# Three-Dimensional Symmetry

## Presentation on theme: "Three-Dimensional Symmetry"— Presentation transcript:

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)

Symmetry axis plus both sets of mirror planes (5m/m)

Symmetry axis plus perpendicular 2-fold axes (52)

Symmetry axis plus mirror planes and perpendicular 2-fold axes (5m2)

The three-dimensional version of glide is called inversion

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

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

The Isometric Classes

The Isometric Classes

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.

Icosahedral Symmetry

Icosahedral Symmetry Without Mirror Planes

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

Malformed Crystals

Why Might Crystals Not Be Symmetrical?
Chemical gradient Temperature gradient Competition for ions by other minerals Stress Anisotropic surroundings

Regardless of Crystal Shape, Face Orientations and Interfacial Angles are Always the Same

We Can Project Face Orientation Data to Reveal the Symmetry

Projections in Three Dimensions are Vital for Revealing and Illustrating Crystal Symmetry