Ad Quadratum Construction and Study of the Regular Polyhedra

We have:
Noting that cos Di
2
65
R s OB and OB OL 2 BL 2
a 1
Now OL
2
a2
2
and BL a1 3 a
2 2
OB a
2 2 1 2
with a 3
3
5 1R
3 2
3 R
OB 3
3 1 2 R
R s
R
3 and sin I i
2
3
2R 3
2
3
(or R s
R 1.776)
1
2 , we can write Rs R
cos Di 2
sin Ii 2
, which we can
define as a growth factor gd1 . If instead, we consider the ratio of Rs to the insphere
radius as will be established for the GSD, we can define a new growth factor :
g d R R cos
s r h
r r
Di 2 cos Ii 2 a cos Ii 2
with cos D i
2
3
b. The Great dodecahedron: (GD)
a 3
and
R 3
Rcos D i
2 cos I i
2
1
a
R cos D i
2
5 1
it comes g d Rs 5 .
r
The same basic geometry will obtain for the GD as for the SSD since the GD can be
considered as made up on the basis of the SSD by extending the planes of the stellated
faces so as to fill the gaps between the star branches, thus forming a convex pentagram
(fig. 46A and 46B). Alternatively, it can be viewed as made up of twelve intersecting
pentagonal planes and the figure of the enveloping icosahedron clearly appears on fig.
47A and 47B.
g d