Ad Quadratum Construction and Study of the Regular Polyhedra

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Note also that three such pentagrams meet at each of the 20 vertices of the convex
dodecahedron to yield a GSD.
So far, by means of the three mutually perpendicular golden rectangles we have
established the convex icosahedron and dodecahedron as well as the four stellated
regular polyhedra.
The octahedron has been implicitly established when we constructed adquadratum
the three mutually perpendicular planes within the sphere. Their mutual
intersections two at a time with the sphere at 6 different points are the vertices of
the octahedron.
For the cube, on one of the three planes we draw through the center of the sphere
two diameters forming angle C ˆ
i.
Four vertices are thereby determined at the
contact point with the sphere. Drawing circles of radius equal to the adquadratum
value of the cube edge on the sphere from each of these four vertices will yield
four new vertices at their intersection. The total of 8 vertices is then available for
the cube.
The tetrahedron will then be easily constructed by selecting one of the cube
vertices as a starting point and further selecting its three second immediate
neighbors and joining these 4 vertices together.
b. By Individual Polyhedra
The three golden rectangles approach yields a top-down generation of the regular
polyhedra so to speak. We go from the complex to the simple, from the
icosahedron and dodecahedron and their stellated forms to the cube and the
tetrahedron. In the present method, we go from the simplest to the more complex,
starting with the individual tetrahedron. We make no direct reference to the
golden rectangle structure.
Given are the sphere of Radius R and the adquadratum diagram providing us with
the internal angle and therefore the edge of each polyhedron inscribed in the
sphere of radius R.
For the Tetrahedron, pick a point at random on the sphere. One can imagine
doing this either externally or internally to the sphere. From that point as center,
draw on the surface of the sphere the circle having for radius the chord of the
tetrahedron internal angle (edge of the tetrahedron) as given on the adquadratum
diagram. From another point at random on this circle and with the same compass
opening, draw another circle on the sphere. It will cut the first circle at two
points. These and the two previous points are the vertices of the tetrahedron. The
six straight line segments joining the 4 vertices together are the edges of the
tetrahedron. Joining the vertices to the center of the sphere gives form to the
Maraldian pyramids of the tetrahedron.