Chemistry Reference
In-Depth Information
Ta b l e 3 . 5
The Platonic
solids and their point groups.
The numbers count the
triangles, squares, and
pentagons, that intersect in a
vertex of the solid
Triangle
Square
Pentagon
3
Tetrahedron (
T
d
)
Cube (
O
h
)
Dodecahedron (
I
h
)
4
Octahedron (
O
h
)
5
Icosahedron (
I
h
)
as
O(
3
)
, which refers to the orthogonal group in three dimensions. This assignment
is based on the one-to-one correspondence between the symmetry operations of the
sphere and the set of 3
3 orthogonal matrices. This will be explained in more detail
in Sect.
7.1
. Only isolated atoms exhibit spherical symmetry. Molecular shapes that
approximate perfect spherical symmetry are based on the
regular polyhedra
,the
building blocks of which are
regular polygons
. These polygons are obtained by
distributing
n
points around a circle in such a way that all points are equivalent, i.e.,
that the distances between any two neighboring points are the same, implying that
the angles subtended by adjacent edges are also the same. In this way the circle may
circumscribe an equilateral triangle, a square, a regular pentagon, etc.
In fact, for any integer
n
with 3
×
≤
≤∞
, a regular
n
-gon can be obtained, though
not all of them can be drawn by use of a ruler and compass. In 3D things are quite
different. Defining regular polyhedra by distributing
n
points over a sphere in such a
way that all vertices, edges, and faces of the resulting structures are the same cannot
be realized in infinitely many ways: quite on the contrary, only five solutions are pos-
sible.
3
These are known as the Platonic solids, and they are listed in Table
3.5
.They
played an important role in Pythagorean tradition, as well as in Eastern philosophy
and religion. The fact that in 3D only five solutions exist was considered to reveal
a fundamental truth about nature, to the extent that Plato, in his
Timaeus
, based his
natural philosophy on these solids. The ancient doctrine of the four elements was
placed in correspondence with four of the solids. The tetrahedron (or 4-plane), with
symmetry
T
d
, has the most acute angles and was associated with fire. The cube, or
hexahedron (6-plane), with symmetry
O
h
, is clearly the most stable structure and
refers to the earth. The icosahedron (20-plane), with symmetry
I
h
, contains twenty
faces and is therefore closest to a globular surface, which symbolizes the most fluid
element, water. Both cube and icosahedron have a dual partner, which is obtained
by replacing vertices by faces and vice versa. The dual of the cube is the octahedron
(8-plane), which figured for air. In Table
3.5
, the octahedron is placed between the
tetrahedron and the icosahedron, and this seemed appropriate for air because it is
intermediate between fire and water in its mobility, sharpness, and ability to pen-
etrate. The dual of the icosahedron is the regular dodecahedron (12-plane). There
being only four elements, the discovery of the fifth solid caused some embarrass-
ment, which found an elegant solution by its being assigned to the substance of the
n
3
The sum of the angles subtended at a vertex of a Platonic solid must be smaller than a full angle
of 2
π
. Hence, no more than five triangles, three squares, or three pentagons can meet in a vertex;
regular hexagons are already excluded since the junction of three such hexagons already gives rise
to an angle of 2
π
at the shared vertex.
