}1)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}2) &(\bar{1}2\bar{1}) &(2\bar{1}\bar{1})\cr}]
3 b .2 [ x,\bar x,0\quad 0,x,\bar x\quad \bar x,0,x] Trigonal prism [(01\bar{1}) \quad(\bar{1}01) \quad(1\bar{1}0)]
        Trigon through origin or [(0\bar{1}1) \quad(10\bar{1}) \quad(\bar{1}10)]
2 a 3. [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o 32 [0,0,0] Point in origin  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Ditrigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z] Truncated trigon [\matrix{(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
        Ditrigonal prism [\matrix{(hki0)&(ihk0) &(kih0)\cr}]
        Truncated trigon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal pyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Hexagon [\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
3 b .m. [x,\bar x,z \quad x,2x,z\quad 2\bar x,\bar x,z ] Trigonal pyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Trigon  
        Trigonal prism [(10\bar{1}0) \quad(\bar{1}100) \quad(0\bar{1}10)]
        Trigon through origin or [(\bar{1}010) \quad(1\bar{1}00) \quad(01\bar{1}0)]
1 a 3m. [0,0,z ] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Ditrigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z] Truncated trigon [\matrix{(khil) &(hikl) &(ikhl)\cr}]
        Ditrigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Truncated trigon through origin [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal pyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagon [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) &(1\bar{1}00) &(\bar{1}010)\cr}]
3 b ..m [x,0,z\quad 0,x,z \quad \bar x,\bar x,z] Trigonal pyramid [\matrix{(hh\overline{2h}l)&(\overline{2h}hhl)& (h\overline{2h}hl)}]
        Trigon  
        Trigonal prism [(11\bar{2}0) \quad(\bar{2}110) \quad(1\bar{2}10)]
        Trigon through origin or [(\bar{1}\bar{1}20) \quad(2\bar{1}\bar{1}0) \quad(\bar{1}2\bar{1}0)]
1 a 3.m [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        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 [x,y,z\quad z,x,y\quad y,z,x ] Ditrigonal pyramid [\matrix{(hkl) &(lhk) &(klh)\cr}]
      [z,y,x\quad y,x,z\quad x,z,y ] Truncated trigon [\matrix{(khl) &(hlk) &(lkh)\cr}]
        Ditrigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h \!+\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]
        Truncated trigon through origin [\matrix{(kh(\overline{h\!+\!k}))&(h(\overline{h \!+\! k})k) &((\overline{h \!+\! k})kh)\cr}]
        Hexagonal pyramid [\matrix{(hk(2k\!-\!h)) &((2k\! -\! h)hk) &(k(2k \!- \!h)h)\cr}]
        Hexagon [\matrix{(kh(2k\!-\!h))&(h(2k\! -\! h)k) &((2k\! - \!h)kh)\cr}]
        Hexagonal prism [\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]
        Hexagon through origin [\matrix{(10\bar{1}) &(0\bar{1}1) &(\bar{1}10)\cr}]
3 b .m [x,y,x \quad x,x,y\quad y,x,x] Trigonal pyramid [\matrix{(hhl) &(lhh) &(hlh)\cr}]
        Trigon  
        Trigonal prism [(11\bar{2}) \quad(\bar{2}11) \quad(1\bar{2}1)]
        Trigon through origin or [(\bar{1}\bar{1}2) \quad(2\bar{1}\bar{1}) \quad(\bar{1}2\bar{1})]
1 a 3m [x,x,x ] Pedion or monohedron [(111) \hbox{ or }(\bar{1}\bar{1}\bar{1})]
        Single point  
        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}1m\cr \bar{3}1{\displaystyle{2 \over m}}\cr}] [D_{3d}] [Scheme scheme42]  
HEXAGONAL AXES  
12 d 1 [\matrix{x,y,z\quad \bar y,x-y,z \quad \bar x+y,\bar x,z\hfill\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z\hfill}] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkil) &(ihkl) &(kihl)\cr (\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z\hfill}] Trigonal antiprism sliced off by pinacoid [\matrix{(\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 (khil) &(hikl)&(ikhl)\cr}]
        Dihexagonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Truncated hexagon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
          [\matrix{(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
          [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagonal prism [\matrix{(0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})\cr}]
          [\matrix{(\bar{h}0h\bar{l}) &(h\bar{h}0\bar{l}) &(0h\bar{h}\bar{l})\cr}]
          [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
6 c ..m [x,0,z \quad 0,x,z \quad \bar x,\bar x,z ] Rhombohedron [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
      [0,\bar x,\bar z \quad \bar x,0,\bar z \quad x,x,\bar z] Trigonal antiprism [\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 [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
