Determining 20 Constellation Points On A Sphere.

 

The solution to this problem is quite simple, but very subtle.  It is solved by considering the solid figure dodecahedron.   A dodecahedron is a solid figure which has 20 vertices and 12 regular pentagonal faces.  This figure can be inscribed within a sphere.

 

It is again 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. 

 

The 20 vertices of the dodecahedron locate the constellation points on the surface of a sphere.  The faces of a dodecahedron are all regular pentagons.  The center or centroid of the pentagonal face, projected upon the surface of the sphere along a radius of the sphere, is one of the 12 primary points.  The compass setting required is the distance from a primary point to a vertex of the dodecahedron.

 

            The procedure is as follows:

 

Mark the 12 primary points as above.  Then erase the circles, keeping the points previously marked.  The circles can be very confusing during further marking out.

 

Set a compass  to a radius equal to

 

, where d is the diameter of the sphere.

 

            For a sphere of 50mm, this measurement is 16.03 mm.

            For a sphere of 62mm, this measurement is 19.87 mm.

 

Place the point of the compass on one of the 12 primary points and draw a circle of radius r.

 

Place the point of the compass on a nearby primary point and draw another circle of radius r.

 

Mark the two points where these circles intersect and from each point of intersection draw another circle of radius r.

 

Continue in this manner until all 20 points have been identified and marked.

 

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.