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The Fascination of Crystals and Symm

Unit 3.1

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Entering the World of Symmetry    E   n  t   e  r  i   n   g  t   h   e     W   o  r  l   d   o  f  S   y    m    m   e  t  r   y

external symmetry of macroscopic objects

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Systematization of Crystal Structures

1. Step: Crystal systemsmetric + symmetry of the UC

2. Step: Bravais latticesprimitive + centered 1

3. Step: Crystal classescrystallographic PG

3

4. Step: Space groups

complete symmetry 23

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Symmetry operation and Symmetry element

is a (geometrical) reorganization/transformation, which maps an object onto itself 

by this operation a congruent image of the motif is generated, i.e. an image which

indistinguishable of the starting point

Symmetry operation (SO)

a = b = c

a = b = g = 90°

90°

a = b = c

a = b = g = 90°

indistinguishable

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Symmetry operation and Symmetry element

a = b = c

a = b = g = 90°

90°

a = b = c

a = b = g = 90°

indistinguishable

axis of rotation 360 °/ 90 ° = 4 4-fold axis of rotation

Symmetry element (SE)

is the geometrical object (point, line, plane) on which the SO is carried out

it comprises at least all invariant spatial points (fixed points) of the operation

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Symmetry operation and Symmetry element

a = b = c

a = b = g = 90°

180°

a = b = c

a = b = g = 90°

indistinguishable

axis of rotation 360 °/ 180 ° = 2 2-fold axis of rotation

Symmetry element (SE)

is the geometrical object (point, line, plane) on which the SO is carried out

it comprises at least all invariant spatial points (fixed points) of the operation

usually on one SE several different SO can be carried out

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Symmetry concerning macroscopic object – external symmetry

Symmetry elements of macroscopic objects

1. Identity

2. Mirror plane

3. Axis of rotation

4. Center of inversion

5. Rotoinversion axis

identity

1-fold axis of ro

(rotation by 3

symbol

even the most asy

objects have at lea

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Mirror symmetry

mirror plane

also called ‘line symmetry’ or ‘reflection symmetry’ or ‘bilateral symmetry’

an object which does not change upon undergoing a reflection has mirror symmetry,

it is mirror symmetric

In 2D there is a line of symmetry or mirror line, in 3D a plane of symmetry or mirror plane

symbol m

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Excursus: Mirror symmetry and Beauty

original right side mirrored left side mirro

Kelly George

Miss Arkansas 2007

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Excursus: Mirror symmetry and Beauty

1995

Prof. Dr. rer. nat. Karl Grammer

University of Vienna

http://evolution.anthro.univie.ac.at/institutes/

urbanethology/staff/grammer.html

Academia

ResearchGate

Google Scholars

2007

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Picture Credits

If not otherwise stated pictures, images, sketches, clip arts are

• self-taken/self-drawn or

• public domain

© alephcomo1 - Fotolia.com CC-BY-SA 3.0 de | US Air Force | Friedrich Graf  with kind perm

Karl Grammer