MATHEMATICS AND ORIGAMI

Mathematics and Origami

18.6 REGULAR POLYHEDRA

Before digging out about them, we shall analyse certain RELATIONS given within the

pentagon, the pentagon-dodecahedron and its conjugate, the icosahedron, to have them at hand

whenever necessary.

18.6.1 RELATIONS 1 (dodecahedron)

Fig. 1 shows two congruent regular pentagons rotated 36º one with respect of the other.

2

cos

α

r a

AC

−

= ;

4

2 tg

β

AB= a

In triangle GIH we have:

cosφ

2

l

l

GI

−

=

2 cosφ

l

= being

2

2

2

2 sen

tg

l

l

l

l−

=

α

φ

2

4 sen 54

=

Summarising:

l = l ; a = 0,6881909 l ; r = 0,8506508 l ; AC = 0,2763932 l ; AB = 0,4472135 l

GI = 1,248606 l ; FD = r – a = 0,1624599 l

In Fig. 2 we can see the relation between side and diagonal of a pentagon:

1 , 618034

2

= 2 sen =

α

d l l

Fig. 3 shows the same pentagon of Fig. 2 associated with another one in which the side

is the former ́s diagonal. Being similar both pentagons, we ́ll have:

d

D

l

d

= ; = = =

2

4 sen^2

(^2) α
l
l
d
D 2.618034 l
Let ́s figure out the value of some singular
segments as a function of the pentagon side l.
l = side of pentagon
r = radius ,, ,,
a = apothem ,,
α = 108 (see Point 11.1) ; 72
5
360
β= =
2
2
2
4
a
l
r = +
2
sen
α
a=r
hence:
2
2
tg
α
l
a= ;
2
2 cos
α
l
r=
1
D
F A C
B
E G
H
I
O
2
4