For their own enjoyment, the reader is encouraged to determine, and discover, each of the following constructions on their own (rubber bands and a sphere, or hoops/rings work well to build constructions), but for the sake of our overarching investigation they will be simply presented here. When dividing the sphere in the most regular way, using four great circles defines a solid called the “cuboctahedron.” (The solid is in red, and the great circles are black.)

Here is the cuboctahedron without the sphere or the great circles:

In using * six *great circles, there is another solid defined by this regular division of the sphere, the “icosidodecahedron.”

Without the sphere:

Again, the reader is encouraged to prove the validity of each of these divisions (3, 4, and 6 great circles) for themselves- and to take the investigation further, by looking into divisions with higher numbers of great circles! The octahedron, cuboctahedron, and icosidodecahedron, are of a single class, defined by great circles that intersect only two at any point, and divide each other evenly. If there are two fully extended straight lines on a sphere (great circles), each one will have to interest the other twice, and only twice, (as long as they do not lie exactly on top of one another); a similar process in a plane will only produce exactly one intersection from two fully extended straight lines. So if we have three great circles, any one will be intersected by the other two four times; so for three great circles we have six intersections, or more generally:

With the three great circles, each one is divided into four parts, defining a square, and the three squares form the octahedron. In a similar fashion, four great circles define regular hexagons, and four hexagons make up the cuboctahedron; six great circles define regular decagons (10 sides), and the icosidodecahedron is made of six decagons. The following three animations have the the polygons in different colors so that they can be distinguished, and, as always, they can be spun around to get a different view.

Octahedron, cuboctahedron, and icosidodecahedron, constructed at a 2007 California LaRouche Youth Movement cadre school.