Ad Quadratum Construction and Study of the Regular Polyhedra

59
and therefore through the vertices of the icosahedron, dual of the original dodecahedron).
What is then seen is a five-pointed star in true size, as shown in figure 42.
Note that each of the pyramids is made up of five isosceles triangles with base a, side of
the pentagon.
We now show that these triangles are golden triangles of type 1. Since the angle at their
base, is
with angle , angle of the pentagon, we can write
3
5 ,
3 2
5 5
and therefore 2 ,
or
5
The pyramids having 5 faces with an apex angle of
, their development will be
5
inscribed in a semi-circle of radius Re , edge of the pyramid.
This radius Re is easily calculated (fig. 43). We can write:
But
where is the golden ratio.
And we finally have
sin
10
a
2
Re But sin 5 1
10 4
a
2R e
or a
R e
5 1
4
5 1
2
5 1
2
1
,