Ad Quadratum Construction and Study of the Regular Polyhedra

platonic forms. They will also lie on the circular planes on which the lines between
vertices (intersection of the dihedral angles) will be the edges of the faces of the
polyhedra.
110
A point to notice is that radial axes along which the spheres are moving are the radii on
which lie the apexes of the dual of the form generated.
Thus for the generation of the cube, the axes will be the radial lines of the octahedron and
conversely. Similarly, for the dodecahedron and the icosahedron. For the tetrahedron,
since the tetrahedron is its own dual, it will be the radial lines of a tetrahedron, oriented
as shown in Kepler’s stella octangula.
We now proceed to calculate the size of the circular interference planes necessary to
build the faces of the five platonic forms. We shall subsequently establish a purely
geometric process based on the ad quadratum method. The assumption is that all platonic
forms will have a common circumsphere. In each case, r will present the radius of the
circular plane bearing the face of a particular polyhedron while a will be the edge of that
face.
R represents the Radius of the common circumsphere. The number of circular planes
required per polyhedron will naturally be equal to the number of faces of that polyhedron.
With circles cut out, the polyhedra models can be assembled by sliding the circles
through the slots as shown on fig. 63.
Tetrahedron (triangle) (fig. 63A)
As we have previously seen: a 2
3 6R
And as established by Euclid 30
or a=1.633R
r
2 2
3 R
r=0.943R
so that a
3
r
for R=3 a 2 6 4.900
30 Heath, Thomas: op. cit. p. 251