6 b ..2 [x,\bar x,0\quad x,2x,0 \quad 2\bar x,\bar x,0] Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
      [ \bar x,x,0 \quad \bar x,2\bar x,0 \quad 2x,x,0 ] Hexagon through origin [\matrix{(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
2 a 3.m [0,0,z \quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3.m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2\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{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\hfill\cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\hfill} ] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkil)\quad (ihkl) \quad(kihl)\cr (khi\bar{l})\quad (hik\bar{l}) \quad(ikh\bar{l})}]
      [\matrix{\bar x,\bar y,\bar z \quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\hfill}] Trigonal antiprism sliced off by pinacoid [\matrix{(\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)}]
        Dihexagonal prism [\matrix{(hki0)\quad (ihk0) \quad(kih0)\cr}]
        Truncated hexagon through origin [\matrix{(khi0)\quad (hik0)\quad (ikh0)\cr}]
          [\matrix{(\bar{h}\bar{k}\bar{i}0)\quad (\bar{i}\bar{h}\bar{k}0)\quad(\bar{k}\bar{i}\bar{h}0)\cr}]
          [\matrix{(\bar{k}\bar{h}\bar{i}0)\quad (\bar{h}\bar{i}\bar{k}0) \quad(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal dipyramid [\matrix{(hh\overline{2h}l)\quad (\overline{2h}hhl)\quad (h\overline{2h}hl)\cr}]
        Hexagonal prism [\matrix{(hh\overline{2h}\,\bar{l})\quad (h\overline{2h}h\bar{l})\quad (\overline{2h}hh\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2h\bar{l})\quad(2h\bar{h}\bar{h}\bar{l})\quad (\bar{h}2h\bar{h}\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2hl)\quad (\bar{h}2h\bar{h}l)\quad (2h\bar{h}\bar{h}l)\cr}]
6 c .m. [x,\bar x,z \quad x,2x,z\quad 2\bar x,\bar x,z ] Rhombohedron [\matrix{(h0\bar{h}l)\quad (\bar{h}h0l)\quad (0\bar{h}hl)\cr}]
      [\bar x,x,\bar z \quad 2x,x,\bar z\quad \bar x,2\bar x,\bar z ] Trigonal antiprism [\matrix{(0h\bar{h}\bar{l})\quad (h\bar{h}0\bar{l})\quad (\bar{h}0h\bar{l})\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0)\quad (\bar{1}100)\quad (0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0)\quad (1\bar{1}00)\quad (\bar{1}010)\cr}]
6 b .2. [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Hexagonal prism [\matrix{(11\bar{2}0)\quad (\bar{2}110)\quad (1\bar{2}10)\cr}]
      [\bar x,0,0 \quad 0,\bar x,0 \quad x,x,0 ] Hexagon through origin [\matrix{(\bar{1}\bar{1}20)\quad (\bar{1}2\bar{1}0)\quad (2\bar{1}\bar{1}0)\cr}]
2 a 3m. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001)\quad (000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3 m.] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox {Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2&&2mm\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 [\matrix{x,y,z\quad z,x,y\quad y,z,x\hfill\cr \bar z,\bar y,\bar x\quad \bar y,\bar x,\bar z\quad \bar x,\bar z,\bar y\hfill }] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkl) &(lhk) &(klh)\cr(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad \bar z,\bar x,\bar y\quad \bar y,\bar z,\bar x\hfill\cr z,y,x\quad y,x,z\quad x,z,y\hfill }] Trigonal antiprism sliced off by pinacoid [\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 [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Truncated hexagon through origin [\matrix{(\bar{k}\bar{h}(h\!+\!k)) &(\bar{h}(h\!+\!k)\bar{k}) &((h\!+\!k)\bar{k}\bar{h})\cr}]
          [\matrix{(\bar{h}\bar{k}(h\!+\!k)) &((h\!+\!k)\bar{h}\bar{k}) &(\bar{k}(h\!+\!k)\bar{h})\cr}]
          [\matrix{(kh(\overline{h\!+\!k})) &(h(\overline{h\!+\!k})k) &((\overline{h\!+\!k})kh)\cr}]
        Hexagonal dipyramid [\matrix{(hk(2k\!-\!h)) &((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]
        Hexagonal prism [\matrix{(\bar{k}\bar{h}(h\!-\!2k)) &(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]
          [\matrix{(\bar{h}\bar{k}(h\!-\!2k)) &((h\!-\!2k)\bar{h}\bar{k}) &(\bar{k}(h\!-\!2k)\bar{h})\cr}]
          [\matrix{(kh(2k\!-\!h)) &(h(2k\!-\!h)k) &((2k\!-\!h)kh)\cr}]
6 c .m [x,y,x\quad x,x,y\quad y,x,x ] Rhombohedron [\matrix{(hhl) &(lhh) &(hlh)\cr}]
      [\bar x,\bar y,\bar x\quad \bar y,\bar x,\bar x\quad \bar x,\bar x,\bar y ] Trigonal antiprism [\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]
        Hexagonal prism [\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}2) &(\bar{1}2\bar{1}) &(2\bar{1}\bar{1})\cr}]
