Symmetry
Symmetry
Symmetry
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Lecture 2:<br />
<strong>Symmetry</strong>, Translation-Free <strong>Symmetry</strong> Elements<br />
Back to symmetry…<br />
How can we describe the movements of a motif to create<br />
a pattern?<br />
Example: What is the difference between the<br />
movements of the motif that created the two patterns<br />
below?<br />
Read PERKINS Chpt 9
SYMMETRY ELEMENTS (“Operators”):<br />
•Used to describe the motif’s movement<br />
• Geometric operations that change the position of a motif, but<br />
NOT its size or shape<br />
SYMMETRY OPERATION<br />
•Application of a symmetry element<br />
Note: Some people have lots of trouble visualizing symmetry & understanding crystallography.<br />
The best way to get a grip on this material is to spend some time in the lab and looking at the<br />
CDROM that came with your textbook…
There are 4 Groups of <strong>Symmetry</strong> Elements<br />
(1) Mirror plane (symmetry element “m”) = Reflection (symmetry<br />
operation):<br />
= move an object perpendicular to an imaginary plane and flip it<br />
(change the “handedness”)<br />
m<br />
m=mirror<br />
plane
How many mirror planes are there in the following 2D objects?<br />
Rectangle<br />
2<br />
Equilateral<br />
Triangle<br />
3<br />
Circle<br />
infinity<br />
How many mirror planes are there in the following 3D objects?<br />
Sphere<br />
infinity<br />
Cube<br />
9 TOTAL<br />
3 to faces<br />
6 at diagonals
There are 4 Groups of <strong>Symmetry</strong> Elements<br />
(2) Proper Rotation Axis (symmetry element “n”) = Rotation<br />
(symmetry operation):<br />
= rotate a motif around an imaginary axis, n times, repeating<br />
motif on each partial rotation<br />
.<br />
1-fold:<br />
Always present<br />
2-fold:<br />
Rotate 180°, repeat motif
(2) Proper Rotation Axis (symmetry element “n”) = Rotation<br />
(symmetry operation):<br />
= rotate a motif around an imaginary axis, n times, repeating<br />
motif on each partial rotation<br />
3-Fold<br />
n=3<br />
Rotate 120°, repeat motif<br />
4-Fold<br />
n=4<br />
Rotate 90°, repeat motif
(2) Proper Rotation Axis (symmetry element “n”) = Rotation<br />
(symmetry operation):<br />
= rotate a motif around an imaginary axis, n times, repeating<br />
motif on each partial rotation<br />
5-fold<br />
≠ in nature<br />
>6-fold<br />
≠ in nature<br />
6-Fold<br />
N=6<br />
Rotate 60°, repeat motif
How many rotation axes (and what kind) are there in the<br />
following 2D objects?<br />
Rectangle<br />
n=2<br />
Square<br />
Eq. Triangle<br />
Circle<br />
n=4 n=3 n=infinity<br />
How many rotation axes are there in the following 3D objects?<br />
Sphere<br />
Cube<br />
3 4-fold (through faces)<br />
6 2-fold (through edges)<br />
4 3-fold (through corners)
(3) Inversion Center (symmetry element “i”) = Inversion (symmetry<br />
operation):<br />
= reflect and invert a motif<br />
i<br />
i
Which of the following 2D objects has an inversion center?<br />
Circle Rectangle Equilateral Triangle “Star of David”<br />
YES YES NO YES<br />
Which of the following 3D objects has an inversion center?<br />
Sphere<br />
YES<br />
Cube<br />
YES<br />
Cylinder<br />
YES
(3) Rotoinversion Axis (symmetry element “bar n”) = Rotoinversion<br />
(symmetry operation):<br />
= rotate and invert a motif<br />
bar1 = i<br />
i<br />
bar2 = m<br />
m
(3) Rotoinversion Center (symmetry element “bar n”) = Rotoinversion<br />
(symmetry operation):<br />
= rotate and invert a motif<br />
bar3 = 3-fold + i<br />
i<br />
bar4<br />
i
(3) Rotoinversion Center (symmetry element “bar n”) = Rotoinversion<br />
(symmetry operation):<br />
= rotate and invert a motif<br />
bar6 = 3-fold perpendicular to m<br />
i
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