Determining 12 Primary Points On A Sphere.

 

This problem is equivalent to determining the vertices of the inscribed icosahedron, a solid figure with 20 faces and 12 vertices.  See Figure 1.

 

It is useful to note that the dodecahedron and the icosahedron are closely related.   The dodecahedron, which has 12 faces and 20 vertices, is the “dual” of the icosahedron. 

 

If the center points of each face of the dodecahedron are projected, along a radial line,  upon the surface of the circumscribed sphere, then these 12 points are the primary points as well as the vertices of the inscribed icosahedron.

 

Set a compass  to a radius equal to:

 

, where R is the radius of the sphere

 

For a sphere of 50mm diameter, this measurement is 26mm.

For a sphere of 62mm diameter, it is 32.6mm.

 

Select a point at random on the sphere and draw a circle of radius r about that point.

 

Place the point of the compass at any point on the first circle, and draw another circle of radius r.

 

Where the circles intersect, place the point of the compass and draw additional circles.

 

Continue in this manner, until all 12 points have been marked by the intersections of circles.

 

Due to small errors in the setting of the compass and in the diameter of the sphere, some of these points must be “fudged” or adjusted by eye.