6 b .2 [x,\bar x,0\quad 0,x,\bar x \quad \bar x,0,x ] Hexagonal prism [\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]
      [\bar x,x,0\quad 0,\bar x,x \quad x,0,\bar x ] Hexagon through origin [\matrix{(0\bar{1}1) &(10\bar{1}) &(\bar{1}10)\cr}]
2 a 3m [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3 m] [0,0,0] Point in origin  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Hexagonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z ] Hexagon [\matrix{(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]
        Hexagonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Hexagon through origin [\matrix{(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
1 a 6.. [0,0,z] Pedion or monohedron [(0001) \hbox{ or }(000\bar{1})]
        Single point  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Trigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z] Trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
3 b m.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0 ] Trigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Trigon through origin  
2 a 3.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6 ..] [0,0,0] Point in origin  
        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 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill \cr Hexagonal\ prism\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\cr x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z}]   [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l}) \cr(\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.. [\matrix{x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0\cr \bar x,\bar y,0\quad y,\bar x+y,0 \quad x-y,x,0 }] [\matrix{\rm Hexagonal\ prism\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
2 a 6.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 6/m.. [0,0,0] Point in origin  
        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 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z \cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z }] [\matrix{{\rm Hexagonal\ trapezohedron}\hfill\cr Twisted\ hexagonal\ antiprism\hfill}] [\matrix{(hkil) &(ihkl)&(kihl) \cr (\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khi\bar{l})&(hik\bar{l}) &(ikh\bar{l}) \cr (\bar{k}\bar{h}\bar{i}\bar{l})&(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
        [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)}]
        [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl) &(h\bar{h}0l) &(0h\bar{h}l)\cr(0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l}) \cr(0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})}]
        [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr(\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr (hh\overline{2h}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l}) \cr(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
6 c ..2 [\matrix{x,\bar x,0\quad x,2x,0\quad 2\bar x,\bar x,0\cr \bar x,x,0\quad \bar x,2\bar x,0 \quad 2x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b .2. [\matrix{x,0,0\quad 0,x,0\quad \bar x,\bar x,0\cr \bar x,0,0\quad 0,\bar x,0\quad x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
2 a 6.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 622 [0,0,0] Point in origin  
        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 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z\cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z }] [\matrix{{\rm Dihexagonal\ pyramid}\hfill\cr Truncated\ hexagon\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khil) &(hikl) &(ikhl) \cr(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
        [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
6 c .m. [\matrix{x,\bar x,z\quad x,2x,z\quad 2\bar x,\bar x,z\cr \bar x,x,z\quad \bar x,2\bar x,z\quad 2x,x,z }] [\matrix{{\rm Hexagonal\ pyramid}\hfill\cr Hexagon\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl) &(h\bar{h}0l) &(0h\bar{h}l)\cr}]
        [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b ..m [\matrix{x,0,z\quad 0,x,z\quad \bar x,\bar x,z\cr \bar x,0,z\quad 0,\bar x,z\quad x,x,z }] [\matrix{{\rm Hexagonal\ pyramid}\hfill\cr Hexagon\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr(\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr}]
        [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
1 a 6mm [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        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 [\matrix{x,y,z \quad \bar y,x-y,z \quad \bar x+y,\bar x,z\hfill}] Ditrigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\matrix{x,y,\bar z\quad \bar y,x-y,\bar z \quad \bar x+y,\bar x,\bar z\hfill}] Edge-truncated trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
