Groups - Group Theory Applied to Chemistry - page 35

Fig. 3.5 Genealogical tree
for the cubic and icosahedral
point groups

heavens. The dodecahedron and its intriguing symmetry were considered worthy of
the stars, with their perfect cosmic order. Besides the three point groups, T d ,O h ,
and I h , represented by the Platonic solids, there are four more groups that belong
to the isotropic family: T h ,T,O , and I . They are subgroups of the octahedral and
icosahedral groups, as shown in Fig. 3.5 .

The Tetrahedron

T d symmetry (Fig. 3.6 (a)) plays a crucial role in chemistry. It is the symmetry of
the valence structure of aliphatic carbon. The attribution of this symmetry to carbon
by Van't Hoff 4 as early as 1874, i.e., well before the modern concept of molecular
structure and some two and a half millennia after Pythagoras, remains a tribute to
the Greek vision that the fundamental structure of matter consists of ideal symmetric
shapes. The tetrahedron is the most fundamental of the solids since it is the simplex
of 3D space. A simplex is a figure consisting of vertices that are all equivalent and
are equidistant from each other. 5

In an n -dimensional space the simplex contains

n 1 vertices, e.g., in 2D Euclidean space exactly three points can be distributed in

such a way that they are equidistant, viz., by occupying the vertices of an equilateral
triangle. In 3D space only four points can be distributed in such a way that they have
this property, the solution being the tetrahedron. The perfect permutational symme-
try of the vertices of the n -simplex implies that the corresponding symmetry group
is isomorphic to the symmetric group S n . We have already seen that the triangular
symmetry of the hydrogens in ammonia could be described by S 3 ;inthesameway
the tetrahedral symmetry group is isomorphic to S 4 . The rotational subgroup of T d
is the group T , which, in turn, is isomorphic to the alternating group A 4 .Anex-
traordinary member of the tetrahedral family is the group T h , which has the same

+

4 Van't Hoff published his findings in 1874 in Utrecht. In the same year, Le Bel came to the same
conclusion, based on the investigation of optical rotatory power. An English translation of the
original papers of both chemists can be found in: [ 1 ].
5 In graph theory the graph of a simplex with n vertices is the complete n -graph, K n .Insuch
a graph, each of the n vertices is connected to all the other (n − 1 ) vertices. There is only one
simplex for each dimension.