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. 742-744

Table 3.2.3.1 

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

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