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

International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 704-707

Table 3.1.2.3 

H. Burzlaffa and H. Zimmermannb

Table 3.1.2.3| top | pdf |
Delaunay types of lattices (`Symmetrische Sorten')

Delaunay–Voronoi typeMetric conditionsSelling tetrahedronProjections along symmetry directionsDirichlet domain in the unit cellTransformation to the conventional cell
K1 V1 [Scheme scheme1] [Scheme scheme2] [Scheme scheme3] [Scheme scheme4] [Scheme scheme5] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
cI
[{4\over m}\overline 3 {2\over m}]
v s s2
K2 V3 [Scheme scheme6] [Scheme scheme7] [Scheme scheme8] [Scheme scheme9] [Scheme scheme10] [\pmatrix{1&-1&1\cr 1&1&1\cr0&0&2}]
cF
[{4 \over m}\overline 3{2\over m}]
v4 v3 v
K3 V5 [Scheme scheme11] [Scheme scheme12] [Scheme scheme13] [Scheme scheme14] [Scheme scheme15] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]
cP
[{4 \over m}\overline 3{2\over m}]
v v3 v2
  [Scheme scheme16] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
H V4 [Scheme scheme17] [Scheme scheme18] [Scheme scheme19] [Scheme scheme20] [Scheme scheme21] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
hP
[{6\over m}{2\over m}{2 \over m}]
s v v2
R1 V1 [2c^2\,\lt\,3a^2] [Scheme scheme22] [Scheme scheme23] [Scheme scheme24] [Scheme scheme25] [\pmatrix{1&0&1\cr-1&1&1\cr0&-1&1}]
hR
[\overline 3 {2\over m}]
s s2
R2 V3 [2c^2\,\gt\,3a^2] [Scheme scheme26] [Scheme scheme27] [Scheme scheme28] [Scheme scheme29] [\pmatrix{1&0&1\cr0&0&3\cr0&1&2}]
hR
[\overline 3 {2\over m}]
v3 v
Q1 V1 [c^2\,\lt\,2a^2] [Scheme scheme30] [Scheme scheme31] [Scheme scheme32] [Scheme scheme33] [Scheme scheme34] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
tI
[{4\over m}{2\over m}{2\over m}]
v v s2
Q2 V2 [c^2\,\gt\,2a^2] [Scheme scheme35] [Scheme scheme36] [Scheme scheme37] [Scheme scheme38] [Scheme scheme39] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
tI
[{4\over m}{2\over m}{2\over m}]
v4 s s2
Q3 V5 [Scheme scheme40] [Scheme scheme41] [Scheme scheme42] [Scheme scheme43] [Scheme scheme44] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
tP
[{4\over m}{2\over m}{2\over m}]
v v v2
  [Scheme scheme45] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]
  [Scheme scheme46] [\pmatrix{0&0&1\cr1&1&0\cr0&1&0}]
O1 V1 [Scheme scheme47] [Scheme scheme48] [Scheme scheme49] [Scheme scheme50] [Scheme scheme51] [\pmatrix{1&-1&1\cr1&1&1\cr0&0&2}]
oF
[{2\over m}{2\over m}{2\over m}]
s2 v s2
O2 V1 [a^2+b^2\,\gt\,c^2] [Scheme scheme52] [Scheme scheme53] [Scheme scheme54] [Scheme scheme55] [Scheme scheme56] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
oI
[{2\over m}{2\over m}{2\over m}]
v v v
O3 V2 [a^2+b^2\,\lt\,c^2] [Scheme scheme57] [Scheme scheme58] [Scheme scheme59] [Scheme scheme60] [Scheme scheme61] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
oI
[{2\over m}{2\over m}{2\over m}]
s s v4
O4 V3 [a^2+b^2=c^2] [Scheme scheme62] [Scheme scheme63] [Scheme scheme64] [Scheme scheme65] [Scheme scheme66] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
oI
[{2\over m}{2\over m}{2\over m}]
v v v4
  [Scheme scheme67] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
O5 V4 [Scheme scheme68] [Scheme scheme69] [Scheme scheme70] [Scheme scheme71] [Scheme scheme72] [\pmatrix{2&0&0\cr1&1&0\cr0&0&1}]
o(AB)C
[{2\over m}{2\over m}{2\over m}]
s v2 v
  [Scheme scheme73] [\pmatrix{1&1&0\cr-1&1&0\cr0&0&1}]
O6 V5 [Scheme scheme74] [Scheme scheme75] [Scheme scheme76] [Scheme scheme77] [Scheme scheme78] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
oP
[{2\over m}{2\over m}{2\over m}]
v v v
  [Scheme scheme79] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]

