Symmetry Operations
Lecture 3 of the “Crystallography” course
1. Symmetry Operations
A) identity - corresponds to rotating the body by 360 degrees, putting the object in the
same state it was in originally
B) reflection (mirroring in a plane) - a plane divides the object into 2 parts that are exact
copies of each other, every point on one side is transferred to the same distance on the
other side
C) inversion (mirroring through a point) - relative to the center of the symmetry, every
point passes through it and is projected at the same distance in the opposite direction
D) rotation (spinning) - rotation around an axis, whereby the object coincides with itself
several times within a single full circle (360 degrees)
2. Symmetry Elements
A) Center of symmetry - performs inversion
B) mirror plane of symmetry - performs mirror reflection
В) axis of symmetry - performs rotation
The center of symmetry and the mirror plane are the only elements capable of creating
enantiomorphic objects - mirror images with reversed signs (directions).
Axes of symmetry determine how the image repeats during rotation. Their order (n)
indicates how many times the figure coincides with itself during a 360-degree rotation, it
is calculated as n= 360:𝛂, where 𝛂 is the angle of repetitions. The possible axes are:
– monogyre at n = 1, the angle is 360
– digyre at n = 2, the angle is 180
– trigyre at n = 3, the angle is 120
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Lecture 3 of the “Crystallography” course
1. Symmetry Operations
A) identity - corresponds to rotating the body by 360 degrees, putting the object in the
same state it was in originally
B) reflection (mirroring in a plane) - a plane divides the object into 2 parts that are exact
copies of each other, every point on one side is transferred to the same distance on the
other side
C) inversion (mirroring through a point) - relative to the center of the symmetry, every
point passes through it and is projected at the same distance in the opposite direction
D) rotation (spinning) - rotation around an axis, whereby the object coincides with itself
several times within a single full circle (360 degrees)
2. Symmetry Elements
A) Center of symmetry - performs inversion
B) mirror plane of symmetry - performs mirror reflection
В) axis of symmetry - performs rotation
The center of symmetry and the mirror plane are the only elements capable of creating
enantiomorphic objects - mirror images with reversed signs (directions).
Axes of symmetry determine how the image repeats during rotation. Their order (n)
indicates how many times the figure coincides with itself during a 360-degree rotation, it
is calculated as n= 360:𝛂, where 𝛂 is the angle of repetitions. The possible axes are:
– monogyre at n = 1, the angle is 360
– digyre at n = 2, the angle is 180
– trigyre at n = 3, the angle is 120
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