      [\matrix{\bar y,\bar x,z\quad \bar x+y,y,z \quad x,x-y,z\hfill}]   [\matrix{(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
      [\matrix{\bar y,\bar x,\bar z\quad \bar x+y,y,\bar z \quad x,x-y,\bar z\hfill}]   [\matrix{(\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l})&(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
        Hexagonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Hexagonal prism [\matrix{(hh\overline{2h}\,\bar{l}) &(\overline{2h}hh\bar{l}) &(h\overline{2h}h\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]
          [\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.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0] Ditrigonal prism [\matrix{(hki0) &(ihk0)&(kih0)\cr}]
      [\bar y,\bar x,0\quad \bar x+y,y,0\quad x,x-y,0] Truncated trigon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
6 c .m. [x,\bar x,z\quad x,2x,z\quad 2\bar x,\bar x,z] Trigonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
      [x,\bar x,\bar z\quad x,2x,\bar z\quad 2\bar x,\bar x,\bar z ] Trigonal prism [\matrix{(h0\bar{h}\bar{l}) &(\bar{h}h0\bar{l}) &(0\bar{h}h\bar{l})\cr}]
3 b mm2 [x,\bar x,0 \quad x,2x,0 \quad 2\bar x,\bar x,0 ] Trigonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Trigon through origin or [\matrix{(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
2 a 3m. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6m2] [0,0,0] Point in origin  
        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 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Ditrigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z] Edge-truncated trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
      [y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z]   [\matrix{(khi\bar{l}) &(hik\bar{l}) &(ikh\bar{l})\cr}]
      [y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z]   [\matrix{(khil) &(hikl) &(ikhl)\cr}]
        Hexagonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagonal prism [\matrix{(h0\bar{h}\bar{l}) &(\bar{h}h0\bar{l}) &(0\bar{h}h\bar{l})\cr}]
          [\matrix{(0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l})\cr}]
          [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
6 d m.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0 ] Ditrigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
      [y,x,0\quad x-y,\bar y,0\quad \bar x,\bar x+y,0] Truncated trigon through origin [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) &(1\bar{1}00) &(\bar{1}010)\cr}]
6 c ..m [x,0,z\quad 0,x,z\quad \bar x,\bar x,z] Trigonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
      [x,0,\bar z\quad 0,x,\bar z\quad \bar x,\bar x,\bar z] Trigonal prism [\matrix{(hh\overline{2h}\,\bar{l}) &(\overline{2h}hh\bar{l}) &(h\overline{2h}h\bar{l})\cr}]
3 b m2m [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Trigonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Trigon through origin or [(\bar{1}\bar{1}20)\quad (2\bar{1}\bar{1}0)\quad(\bar{1}2\bar{1}0)]
2 a 3.m [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6 2 m] [0,0,0] Point in origin  
        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 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\hfill\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z\hfill \cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\hfill\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z\hfill \cr \cr\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr x,y,\bar z\quad \bar y,x-y,\bar z \quad \bar x+y,\bar x,\bar z\hfill \cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\hfill\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z\hfill \cr}] [\matrix{{\rm Dihexagonal\ dipyramid}\hfill\cr Edge\hbox{-}truncated\ hexagonal\ prism\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khi\bar{l}) &(hik\bar{l}) &(ikh\bar{l}) \cr(\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr\cr (\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 (hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr (\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l) \cr(khil) &(hikl) &(ikhl)\cr}]
12 f m.. [\matrix{x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0\cr \bar x,\bar y,0\quad y,\bar x+y,0 \quad x-y,x,0 \cr y,x,0 \quad x-y,\bar y,0\quad \bar x,\bar x+y,0 \cr \bar y,\bar x,0\quad \bar x+y,y,0\quad x,x-y,0}] [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0)&(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)}]
