1. Derivation of the 32 Point Groups
Crystal Structure and Crystallography of Materials
Chapter 8:
Derivation of the 32 Point Groups
(1st Method)

2. Repetition: Derivation of the 32 Crystal Classes:
(basis for the lattice structure)
produced by a combination of the geometrical motions of
a rotation, a translation, and any operation producing
enantimorphous figure.
→ improper rotation
Now consider the particular kind of repetitions than can be produced
if translations are omitted.
→ proper and improper rotations are used.
rotations about axes should intersect. →

3. Derivation of the 32 Crystal Classes:
What is the nature of repetitions which can be produced by rotations about
intersecting axes?

4. Derivation of the 32 Crystal Classes:
• A set of equivalent points is confined to the surface of a sphere
The same is true for planes (geometrical)
→ repeated planes constitutes a polyhedra.
∴ The possible sets of intersecting rotational axis are the possible
symmetries which polyhedra can have.
→ All repetitions not involving translations can be produced by
rotations, proper and improper, all symmetries not involving
translations can be described by rotations, proper and improper.

5. Plane lattices n-fold symmetry 1, 2, 3, 4, and 6
Principles for Discovering the Permissible Crystal Symmetries:
Plane lattices n-fold symmetry 1, 2, 3, 4, and 6
n-fold symmetry 1, 2, 3, 4, and 6
(since all lattices are inherently centrosymmetrical)

6. The 32 Crystal Classes:
• Crystallographically permissible symmetries involving sets of
axes all of which intersect in a common point
→ 32 point groups
→ 32 crystal classes (solid polyhedral body)
if a
• Plane lattice net has n-fold :
symmetry about that axis can be either
n, n, or both n, n simultaneously.

7. 6 permissible combinations :
The 32 Crystal Classes:
Therefore, all the axial symmetries can be derived by
imposing these three alternatives on
a) on each of the 5 possible individual symmetry axes
such as 1, 2, 3, 4, and 6
b) on each of the 6 permissible combinations of intersecting sets
of axes, but within the limitations that these combinations be
consistent.
6 permissible combinations :
222, 322, 422, 622, 332, 432

8. Coincident Rotation and Rotoinversion Axes:
• Lattice can be consistent with an axis n and an axis n simultaneously.
→ n-fold pure rotation axis coincident with an n-fold
rotoinversion axes.
→ written as
Reduction of
n=1, =
n=2, =
n=3, =
n=4, =
n=6, =

9. Derivation of the Crystal Classes:
The monoaxial classes:
single n-fold axis can be either n, n, or

10. Permissible Combinations of Proper and Improper Rotations:
Consider a connected set of rotation axes ABC.
When two of these are proper rotations?
A requires 1R → 2R
B requires 2R → 3R
C requires 1R → 3R
Thus PP = P (P : proper rotations)
When two of these (A and B) be improper rotations.
A requires 1R → 2L
B requires 2L → 3R
Thus II = P (I : improper rotations)
Thus, PPP
IIP Permissible Sets.
IPI III is impossible.
PII

11. The Polyhedral Symmetry Combinations:

12. Derivation of the Crystal Classes:
a) Proper combinations
b) Improper combinations
1) In the event that two of the three are symmetrically equivalent,
as in 332, obviously these must always be the same.
→ if the first 3 is I, so is the second.
2) n22 and n22 are not distinct

13. 3 322 Derivation of the Crystal Classes: e.g. Why not 322 ?
Should be 2. not 2

14. • The two numbers within each pair of parentheses are
The Polyaxial Classes Conforming to both Proper and Improper Combinations:
When n1, n2, n3 is combined with n1, n2, n3
• The two numbers within each pair of parentheses are
coincident axes.
→ the lattice is simultaneously consistent with all these axis.
→ impossible if we choose
( )(n2)(n3)
Unless there is n1

15. The Polyaxial Classes Conforming to both Proper and Improper Combinations:
Thus the only permissible
complete combinations must
have the form

16. Derivation of the 32 Point Groups: