Internal Order and Symmetry
GLY 4200
Fall, 2012
1
Symmetry
• The simple symmetry operations not
involving displacement are:
 Rotation
 Reflection
 Inversion
2
Symmetry Elements
• Each symmetry operation has an associated
symmetry element
 Rotation about an axis (A2, A3, A4, or A6 – in
combination we use 2, 3, 4 or 6)
 Reflection across a mirror plane
 Inversion through a point, the center of
symmetry
3
Rotation Around An Axis
• Rotation axes of a
cube
• Note that the labels are
points, not the fold of
the axis
4
Reflection Across a Plane
• The shaded plane is
known as a mirror
plane
5
Inversion Center
• Inversion through a
point, called the center
of symmetry
6
Symmetry Operation
• Any action which, when performed on an
object, leaves the object in a manner
indistinguishable from the original object
• Example – sphere
 Any action performed on a sphere leaves the sphere
in a manner identical to the original
 A sphere thus has the highest possible symmetry
7
Identity Operation
• All groups must have an identity operation
• We choose an A1 rotation as the identity
operation
• A1 involves rotation by 360º/n, where n is
the fold of the axis
• Therefore A1 = 360º/1 = 360º
8
Combinations of Simple Operations
• We may combine our simple symbols in
certain ways
• 2/m means a two-fold rotation with a mirror
plane perpendicular to it
• Similarly 4/m and 6/m
9
Parallel Mirror Planes
• 2mm 2 fold with two parallel mirror planes
• 3m 3 fold with 3 parallel mirror planes
• 4m 4 fold with 2 sets of parallel mirror
planes
• 6mm 6 fold with 2 sets of parallel mirror
planes
10
Special Three Fold Axis
• 3/m 3 fold with a perpendicular mirror
plane
• Equivalent to a 6 fold rotation inversion
11
2/m 2/m 2/m
• May be written 2/mmm
• Three 2-fold axes, mutually perpendicular,
with a mirror plane perpendicular to each
12
4/m 2/m 2/m
• A four fold axis has a mirror plane
perpendicular to it
• There is a two-fold axis, with a ⊥ mirror
plane, ⊥ to the four-fold axis – the A4
duplicate the A2 90º away
• There is a second set of two-fold axes, with
⊥ mirror planes, ⊥ to the four-fold axis –
the A4 duplicate the A2’s 90º away
13
Ditetragonal-dipyramid
• Has 4/m 2/m 2/m
symmetry
14
Derivative Structures
• Stretching or
compressing the
vertical axis
15
Hermann – Mauguin symbols
• The symbols we have been demonstrating
are called Hermann – Mauguin (H-M)
symbols
• There are other systems in use, but the H-M
symbols are used in mineralogy, and are
easy to understand than some of the
competing systems
16
Complex Symmetry Operations
• The operations defined thus far are simple
operations
• Complex operations involve a combination
of two simple operations
• Two possibilities are commonly used
 Roto-inversion
 Roto-reflection
• It is not necessary that either operation exist
separately
17
Roto-Inversion
• This operation involves rotation through a
specified angle around a specified axis,
followed by inversion through the center of
symmetry
• The operations are denoted bar 1, bar 2, bar
3, bar 4, or bar 6
18
Bar 2 Axis
• To what is a twofold roto-inversion
equivalent?
19
Bar 4 Axis
• A combination of an A4
and an inversion center
• Note that neither
operation exists alone
• Lower figure – A1
becomes A1’, which
becomes A2 upon
inversion
20
Hexagonal Scalenohedron
• This was model #11 in the plastic set
• The vertical axis is a barA3, not an A6
• Known as a scalenohedron because
each face is a scalene triangle
• The red axes are A2
• There are mp’s  to the A2 axes
• The H-M symbol is bar3 2/m
21
Roto-Inversion Symbols
• The symbols shown are used to represent
roto-inversion axes in diagrams
22
Roto-Reflection
• A three-fold roto-reflection
• Starting with the arrow #1
pointing up, the first
operation of the
rotoreflection axis generates
arrow #2 pointing down
• The sixth successive
operation returns the object
23
to its initial position