# Three-Dimensional Symmetry - PowerPoint PPT Presentation

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## Three-Dimensional Symmetry

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### Three-Dimensional Symmetry How can we put dots on a sphere? The Seven Strip Space Groups Simplest Pattern: motifs around a symmetry axis (5) Equivalent to wrapping a ... – PowerPoint PPT presentation

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Title: Three-Dimensional Symmetry

1
Three-Dimensional Symmetry
• How can we put dots on a sphere?

2
The Seven Strip Space Groups
3
Simplest Pattern motifs around a symmetry
axis (5) Equivalent to wrapping a strip around a
cylinder
4
Symmetry axis plus parallel mirror planes (5m)
5
Symmetry axis plus perpendicularmirror plane (5/m)
6
Symmetry axis plus both sets of mirror planes
(5m/m)
7
Symmetry axis plus perpendicular 2-fold axes (52)
8
Symmetry axis plus mirror planes and
perpendicular 2-fold axes (5m2)
9
The three-dimensional version of glide is called
inversion
10
Axial Symmetry
• (1,2,3,4,6 fold symmetry) x 7 types 35
• Only rotation and inversion possible for 1-fold
symmetry (35 - 5 30)
• 3 other possibilities are duplicates
• 27 remaining types

11
Isometric Symmetry
• Cubic unit cells
• Unifying feature is surprising four diagonal
3-fold symmetry axes
• 5 isometric types 27 axial symmetries 32
crystallographic point groups
• Two of the five are very common, one is less
common, two others very rare

12
The Isometric Classes
13
The Isometric Classes
14
Non-Crystallographic Symmetries
• There are an infinite number of axial point
groups 5-fold, 7-fold, 8-fold, etc, with mirror
planes, 2-fold axes, inversion, etc.
• In addition, there are two very special 5-fold
isometric symmetries with and without mirror
planes.
• Clusters of atoms, molecules, viruses, and
biological structures contain these symmetries
• Some crystals approximate these forms but do not
have true 5-fold symmetry, of course.

15
Icosahedral Symmetry
16
Icosahedral Symmetry Without Mirror Planes
17
Why Are Crystals Symmetrical?
• Electrostatic attraction and repulsion are
symmetrical
• Ionic bonding attracts ions equally in all
directions
• Covalent bonding involves orbitals that are
symmetrically oriented because of electrostatic
repulsion

18
Malformed Crystals
19
Why Might Crystals Not Be Symmetrical?
• Competition for ions by other minerals
• Stress
• Anisotropic surroundings

20
Regardless of Crystal Shape, Face Orientations
and Interfacial Angles are Always the Same
21
We Can Project Face Orientation Data to Reveal
the Symmetry
22
Projections in Three Dimensions are Vital for
Revealing and Illustrating Crystal Symmetry
23
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