Ad Quadratum Construction and Study of the Regular Polyhedra

Tetrahedron:
For the tetrahedron, as is well-known:
a 2
3
25
6R and r a
12
so that r 2 6 1
R 3 12 3 ;
and we note that 1
cos C ˆ i
3 r
cos C ˆ i .
R
Note also that when R = 3 (as in our Ad Quadratum construction), r = 1;
While the height of the tetrahedron given by
h a
3
6 2
3
6R 6
3
6 ,
2 6 4
R
9 3 R
So that for R = 3, h = 4.
Having started with 1, 2, and 3, we find 4 and with it, the 3 rd Dimension. Socrates would
be pleased!
Cube:
Octahedron:
2 3
a
3
R
a
and r
2
r 3
3 cosCˆ
i
R 3
noting that 3
2
r
R
a
R
3
3
cos
6 ,
2cos cosC ˆ i
6
2
R