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Three-Dimensional Symmetry

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 strip around a cylinder. Symmetry axis plus parallel mirror planes (5m). Symmetry axis plus perpendicularmirror plane (5/m).

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Three-Dimensional Symmetry

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  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

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