MATHEMATICS AND ORIGAMI

Mathematics and Origami

• the value of VV which is the dodecahedron ́s diameter; its mid-point O is, obvi-
  ously, the polyhedron ́s center
  VV= D^2 +l^2 =l 2 , 6180342 + 1 = 2 , 8025171 l

Hence the dodecahedron ́s radius is l

VV

1 , 4012586

2

=

• the proof that distance XY is equal to D

XY l A l = l=D











= + = + × 2 , 618034

2

116 , 56505

1 2 1 , 5388418 cos

2

2 cos

ε

In the same figure we can verify that VZ = r – a has the same value as FD in Fig. 1.

18.6.2 RELATIONS 2 (icosahedron)

Let ́s note in Fig. 2 how the icosahedron of Fig. 1 is constructed: it is made up, in first

place, by two domes (the upper and the lower one) same as that in Fig. 2 of Point 18.2.3; one is
rotated with respect to the other an angle of 36º (see Fig. 1, Point 18.6.1). In second place, by a
belt of 10 equilateral triangles: its upper and lower pentagonal bases coincide with the respec-
tive bases of the associated domes.
In Fig. 3 we can get the value of the dihedral angle formed by two adjacent faces of the
icosahedron: it is angle BAC, being BC the diagonal of the dome ́s base pentagon and AB =
AC the altitude of one face of said icosahedron.

AC

BC

AngBAC Arc

2

. = 2 sen

as BC = 618034 1 , l (see Point 18.6.1), and AC =
2

l 3

, the result is:

Ang. BAC = 138,18971º
Either in Figs. 1 or 2 we can see that VV, the icosahedron ́s diameter is the sum of: two
domes ́ altitude (h) plus the belt altitude. Fig. 3 also shows that the dome ́s altitude is the verti-
cal leg of a right triangle whose other leg is the radius of its pentagonal base, and its hypote-
nuse is one side of the icosahedron.

h= l^2 − 0 , 850650812 l^2 = 0 , 5257311
Fig. 4 gives the belt ́s altitude HF which is the great leg of right ∆HFD whose hypote-
nuse HD is the altitude of one face of the icosahedron; the small leg FD has the same length as
FD in Fig. 1, Point 18.6.1.