Three Dimensional Symmetry

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Three Dimensional Symmetry

• Symmetry Elements

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Rotation Axis
Operation Rotation about an axis International
Notation n Schoenflies Notation Cn
4 (C4)
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Mirror Plane
Operation   Reflection across a
plane International notation  m Schoenflies
notation  s
m (s)
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Inversion Center
Operation   Inversion through origin Internationa
l notation  Schoenflies notation  i
(i)
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Rotoinversion Axis
Operation   Rotation about an axis followed by
inversion through origin International notation 
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Roto-Reflection Axis
Operation   Rotation about an axis followed by
reflection through plane perpendicular to the
axis Schoenflies notation  Sn
S3
7
S4
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Improper Rotation Axes
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Symmetry Operations and Matrices
An initial set of co-ordinates x, y, z is
transformed into a new set x, y, z by one of
the symmetry operations
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Symmetry Operations and Matrices
For a rotation axis parallel to the c-axis
Thus for a 2-fold rotor, 2(C2), parallel to c (q
180)
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Inversion Center
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ReciprocalInversion Center
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Mirrors
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Mirror
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ReciprocalMirror
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2-Fold Rotation
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2-Fold Rotation
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4-Fold Rotation
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4-Fold Rotation
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4-Fold Rotation
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Rotation Reciprocals
To reverse the effect of a 180 rotation a second
180 rotation is required.
On the other hand, to reverse the effect of a 90
rotation a 270 rotation is required
In general