32 Crystallographic Point Groups

Point Groups The 32 crystallographic point groups (point groups consistent with translational symmetry) can be constructed in one of two ways: 1.From 11 initial pure rotational point groups, inversion centers can be added to produce an additional 11 centrosymmetric point groups. From the centrosymmetric point groups an additional 10 symmetries can be discovered. 2.The Schoenflies approach is to start with the 5 cyclic groups and add or substitute symmetry elements to produce new groups.

Cyclic Point Groups 5

Cyclic + Horizontal Mirror Groups +5 = 10

Cyclic + Vertical Mirror Groups +4 = 14

Rotoreflection Groups +3 = 17

17 of 32? Almost one-half of the 32 promised point groups are missing. Where are they? We have not considered the combination of rotations with other rotations in other directions. For instance can two 2-fold axes intersect at right angles and still obey group laws?

The Missing 15 Combinations of Rotations

Moving Points on a Sphere

 = "throw" of axis i.e. 2-fold has 180° throw Euler Investigate: 180°, 120°, 90°, 60°

Possible Rotor Combinations

Allowed Combinations of Pure Rotations

Rotations + Perpendicular 2-folds Dihedral (D n ) Groups +4 = 21

Dihedral Groups +  h +4 = 25

Dihedral Groups +  d +2 = 27

Isometric Groups Roto-Combination with no Unique Axis

T Groups +3 = 30

T Groups

O Groups +2 = 32

O Groups

Flowchart for Determining Significant Point Group Symmetry