Determining The 6 Main Points On A Sphere.

 

This problem is equivalent to determining the vertices of the inscribed octahedron, a solid figure with 6 vertices.  See Figure 1.

 

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 35 mm.

For a sphere of 62mm diameter, this measurement is 44 mm

 

Select a point at random on the sphere and draw a circle with the above radius r.

 

Place the point of the compass at a random point on the circle just drawn and draw another circle of radius r. 

 

Place the point of the compass on the intersection of these two circles and draw a third circle of radius r.  This will create 6 points where the circles intersect.  These points will be equal distances apart.