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
perpendicular
mirror 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 threedimensional
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 1fold 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