Symmetry

Symmetry plays a very important role in nature and there is certainly no exception at the atomic scale. Atoms themselves are highly symmetric (spheres) and most of the molecules and all crystalline materials have some elements of symmetry - centers (inversion), axes (rotation), planes (reflection), translation and a general combination of them all. For example, a spiral; a fascinating feature of the DNA molecule, is a combination of rotation around and translation along the symmetry axis.

Each molecule is assigned to a certain point group: a set of symmetry elements which can be applied to this molecule. Completely asymmetric molecules belong to the group C₁, which has only one symmetry element: identity (E). Obviously, all other molecules have identity as well and thus all point groups have element E present.

Here are some examples of well known small molecules which have symmetry:

Water (C2v )	Benzene (D6h )	Methane (Td )	Hydrogen (D∞h )

The water molecule (H₂O) has four symmetry elements: identity (E), 2^nd order axis of symmetry (C₂) and two symmetry planes (?_v). Higher symmetry molecules, such as benzene (C₆H₆) and methane (CH₄), have more elements of symmetry. Methane has five and benzene has twelve different elements of symmetry, while the total number of symmetry elements: twenty four - is equal for both molecules. Hydrogen, a linear molecule with two equivalent atoms, has an infinite number of symmetry elements such as rotations (C₂ and C_∞), rotation-reflections (S_∞) and planes (C_v).

Carbon, the major element in organic molecules, can form linear polymers (such as hydrocarbons (C_nH_m), two-dimensional sheets (graphite) and three-dimensional crystals (diamond). Fullerenes and nanotubes, which are very popular in nanotechnology research, are also made from carbon as pictured below:

Crystals, macroscopic periodic systems, made from atoms and molecules have fascinating symmetry, which in combination with color is a working horse of the big industry sector - jewelry.

There are fourteen 3D crystal lattices with different symmetries (Bravais lattices): which are characterized by the relations between unit cell parameters and the presence of the additional lattice point in the center of the unit cell or on its faces. See below:

There are seven crystal systems which have one or more different types of unite cell (P, I,C, or F):

There are 230 space groups which are made of 32 crystallographic point groups (crystal classes) compatible with translation symmetry.

More information on molecular and crystal symmetry is available at the following web sites:

• Molecular Symmetry Lesson
• Molecular Symmetry Examples
• Molecular Symmetry: An Alternative Guide
• Point Group Symmetry
• Introduction to Crystallography
• Crystal Packing
• Mineralogy Data Base
• Crystal Lattice: Structures
• Bilbao Crystallographic server
• Cambridge Crystallographic Data Center
• Art of M.C. Escher

Atomic scale images are created with SageMD2 program and DNA image is created by Anastassia Alexandrova.