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This implies that
k
3, but
in the graphs coming from regular polyhedra
k
and
g
are at least 3, In fact, a face
needs at least three edges (triangle) and a graph with all nodes with degree 2 is a
cycle (only one face).
For
k
<
6. In fact, for
k
≥
6 the above inequality implies
g
<
=
3 Eq. (7.30) gives:
(
6
−
g
)
V
=
12
therefore,
g
can assume only the values 3, 4, and 5, which correspond to the first
three lines of following Table 7.9, where the values of
F
correspond to the number
of regular polygons of a polyhedron.
From Eq. (7.30), for
k
5 the only possible value of
g
is 3, and we
get the last two lines of Table 7.9. We remark that the tetrahedron is self-dual and
that the hexahedron is dual of the octahedron, while the dodecahedron is dual of the
Icosahedron. No other possibility is allowed.
=
4and
k
=
Ta b l e 7 . 9
ThefivePlatonicsolids
g=3
V = 4
F= 4 (3F = 3V=12)
k=3 Tetrahedron
g=4
V = 6
F = 8 (3F = 4V=24)
k=3 Octahedron
g=5
V = 12
F = 20 (3F = 5V=60)
k=3 Icosahedron
g=3
V = 8
F =6 (dual of Octahedron)
k=4 Hexahedron
g=3
V = 20
F = 12 (dual of Icosahedron)
k=5 Dodecahedron
It is interesting to consider the constructions of the two more complex Platonic
polyhedra, the dodecahedron and the icosahedron. For dodecahedron we can start
from a pentagon surrounded by other five pentagons (each of them shares an edge
with the central one). If the peripheral pentagons share two edges with their neigh-
bors, then they form a sort of cup, a
6-pentagon cup
. By joining two 6-pentagon
cups, we get a dodecahedron.
For an icosahedron we consider a 5-triangle cup with a vertex common to all
triangles and a pentagon as basis. If at each edge of this pentagon a triangle is put
(equal to the triangles of the cup), then between two triangles put at consecutive
edges an equal triangle is comprised, sharing an edge with each of them (and the
vertex common to these edges). In this way 15 triangles are connected providing
a pentagonal border. Then, by adding to this border a reversed 5-triangle cup the
icosahedron is formed having 20 triangles. A planar representation of icosahedron
is given in Fig. 7.20. Platonic solids occur in natural forms, for example, in capsids
of virus. The capsid of Adenovirus, Reovirus, Papovavirus, and Picornavirus is an
icosahedron (the regular polyhedron more similar to a sphere, where the 12 vertexes
are equipped with extruding fibers used to bind target cells).
From Eq. (7.30), it follows that no regular solid can be constructed with hexagons.
However, if 12 pentagons are mixed with hexagons, then convex polyhedra can
be constructed. The minimum of such polyhedron, having 20 hexagons and 12
