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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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Date added: 16 September 2019
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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?
  • Chemical gradient
  • Temperature gradient
  • 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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24
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