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SymmetrySymmetry
Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern
Operation: some act that reproduces the motif to create the pattern
Element: an operation located at a particular point in space
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
= 360o/2 rotation
to reproduce a motif in a symmetrical pattern
6
6
A Symmetrical PatternA Symmetrical Pattern
Symmetry Elements
1. Rotation
a. Two-fold rotation
= 360o/2 rotation
to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation
Motif
Element
OperationOperation
6
62-D Symmetry2-D Symmetry
6
6
first operation step
second operation step
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
= 360o/2 rotation
to reproduce a motif in a symmetrical pattern
= the symbol for a two-fold rotation
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar objects have an intrinsic symmetry
180o rotation makes it coincident
What’s the motif here??
Second 180o brings the object back to its original position
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
b. Three-fold rotation
= 360o/3 rotation
to reproduce a motif in a symmetrical pattern
66
6
2-D Symmetry2-D Symmetry
6
66
step 1
step 2
step 3
2-D Symmetry2-D Symmetry
Symmetry Elements
1. Rotation
b. Three-fold rotation
= 360o/3 rotation
to reproduce a motif in a symmetrical pattern
Symmetry Elements
1. Rotation
6
6
6
6
6
6 6
6
6
6
6
6
6
6
6
6
1-fold 2-fold 3-fold 4-fold 6-fold
Z
5-fold and > 6-fold rotations will not work in combination with translations in crystals (as we shall see later). Thus we will exclude them now.
aidentity
Objects with symmetry:
2-D Symmetry2-D Symmetry
4-fold, 2-fold, and 3-fold 4-fold, 2-fold, and 3-fold rotations in a cuberotations in a cube
Click on image to run animation
Symmetry Elements
2. Inversion (i)
inversion through a center to reproduce a motif in a symmetrical pattern
= symbol for an inversion centerinversion is identical to 2-fold rotation in 2-D, but is unique in 3-D (try it with your hands)
6
62-D Symmetry2-D Symmetry
Symmetry Elements
3. Reflection (m)
Reflection across a “mirror plane” reproduces a motif
= symbol for a mirror
plane
2-D Symmetry2-D Symmetry
We now have 6 unique 2-D symmetry operations:
1 2 3 4 6 m
Rotations are congruent operations
reproductions are identical
Inversion and reflection are enantiomorphic operations
reproductions are “opposite-handed”
2-D Symmetry2-D Symmetry
Combinations of symmetry elements are also possible
To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements
In the interest of clarity and ease of illustration, we continue to consider only 2-D examples
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
Is that all??
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required
2-D Symmetry2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
The result is Point Group 2mm
“2mm” indicates 2 mirrors
The mirrors are different
(not equivalent by symmetry)
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 1
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 2
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 3
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Yes, two more mirrors
Any other elements?
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Point group name??
Yes, two more mirrors
Any other elements?
2-D Symmetry2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
4mm
Point group name??
Yes, two more mirrors
Any other elements?
2-D Symmetry2-D Symmetry
Why not 4mmmm?
3-fold rotation axis with a mirror creates point group 3m
Why not 3mmm?
2-D Symmetry2-D Symmetry
6-fold rotation axis with a mirror creates point group 6mm
2-D Symmetry2-D Symmetry
All other combinations are either:
Incompatible
(2 + 2 cannot be done in 2-D)
Redundant with others already tried
m + m 2mm because creates 2-fold
This is the same as 2 + m 2mm
2-D Symmetry2-D Symmetry
The original 6 elements plus the 4 combinations creates 10 possible 2-D Point Groups:
1 2 3 4 6 m 2mm 3m 4mm 6mm
Any 2-D pattern of objects surrounding a point must conform to one of these groups
2-D Symmetry2-D Symmetry
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
3-D Symmetry3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
Step 2: invert
This is the same as i, so not a new operation
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Note: this is a temporary step, the intermediate motif element does not exist in the final pattern
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
The result:
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
This is the same as m, so not a new operation
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Step 1: rotate 360o/3
Again, this is a temporary step, the intermediate motif element does not exist in the final pattern
1
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Step 2: invert through center
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Completion of the first sequence
1
2
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Invert through center
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Complete second step to create face 3
1
2
3
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Third step creates face 4
(3 (1) 4)
1
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5 (4 (2) 5)
1
2
5
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6
(5 (3) 6)
Sixth step returns to face 1
1
6
5
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
This is unique1
6
5
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental representative of the pattern
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Begin with this framework:
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 ) 1
3-D Symmetry3-D Symmetry
1
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
3
4
2
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
1
2
3
4
5
6
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane
(combinations of elements follows)
Top View
3-D Symmetry3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
A simpler pattern
Top View
3-D Symmetry3-D SymmetryWe now have 10 unique 3-D symmetry operations:
1 2 3 4 6 i m 3 4 6
Combinations of these elements are also possible
A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements
3-D Symmetry3-D Symmetry
3-D symmetry element combinations
a. Rotation axis parallel to a mirrorSame as 2-D
2 || m = 2mm
3 || m = 3m, also 4mm, 6mm
b. Rotation axis mirror2 m = 2/m
3 m = 3/m, also 4/m, 6/m
c. Most other rotations + m are impossible2-fold axis at odd angle to mirror?
Some cases at 45o or 30o are possible, as we shall see
3-D Symmetry3-D Symmetry
3-D symmetry element combinations
d. Combinations of rotations
2 + 2 at 90o 222 (third 2 required from combination)
4 + 2 at 90o 422 ( “ “ “ )
6 + 2 at 90o 622 ( “ “ “ )
3-D Symmetry3-D Symmetry
As in 2-D, the number of possible combinations is limited only by incompatibility and redundancy
There are only 22 possible unique 3-D combinations, when combined with the 10 original 3-D elements yields the 32 3-D Point Groups
3-D Symmetry3-D Symmetry
But it soon gets hard to visualize (or at least portray 3-D on paper)
Fig. 5.18 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
3-D Symmetry3-D SymmetryThe 32 3-D Point Groups
Every 3-D pattern must conform to one of them.
This includes every crystal, and every point within a crystal
Rotation axis only 1 2 3 4 6
Rotoinversion axis only 1 (= i ) 2 (= m) 3 4 6 (= 3/m)
Combination of rotation axes 222 32 422 622
One rotation axis mirror 2/m 3/m (= 6) 4/m 6/m
One rotation axis || mirror 2mm 3m 4mm 6mm
Rotoinversion with rotation and mirror 3 2/m 4 2/m 6 2/m
Three rotation axes and mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m
Additional Isometric patterns 23 432 4/m 3 2/m
2/m 3 43m
Increasing Rotational Symmetry
Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
3-D Symmetry3-D SymmetryThe 32 3-D Point Groups
Regrouped by Crystal System (more later when we consider translations)
Crystal System No Center Center
Triclinic 1 1
Monoclinic 2, 2 (= m) 2/m
Orthorhombic 222, 2mm 2/m 2/m 2/m
Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m
Hexagonal 3, 32, 3m 3, 3 2/m
6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m
Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
3-D Symmetry3-D SymmetryThe 32 3-D Point Groups
After Bloss, Crystallography and Crystal Chemistry. © MSA
+c
+a
+b
Axial convention:“right-hand rule”
3-D Symmetry3-D SymmetryCrystal Axes
3-D Symmetry3-D SymmetryCrystal Axes
3-D Symmetry3-D SymmetryCrystal Axes
3-D Symmetry3-D SymmetryCrystal Axes
3-D Symmetry3-D SymmetryCrystal Axes