12 e .m. [\matrix{x,2x,z\quad 2\bar x,\bar x,z\quad x,\bar x,z\cr \bar x,2\bar x,z \quad 2x,x,z\quad \bar x,x,z \cr 2x,x,\bar z \quad \bar x,2\bar x,\bar z \quad \bar x,x,\bar z\cr 2\bar x,\bar x,\bar z\quad x,2x,\bar z\quad x,\bar x,\bar z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl)&(h\bar{h}0l) &(0h\bar{h}l)\cr (0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l}) \cr (0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})\cr}]
12 d ..m [\matrix{x,0,z\quad 0,x,z\quad \bar x,\bar x,z\cr \bar x,0,z\quad 0,\bar x,z \quad x,x,z \cr 0,x,\bar z\quad x,0,\bar z \quad \bar x,\bar x,\bar z \cr 0,\bar x,\bar z \quad \bar x,0,\bar z\quad x,x,\bar z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr (\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr (hh\overline{2h}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l}) \cr (\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
6 c mm2 [\matrix{x,2x,0 \quad 2\bar x,\bar x,0\quad x,\bar x,0\cr \bar x,2\bar x,0\quad 2x,x,0 \quad \bar x,x,0 \cr}] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b m2m [\matrix{x,0,0\quad 0,x,0 \quad \bar x,\bar x,0 \cr \bar x,0,0\quad 0,\bar x,0\quad x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr (\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
2 a 6mm [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 6mmm [0,0,0] Point in origin  
        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{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill\cr z,x,y \quad z,\bar x,\bar y\quad \bar z,\bar x,y \quad \bar z,x,\bar y \hfill\cr y,z,x\quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad\bar y,\bar z,x\hfill}] [\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)\hfill\cr}] [\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\}] [\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}] [\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}] [\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{ x,0,0 \quad \bar x,0,0\hfill\cr 0,x,0\quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\openup-1pt\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
4 a .3. [x,x,x \quad \bar x,\bar x,x\quad \bar x,x,\bar x\quad x,\bar x,\bar x] [\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\hfill\cr}] [\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]
1 o 23. [0,0,0] Point in origin  
        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{x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z \hfill\cr z,x,y\quad z,\bar x,\bar y \quad \bar z,\bar x,y\quad \bar z,x,\bar y\hfill\cr y,z,x \quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad \bar y,\bar z,x\hfill\cr \noalign{\vskip10pt}\bar x,\bar y,\bar z \quad x,y,\bar z \quad x,\bar y,z\quad \bar x,y,z \hfill\cr \bar z,\bar x,\bar y\quad \bar z,x,y\quad z,x,\bar y\quad z,\bar x,y\hfill\cr \bar y,\bar z,\bar x \quad y,\bar z,x\quad \bar y,z,x \quad y,z,\bar x\hfill}] [\matrix{\hbox{Didodecahedron or diploid}\hfill\cr\hbox{or dyakisdodecahedron}\hfill\cr Cube\ \&\ octahedron\ \&\hfill\cr pentagon\hbox{-}dodecahedron\hfill\cr}] [\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\}] [\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{0,y,z\quad 0,\bar y,z \quad 0,y,\bar z \quad 0,\bar y,\bar z \hfill\cr z,0,y \quad z,0,\bar y \quad \bar z,0,y \quad \bar z,0,\bar y\hfill\cr y,z,0 \quad \bar y,z,0 \quad y,\bar z,0\quad \bar y,\bar z,0\hfill}] [\matrix{\hbox{Pentagon-dodecahedron}\hfill\cr \hbox{or dihexahedron or pyritohedron}\hfill\cr Irregular\ icosahedron\hfill\cr (\!= pentagon\hbox{-}dodecahedron + octahedron)\hfill\cr}] [\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}] [\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{ x,x,x\quad \bar x,\bar x,x \quad \bar x,x,\bar x\quad x,\bar x,\bar x\hfill\cr \bar x,\bar x,\bar x \quad x,x,\bar x \quad x,\bar x,x \quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}\hfill] [\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{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0 \quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr(001) &(00\bar{1})\cr}]
1 o [m\bar3 .] [0,0,0] Point in origin  
        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{x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z \hfill\cr z,x,y\quad z,\bar x,\bar y \quad \bar z,\bar x,y\quad \bar z,x,\bar y\hfill\cr y,z,x\quad \bar y,z,\bar x\quad y,\bar z,\bar x\quad \bar y,\bar z,x\hfill\cr \cr y,x,\bar z \quad \bar y,\bar x,\bar z \quad y,\bar x,z \quad \bar y,x,z\hfill\cr x,z,\bar y \quad \bar x,z,y \quad \bar x,\bar z,\bar y\quad x,\bar z,y\hfill\cr z,y,\bar x \quad z,\bar y,x \quad \bar z,y,x \quad \bar z,\bar y,\bar x\hfill}] [\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)\hfill\cr}] [\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\cr (kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr(hl\bar{k}) &(\bar{h}lk) &(\bar{h}\bar{l}\bar{k}) &(h\bar{l}k)\cr (lk\bar{h}) &(l\bar{k}h) &(\bar{l}kh) &(\bar{l}\bar{k}\bar{h})\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\}] [\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\cr (hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (hl\bar{h}) &(\bar{h}lh) &(\bar{h}\bar{l}\bar{h}) &(h\bar{l}h)\cr (lh\bar{h}) &(l\bar{h}h) &(\bar{l}hh) &(\bar{l}\bar{h}\bar{h})\cr}]
        [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}] [\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\cr (k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (0l\bar{k}) &(0lk) &(0\bar{l}\bar{k}) &(0\bar{l}k)\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)\cr}]
12 c ..2 [\matrix{0,y,y \quad 0,\bar y,y \quad 0,y,\bar y \quad 0,\bar y,\bar y \hfill\cr y,0,y \quad y,0,\bar y \quad \bar y,0,y \quad \bar y,0,\bar y\hfill\cr y,y,0 \quad \bar y,y,0 \quad y,\bar y,0 \quad \bar y,\bar y,0\hfill}] [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\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 .3. [\matrix{x,x,x\quad \bar x,\bar x,x\quad \bar x,x,\bar x \quad x,\bar x,\bar x \hfill\cr x,x,\bar x \quad \bar x,\bar x,\bar x \quad x,\bar x,x \quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}] [\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\cr (11\bar{1}) &(\bar{1}\bar{1}\bar{1})&(1\bar{1}1) &(\bar{1}11)\cr}]
6 a 4.. [\matrix{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0\quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
1 o 432 [0,0,0] Point in origin  
        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{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill \cr z,x,y \quad z,\bar x,\bar y \quad \bar z,\bar x,y \quad \bar z,x,\bar y\hfill \cr y,z,x \quad \bar y,z,\bar x\quad y,\bar z,\bar x\quad \bar y,\bar z,x\hfill \cr \cr y,x,z\quad \bar y,\bar x,z \quad y,\bar x,\bar z \quad \bar y,x,\bar z\hfill\cr x,z,y \quad\bar x,z,\bar y \quad\bar x,\bar z,y\quad x,\bar z, \bar y\hfill \cr z,y,x \quad z,\bar y,\bar x \quad \bar z,y,\bar x\quad \bar z,\bar y,x\hfill}] [\matrix{\hbox{Hexatetrahedron}\hfill\cr \hbox{or hexakistetrahedron}\hfill\cr Cube\ truncated\ by\hfill\cr two\ tetrahedra\hfill\cr}] [\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\cr (khl) &(\bar{k}\bar{h}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr (hlk) &(\bar{h}l\bar{k}) &(\bar{h}\bar{l}k) &(h\bar{l}\bar{k})\cr (lkh) &(l\bar{k}\bar{h}) &(\bar{l}k\bar{h}) &(\bar{l}\bar{k}h)\cr}]
        [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}] [\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\cr (k0l) &(\bar{k}0l) &(k0\bar{l}) &(\bar{k}0\bar{l})\cr (0lk) &(0l\bar{k}) &(0\bar{l}k) &(0\bar{l}\bar{k})\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)}]
12 c ..m [\matrix{x,x,z \quad \bar x,\bar x,z \quad \bar x,x,\bar z \quad x,\bar x,\bar z \hfill\cr z,x,x \quad z,\bar x,\bar x \quad \bar z,\bar x,x \quad \bar z,x,\bar x\hfill\cr x,z,x \quad \bar x,z,\bar x \quad x,\bar z,\bar x \quad \bar x,\bar z,x\hfill}] [\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\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\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})\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}] [\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{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0 \quad 0,\bar x,0 \hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
4 a .3m [x,x,x \quad \bar x,\bar x,x\quad \bar x,x,\bar x\quad x,\bar x,\bar x] [\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\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}]
1 o [\bar 4 3 m] [0,0,0] Point in origin  