Delaunay–Voronoi typeMetric conditionsSelling tetrahedronProjections along symmetry directionsDirichlet domain in the unit cellTransformation to the conventional cell
M1 V1 [b^2\,\gt\,p^2] [Scheme scheme80] [Scheme scheme81] [Scheme scheme82] [Scheme scheme83] [Scheme scheme84] [\pmatrix{-1&1&0\cr-1&-1&0\cr-1&0&1}]
m(AC)I
[{2\over m}]
s2
  [A\colon\ b^2\,\gt\,c^2] [C\colon\ b^2\,\gt\,a^2] [I\colon\ b^2\,\gt\,f^2]
M2 V1 [p^2\,\gt\,b^2], [b^2\,\gt\,r^2-q^2] [Scheme scheme85] [Scheme scheme86] [Scheme scheme87] [Scheme scheme88] [Scheme scheme89] [\pmatrix{0&1&-1\cr1&1&0\cr1&0&-1}]
m(AC)I
[{2\over m}]
v
  [A\colon c^2\,\gt\,b^2\,\gt\,f^2-a^2] [C\colon a^2\,\gt\,b^2\,\gt\,f^2-c^2] [I\colon f^2\,\gt\,b^2\,\gt\,c^2-a^2]
M3 V2 [r^2-q^2\,\gt\,b^2] [Scheme scheme90] [Scheme scheme91] [Scheme scheme92] [Scheme scheme93] [Scheme scheme94] [\pmatrix{-1&0&1\cr-1&1&0\cr-2&0&0}]
m(AC)I
[{2\over m}]
s
  [A\colon\ f^2-a^2\,\gt\,b^2] [C\colon\ f^2-c^2\,\gt\,b^2] [I\colon\ c^2-a^2\,\gt\,b^2]
M4 V4 [b^2=p^2] [Scheme scheme95] [Scheme scheme96] [Scheme scheme97] [Scheme scheme98] [Scheme scheme99] [\pmatrix{0&1&-1\cr1&1&0\cr1&0&-1}]
m(AC)I
[{2\over m}]
s2
  [Scheme scheme100] [A\colon\ b^2=c^2] [C\colon\ b^2=a^2] [I\colon\ b^2=f^2] [\pmatrix{-1&1&0\cr-1&-1&0\cr-1&0&1}]
       
M5 V3 [b^2=r^2-q^2] [Scheme scheme101] [Scheme scheme102] [Scheme scheme103] [Scheme scheme104] [Scheme scheme105] [\pmatrix{-1&0&1\cr-1&1&0\cr-2&0&0}]
m(AC)I
[{2\over m}]
v
  [Scheme scheme106] [A\colon\ b^2=f^2-a^2] [C\colon\ b^2=f^2-c^2] [I\colon\ b^2=c^2-a^2] [\pmatrix{1&0&-1\cr1&-1&0\cr0&-1&-1}]
       
M6 V4 [Scheme scheme107] [Scheme scheme108] [Scheme scheme109] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
mP
[{2\over m}]
s
T1 V1 [Scheme scheme110]   [Scheme scheme111] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
aP
1
T2 V2 [{\bf a}\cdot{\bf b}=0] [Scheme scheme112]   [Scheme scheme113] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
aP
1
T3 V3 [{\bf a}\cdot{\bf b}=0] [Scheme scheme114]   [Scheme scheme115] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
aP [({\bf a}+{\bf b}+{\bf c})\cdot{\bf c}]
1 = 0