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 ( = 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