        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 1 [\matrix{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill\cr z,x,y \quad z,\bar x,\bar y \quad \bar z,\bar x,y \quad \bar z,x,\bar y\hfill\cr y,z,x \quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad \bar y,\bar z,x\hfill\cr\cr y,x,\bar z \quad \bar y,\bar x,\bar z \quad y,\bar x,z \quad\bar y,x,z \hfill\cr x,z,\bar y \quad \bar x,z,y \quad \bar x,\bar z,\bar y \quad x,\bar z,y\hfill\cr z,y,\bar x \quad z,\bar y,x \quad \bar z,y,x \quad\bar z,\bar y,\bar x\hfill\cr \cr \bar x,\bar y,\bar z \quad x,y,\bar z \quad x,\bar y,z \quad \bar x,y,z\hfill\cr \bar z,\bar x,\bar y \quad \bar z,x,y \quad z,x,\bar y \quad z,\bar x,y\hfill\cr \bar y,\bar z,\bar x \quad y,\bar z,x \quad \bar y,z,x \quad y,z,\bar x\hfill\cr\cr \bar y,\bar x,z \quad y,x,z \quad \bar y,x,\bar z \quad y,\bar x,\bar z\hfill\cr \bar x,\bar z,y \quad x,\bar z,\bar y \quad x,z,y \quad \bar x,z,\bar y\hfill\cr \bar z,\bar y,x \quad \bar z,y,\bar x \quad z,\bar y,\bar x \quad z,y,x\hfill}] [\matrix{\hbox{Hexaoctahedron}\hfill\cr \hbox{or hexakisoctahedron}\hfill\cr Cube\ truncated\ by\hfill\cr octahedron\ and\ by\ rhomb\hbox{-}\hfill\cr dodecahedron\hfill\cr}] [\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\cr (kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr (hl\bar{k}) &(\bar{h}lk) &(\bar{h}\bar{l}\bar{k}) &(h\bar{l}k)\cr (lk\bar{h}) &(l\bar{k}h) &(\bar{l}kh) &(\bar{l}\bar{k}\bar{h})\cr\cr (\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\cr (\bar{k}\bar{h}l) &(khl) &(\bar{k}h\bar{l}) &(k\bar{h}\bar{l})\cr (\bar{h}\bar{l}k) &(h\bar{l}\bar{k}) &(hlk) &(\bar{h}l\bar{k})\cr (\bar{l}\bar{k}h) &(\bar{l}k\bar{h}) &(l\bar{k}\bar{h}) &(lkh)\cr}]
24 e ..m [\matrix{ x,x,z \quad \bar x,\bar x,z \quad \bar x,x,\bar z \quad x,\bar x,\bar z\hfill\cr z,x,x \quad z,\bar x,\bar x \quad \bar z,\bar x,x \quad \bar z,x,\bar x\hfill\cr x,z,x \quad \bar x,z,\bar x \quad x,\bar z,\bar x \quad \bar x,\bar z,x\hfill\cr\cr x,x,\bar z \quad \bar x,\bar x,\bar z \quad x,\bar x,z \quad \bar x,x,z \hfill \cr x,z,\bar x \quad \bar x,z,x \quad \bar x,\bar z,\bar x \quad x,\bar z,x\hfill\cr z,x,\bar x \quad z,\bar x,x\quad \bar z,x,x \quad \bar z,\bar x,\bar x\hfill}] [\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\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\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}) \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\cr (hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (hl\bar{h}) &(\bar{h}lh) &(\bar{h}\bar{l}\bar{h}) &(h\bar{l}h)\cr (lh\bar{h}) &(l\bar{h}h) &(\bar{l}hh) &(\bar{l}\bar{h}\bar{h})\cr}]
24 d m.. [\matrix{ 0,y,z \quad 0,\bar y,z \quad 0,y,\bar z \quad 0,\bar y,\bar z\hfill\cr z,0,y \quad z,0,\bar y \quad \bar z,0,y \quad \bar z,0,\bar y\hfill\cr y,z,0 \quad \bar y,z,0 \quad y,\bar z,0 \quad \bar y,\bar z,0\hfill\cr\cr y,0,\bar z \quad \bar y,0,\bar z \quad y,0,z \quad \bar y,0,z \hfill\cr 0,z,\bar y \quad 0,z,y \quad 0,\bar z,\bar y \quad 0,\bar z,y\hfill\cr z,y,0 \quad z,\bar y,0 \quad \bar z,y,0 \quad \bar z,\bar y,0\hfill}] [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\hfill\cr by\ cube\hfill\cr}] [\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\cr (k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (0l\bar{k}) &(0lk) &(0\bar{l}\bar{k}) &(0\bar{l}k)\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)\cr}]
12 c m.m2 [\matrix{0,y,y \quad 0,\bar y,y \quad 0,y,\bar y \quad 0,\bar y,\bar y\hfill\cr y,0,y \quad y,0,\bar y \quad \bar y,0,y \quad \bar y,0,\bar y\hfill\cr y,y,0 \quad \bar y,y,0 \quad y,\bar y,0 \quad \bar y,\bar y,0\hfill}] [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\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{x,x,x\quad \bar x,\bar x,x \quad \bar x,x,\bar x \quad x,\bar x,\bar x \hfill\cr x,x,\bar x \quad \bar x,\bar x,\bar x \quad x,\bar x,x\quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}] [\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\hfill\cr (11\bar{1}) &(\bar{1}\bar{1}\bar{1}) &(1\bar{1}1) &(\bar{1}11)\hfill\cr}]
6 a 4m.m [\matrix{x,0,0 \quad \bar x,0,0\hfill\cr 0,x,0 \quad 0,\bar x,0 \hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
1 o [m\bar3m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&6mm&&2mm\cr}]  

Table 3.2.3.3| top | pdf |
The two icosahedral point groups

Each point group is specified by its Hermann–Mauguin and Schoenflies symbol. For each point group, the stereographic projections show (on the left) the general position and (on the right) the symmetry elements. The list of the Wyckoff positions includes: Columns 1 to 3: multiplicity, Wyckoff letter, oriented site-symmetry symbol; Under the left stereographic projection: face forms (in roman type) and point forms (in italics), corresponding to the values for the Miller indices and coordinates listed in the last column; only `initial' Miller indices and coordinates are given (see text).

