Ad Quadratum Construction and Study of the Regular Polyhedra

Dodecahedron:
11
For the dodecahedron, we proceed as for the cube calling in this case its internal angle
D ˆ i , the angle between two consecutive diagonals. These diagonals constitute the sides of
a triangle, itself side of a pyramid with apex at the center of the circumsphere and having
for base a regular pentagon, face of the dodecahedron.
Though neither equal nor similar, the isosceles triangle, face of the pyramid just
described, has the same geometry as that for the cube previously examined (fig. 7). We
can therefore write:
ˆ
2
ˆ
2 Di
a
cos Di
1
2sin
1
2
2 2R
Here, however:
and therefore:
or:
a 3
3
5 1R
a 2 1
62 5R
3 2
a 2
5
2 1
2R 3
cos ˆ D i 1 1 5
5
3
3
In triangle ACH we therefore have:
AC R; CH 5
3 R; and AH 2 AC 2 CH 2 ,
or: AH 2 R 2 5
9 R2 R 2 1 5
R
9
24
9
AH
2
R ;
3
where from sin ˆ D i AH 2
AC 3
Icosahedron:
sin ˆ
D i 2
3
Proceeding similarly for the icosahedron, we have:
cos ˆ
2 I ˆ i
I i 12sin
2
1 a 2
2R 2 ,