Information Technology Reference
In-Depth Information
2
E
=
kF
(7.29)
In fact,
gV
counts edges twice because each edge is common to two nodes, and also
kF
counts edges twice because each edge is common to two faces.
From Euler's formula
V
−
+
=
=
−
+
E
F
2wehave
F
E
V
2, therefore we have:
2
E
=
k
(
E
−
V
+
2
)
that is:
(
k
−
2
)
E
=
kV
−
2
k
whence:
2
(
k
−
2
)
E
=
2
kV
−
4
k
but from Eq. (7.28) we can replace 2
E
by
gV
, therefore:
(
k
−
2
)
gV
=
2
kV
−
4
k
and finally:
[
2
(
k
+
g
)
−
kg
]
V
=
4
k
.
(7.30)
In order to have a positive value 4
k
, we need to have:
2
k
+
2
g
>
kg
that is:
2
k
g
<
k
−
2
Fig. 7.17
Platonic solids