235 I [Scheme scheme58]
60 d 1 Pentagon-hexecontahedron (hkl)
      Snub pentagon-dodecahedron (= pentagon-dodecahedron + icosahedron + pentagon-hexecontahedron) x, y, z
      [\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\ icosahedron\hfill\cr (\hbox{poles between axes 2 and 3})\hfill\cr \cr\hbox{Deltoid\hbox{-}hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \&\ \hfill\cr pentagon\hbox{-}dodecahedron\ \&\ icosahedron\hfill\cr (\hbox{poles between axes 3 and 5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\hfill\cr pentagon\hbox{-}dodecahedron\hfill\cr (\hbox{poles between axes 5 and 2)}\hfill\cr}\right.] [\matrix{(0kl)\hbox{ with } |l| \,\lt\, 0.382 |k|\hfill\cr 0,y,z\ with\ |z| \,\lt\, 0.382 |y|\hfill\cr \cr\cr (0kl)\hbox{ with }0.382 |k| \,\lt\, |l| \,\lt\, 1.618 |k|\hfill\cr 0,y,z\ with\ 0.382 |y| \,\lt\, |z| \,\lt\, 1.618 |y|\hfill\cr\cr\cr\cr (0kl)\hbox{ with }|l| \,\gt\, 1.618 |k|\hfill\cr 0,y,z\ with\ |z| \,\gt\, 1.618 |y|\hfill\cr\cr\cr}]
30 c 2.. Rhomb-triacontahedron (100)
      Icosadodecahedron (= pentagon-dodecahedron [\&] icosahedron) x, 0, 0
20 b .3. Regular icosahedron (111)
      Regular pentagon-dodecahedron x, x, x
12 a ..5 [\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr {Regular\ icosahedron}\hfill\cr}] [\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with } \tau = {\textstyle{1 \over 2}}(\sqrt{5} + 1) = 1.618]
1 o 235 Point in origin 0, 0, 0
      Symmetry of special projections
      [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&3m&&5m\cr}]
[\matrix{m\bar{3}\bar{5}\hfill\cr\cr\displaystyle{2 \over m}\bar{3}\bar{5}\hfill\cr}] [I_{h}] [Scheme scheme59]
120 e l Hecatonicosahedron or hexaicosahedron (hkl)
      Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron x, y, z
60 d m.. [\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\hfill\cr icosahedron\ \hbox{(poles between axes 2 and }\overline{3})\hfill\cr \cr \hbox{Deltoid-hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \& \ pentagon\hbox{-}\hfill\cr dodecahedron\ \& \ icosahedron\hfill\cr \hbox{(poles between axes }\overline{3} \hbox{ and } \overline{5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\ pentagon\hbox{-}\hfill\cr dodecahedron\ \hbox{(poles between axes }\overline{5} \hbox{ and } 2)\hfill\cr}\right.] [\matrix{(0kl) \hbox{ with } |l| \,\lt\, 0.382 |k|\hfill\cr 0,y,z\ with\ |z| \,\lt\, 0.382 |y|\hfill\cr \cr\cr (0kl) \hbox{ with } 0.382 |k| \,\lt\, |l| \,\lt\, 1.618 |k|\hfill\cr 0,y,z\ with\ 0.382 |y| \,\lt\, |z| \,\lt\, 1.618 |y|\hfill\cr \cr\cr\cr (0kl) \hbox{ with } |l| \,\gt\, 1.618 |k|\hfill\cr 0,y,z\ with\ |z| \,\gt\, 1.618 |y|\hfill\cr\cr}]
30 c 2mm.. Rhomb-triacontahedron (100)
      Icosadodecahedron (= pentagon-dodecahedron [\&] icosahedron) x, 0, 0
20 b 3m (m3.) Regular icosahedron (111)
      Regular pentagon-dodecahedron x, x, x
12 a 5m (m.5) [\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr Regular\ icosahedron\hfill\cr}] [\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with }\tau = {1 \over 2}(\sqrt{5} + 1) = 1.618]
1 o [2/m\,\overline{3}\,\overline{5}] Point in origin 0, 0, 0
      Symmetry of special projections
      [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&6mm&&10mm\cr}]

There is a physical difference between the point groups of crystals and those of molecules. For a molecule, the point group is the set of all symmetry operations that map its atoms onto one another. Macroscopic crystals, however, hardly ever exhibit their ideal symmetry because of the defects that occur during crystal growth. For a crystal, its point group is the set of all symmetry operations that map the set of the vectors normal to the crystal faces onto one another; it does not operate in point space, but in vector space.

In Tables 3.2.3.1[link] and 3.2.3.2[link] the point-group types are presented for crystals as well as for molecules. However, parts of the tables concern either crystals only or molecules only. The names of crystal forms in the fifth column (in roman type) and the hkl face indices in the last column are only relevant for crystals. The names of the point forms in the second line of each pair of entries in the fifth column (given in italics) and the other data (multiplicities, Wyckoff letters, site symmetries and sets of symmetry-equivalent coordinates) concern both crystals and molecules. However, for point groups with an origin fixed by symmetry, the Wyckoff position with the Wyckoff letter o is only of interest for molecules.

Because the meanings of the entries are not identical for crystals and for molecules, they are not explained here, but in Sections 3.2.1[link] (for crystals) and 3.2.4[link] (for molecules).








































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