MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Fig. 4 will allow us to find the dihedral angle ε formed by two adjacent faces of a do-

decahedron, as well as its diameter. At left we have two pentagons like those in Fig. 3 and, as-

sociated with them, segments l, d, D and A. The value of the latter is:

A = a + r = 0,6881909 l + 0,8506508 l = 1,5388417 l

The figure at right is a hemispherical section of a dodecahedron; in it, D is the diagonal

of the great pentagon at left (see Fig. 3, Point 18.6.3). To draw that section we shall start by

∆YVW whose three sides are given. In it we get ε:

A

D

2 2

sen =

ε

; ε = 116,56505 º

The figure at right is symmetric with respect to XY; points V in it are vertices of the do-

decahedron. After all that we can deduce:

• the angle γ in the irregular hexagon with sides A, l:

180 (6 – 2 ) = 2 ε + 4 γ ; 121 , 71748

4

720 2 116 , 56505

=

− ×

γ =

• the proof that angle VWV is a right one:

Ang. VWV = 90

2

180 116 , 56505

121 , 71748

2

180

=

−

= −

−

−

ε

γ

A

l

B d C

D

d

d

4

A

D A

V

V

l

l

A

A

W

X

Z Y

O

2

d

2

l

d

3

D

A l B d C