See also: The 25 Great Circles of the VE (new illustrations).
“The vector equilibrium’s 25 great circles, all of which pass through the 12 vertexes, represent the only most economical lines of energy travel from one sphere to another. The 25 great circles constitute all the possible most economical railroad tracks of energy travel from one atom to another of the same chemical elements. Energy can and does travel from sphere to sphere of closest-packed sphere agglomerations only by following the 25 surface great circles of the vector equilibrium, always accomplishing the most economical travel distances through the only 12 points of closest-packed tangency.”
— R. Buckminster Fuller, Synergetics, 452.02
In Fuller’s energetic geometry, the great circles of the VE and icosahedron model energy transference and containment. He described them as “railroad tracks” which either move energy from one domain to another, or divert energy into into loops or holding patterns. Communication occurs when a great circle of one sphere crosses a great circle of an adjacent sphere, i.e. crosses a point of contact or, in Fuller’s terminology, a “kissing point.” All of the 25 great circles of the VE intersect at least two points of contact with neighboring spheres. This is consistent with the role of the VE in the isotropic vector matrix and what Fuller called “nature’s coordinate system.” Of the 31 great circles of the icosahedron, only 7 intersect with the isotropic vector matrix, and these are identical with the sets of 3 and 4 great circles of the VE and the seven axes of symmetry. This isolation is consistent with Fuller’s conception of the icosahedron as a model of potential energy or charge. Of the remaining 24 great circles of the icosahedron, 18 intersect with adjacent spheres only when close-packed as icosahedral shells, and 6 are true holding patterns, communicating with neither the isotropic vector matrix of radially close-packed spheres, nor with the icosahedron shells of laterally close-packed spheres. (See: Close-Packing of Spheres)
The 25 great circles of the vector equilibrium (VE) are derived from its 25 axes of spin: 12 edge-to-edge axes; 6 vertex-to-vertex axes, 4 triangular-face-to-triangular-face axes; and 3 square-face-to-square-face axes.
The 25 great circles disclose the following polyhedra: the octahedon; the vector equilibrium (VE); the tetrahedron; the rhombic dodecahedron; and the cube. See VE: Spherical Polyhedra Disclosed by Great Circles. The sets of 6 and 3 great circles divide the sphere’s surface into the 48 Basic Equilibrium LCD Triangles.
All 25 great circles of the VE pass through one or more points of contact between the radially close-packed spheres of the isotropic vector matrix. In the figures below, the referenced great circles are indicated by the thin black lines, the points of contact, or the vertices of the VE, are indicated by red dots. The vertices of the regular icosahedron are indicated by black dots, and the paths the vertices follow in the jitterbug transformation are indicated by gray lines.
Set of 12 (Axes Connect Opposite Edges of VE)
The set of 12 great circles of the VE create a spider web pattern that appears more random than the other great circle sets. In combination with the sets 6 and 4 great circles, the 12 great circles describe an alternate octahedron, but otherwise disclose no other recognizable patterns. Combined, the 12 great circles pass through all 12 points of tangency with neighboring spheres in the isotropic vector matrix.
Set of 6 (Axes Connect Opposite Vertices of VE)
The set of 6 great circles of the VE describe the spherical rhombic dodecahedron, cube, and tetrahedron. Combined, the 6 great circles pass through all 12 points of tangency with neighboring spheres in the isotropic vector matrix.
Set of 4 (Axes Connect Opposing Triangular Faces of VE)
The set of 4 great circles of the VE describe the spherical vector equilibrium. Each intersects with 6 points of tangency with neighboring spheres in the isotropic vector matrix, and the set as a whole passes through all 12 points of tangency. The set of 4 great circles in combination with the set of 3 great circles of the VE comprise the seven axes of symmetry.
Set of 3 (Axes Connect Opposing Square Faces of VE)
The set of 3 great circles of the VE describe the spherical octahedron. Each intersects with 4 points of tangency with the isotropic vector matrix, and the set as a whole intersects with all 12 points of tangency. The set of the 3 great circles in combination with the set of 4 great circles of the VE comprise the seven axes of symmetry.
For a discussion of the surface and central angles formed by the 25 great circles of the VE, see, The Basic Equilibrium LCD Triangle.
The Geometry of Thinking 