Isometric
Hexagonal
Trigonal
Tetragonal
Four 3-fold
or 3 axis
One 6-fold
or 6 axis
One 3-fold
or 3 axis
One 4-fold
or 4 axis
Orthorhombic
Three 2-fold
or 2 axis
Monoclinic
Triclinic
One 2-fold
or 2 axis
One 1-fold or
1 axis
6
3
4
23
622
32
422
432
6 = 3/m
3
4
m=2
4/m
2/m
6/m
2
1
222
1=i
(2nd Setting)
-43m
2m3
Symmetry Element Symbols
6mm
-6m2 = 3 m2
m
4- 2
m3m
6 2 2
mmm
a=b=c
� = � = � = 90o
a=b
� = � = 90o, � = 120o
3m
-3
2
m
4mm
-
42m
4 2 2
mmm
a=b
� = � = 90o, � = 120o
2mm
a=b
� = � = � = 90o
Inversion center
2-fold rotation axis
3-fold rotation axis
4-fold rotation axis
6-fold rotation axis
Mirror plane (= 2 axis)
3-fold rotoinversion axis
4-fold rotoinversion axis
6-fold rotoinversion axis
2 2 2
mmm
� = � = � = 90o
� = � = 90o
Lattice
Constraints
Space Groups
A particular Bravais lattice is compatible with only certain space point groups. In general, the symmetry of
a lattice must be greater than or equal to the point symmetry of the motif it repeats. When the fourteen
Bravais lattices are combined with space point groups having the appropriate symmetry, one arrives at 230
space groups. These space groups are like the plane groups discussed in the previous lecture: they
represent the different ways in which motifs can be arranged in space. These motifs may be abstract
patterns or the atoms in a crystal.
Summary of Symmetry Groups
Symmetry
Group
2D point
2D plane
3D point
3D space
Example
Translation?
Rotation?
Reflection?
Inversion?
Roadkill
Floor tiles
Natural objects
crystals
no
yes
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
n.a.
yes
yes
yes
Number in
Group
10
17
32
230