three-dimensional symmetry how can we put dots on a sphere?
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 (35 - 5 = 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
