Amy C. Edmondson A Fuller Explanation

Chapter 14, Cosmic Railroad Tracks: Great Circles pages 230 through 231

Inventory: Seven Unique Cosmic Axes of Symmetry
The VE's 25 great circles incorporate those of the rhombic dodecahedron, octahedron, cube, and tetrahedron. The icosahedral 31 belong to a different family of symmetries. Both groups together constitute seven sets of axes of symmetry: four contributed by the VE's vertices, edges, and two types of faces, and three by the icosahedron's vertices, edges, and faces.
1042.05 The seven unique cosmic axes of symmetry describe all of crystallography. They describe the all and only great circles foldable into bow ties, which may be reassembled to produce the seven, greatcircle, spherical sets... .
We have a list of symmetrical possibilities. With this inventory, Fuller integrates specific information about inherent spatial characteristics with concepts of energy behavior, to gain insights about structuring in nature.
Excess of One
It is interesting to note that the number of great circles associated with each polyhedron is always one more than the number of its edges. For example:

Edges  Great Circles  
Tetrahedron  6  7  
Octahedron  12  13  
Cube  12  13  
VE  24  25  
Icosahedron  30  31  
Fuller quickly identifies this constant "excess of one great circle" (and its implied two poles) with the "excess two polar vertices characterizing all topological systerns" (1052.31). However, this discovery is not a new bit of magic, but rather follows directly from Euler's law. Recalling the way in which great circles are generated, we realize that each circle corresponds to a pair of either vertices, edges, or faces. Therefore, the number of great circles can be tallied by counting half the total number of topological aspects, or ½( E + F + V). We can now write an equation stating that one less than the number of great circles is equal to the number of edges:
½(E + F + V)  1 = E.
Multiplying both sides by 2, we have
E + F + V  2 = 2E,
or
F + V  2 = E
and finally
F + V = E + 2,
which is Euler's law.
Even if Fuller's cosmic railroad tracks leave you skeptical, great circles provide fascinating geometric patterns, which introduce a new system of classifying and comparing the topology and symmetry of various polyhedra. Part of their fascination lies in the surprisingly limited number of variations among the greatcircle sets generated by our cast of polyhedra. But perhaps most importantly, experiments with great circles—building wire models—provided the impetus and the clues for Fuller's subsequent journey into geodesics.
