154
6 Interactions
Note especially the fibre modification of the edge term. The maximal local symmetry of an edge is C 2v . The arrow along the edge transforms as b 1 , while the
pseudo-scalar irrep in C 2v is a 2 . The product b 1 × a 2 produces b 2 , which is precisely
the symmetry of an arrow, tangent to the surface of the polyhedron, but directed
perpendicular to the edge. Multiplication with the pseudo-scalar irrep thus has the
effect of rotating the edges through 90 ◦ .InEq.(6.136) the resulting representation
is denoted as Γ ⊥ (e).
Deltahedra Deltahedra are polyhedra that consist entirely of triangular faces.
Three of the Platonic solids are deltahedra: the tetrahedron, the octahedron and
the icosahedron. In a convex deltahedron the bond stretches (i.e. stretchings of the
edges) span precisely the representation of the internal vibrations. In other words,
a convex deltahedron cannot vibrate if it is made of rigid rods. This is the Cauchy
theorem:
Theorem 17 Convex polyhedra in three dimensions with congruent corresponding
faces must be congruent to each other. In consequence, if a polyhedron is made up
of triangles with rigid rods, the angles between the triangular faces are fixed.
This result can be cast in the language of induced representations. The stretchings
of the edges correspond to scalar changes of edge lengths and transform as σ -type
objects, and hence will correspond to Γ σ (e). On the other hand, the internal vibrations span the mechanical representation, which can be written as a bundle of the
translation, minus the spurious modes of translation and rotation. The symmetries
of these will be denoted as Γ T and Γ R , respectively. One thus has:
Deltahedron: Γ σ (v) × Γ T − Γ T − Γ R = Γ σ (e)
(6.138)
Trivalent Polyhedra The dual of a deltahedron is a trivalent polyhedron, meaning
that every vertex is connected to three nearest neighbours. The fullerene networks of
carbon are usually trivalent polyhedra. This reflects the sp 2 hybridization of carbon,
which can form three σ -bonds. Also in this case several specialized forms of the
Euler symmetry theorem can be formulated. We may start from Eq. (6.138) and
replace vertices by faces. The edge terms remain the same since they are totally
symmetric under the local symmetries of the edges. Rotations of the edges by 90 ◦
will thus not affect these terms.
Trivalent: Γ σ (f ) × Γ T − Γ T − Γ R = Γ σ (e)
(6.139)
Furthermore, by multiplying the vertices in a trivalent polyhedron by three, we have
accounted for all the edges twice, since each edge is linked to two vertices, hence:
Trivalent: 3v = 2e
(6.140)
The 3v in this formula suggests once again taking the fibre representation
Γ σ (v) × Γ T . In doing so we have considered on each vertex one σ and two π
6 Interactions
Note especially the fibre modification of the edge term. The maximal local symmetry of an edge is C 2v . The arrow along the edge transforms as b 1 , while the
pseudo-scalar irrep in C 2v is a 2 . The product b 1 × a 2 produces b 2 , which is precisely
the symmetry of an arrow, tangent to the surface of the polyhedron, but directed
perpendicular to the edge. Multiplication with the pseudo-scalar irrep thus has the
effect of rotating the edges through 90 ◦ .InEq.(6.136) the resulting representation
is denoted as Γ ⊥ (e).
Deltahedra Deltahedra are polyhedra that consist entirely of triangular faces.
Three of the Platonic solids are deltahedra: the tetrahedron, the octahedron and
the icosahedron. In a convex deltahedron the bond stretches (i.e. stretchings of the
edges) span precisely the representation of the internal vibrations. In other words,
a convex deltahedron cannot vibrate if it is made of rigid rods. This is the Cauchy
theorem:
Theorem 17 Convex polyhedra in three dimensions with congruent corresponding
faces must be congruent to each other. In consequence, if a polyhedron is made up
of triangles with rigid rods, the angles between the triangular faces are fixed.
This result can be cast in the language of induced representations. The stretchings
of the edges correspond to scalar changes of edge lengths and transform as σ -type
objects, and hence will correspond to Γ σ (e). On the other hand, the internal vibrations span the mechanical representation, which can be written as a bundle of the
translation, minus the spurious modes of translation and rotation. The symmetries
of these will be denoted as Γ T and Γ R , respectively. One thus has:
Deltahedron: Γ σ (v) × Γ T − Γ T − Γ R = Γ σ (e)
(6.138)
Trivalent Polyhedra The dual of a deltahedron is a trivalent polyhedron, meaning
that every vertex is connected to three nearest neighbours. The fullerene networks of
carbon are usually trivalent polyhedra. This reflects the sp 2 hybridization of carbon,
which can form three σ -bonds. Also in this case several specialized forms of the
Euler symmetry theorem can be formulated. We may start from Eq. (6.138) and
replace vertices by faces. The edge terms remain the same since they are totally
symmetric under the local symmetries of the edges. Rotations of the edges by 90 ◦
will thus not affect these terms.
Trivalent: Γ σ (f ) × Γ T − Γ T − Γ R = Γ σ (e)
(6.139)
Furthermore, by multiplying the vertices in a trivalent polyhedron by three, we have
accounted for all the edges twice, since each edge is linked to two vertices, hence:
Trivalent: 3v = 2e
(6.140)
The 3v in this formula suggests once again taking the fibre representation
Γ σ (v) × Γ T . In doing so we have considered on each vertex one σ and two π