450.00 Great Circles of the Vector Equilibrium and Icosahedron

450.10 Great Circles of the Vector Equilibrium (see also drawings section )

450.11 Four Sets of Axes of Spin: The omni-equi-edged and radiused vector equilibrium is omnisymmetrical, having 12 vertexes, six square faces, eight triangular faces, and 24 edges for a total of 50 symmetrically positioned topological features. These four sets of unique topological aspects of the vector equilibrium provide four different sets of symmetrically positioned polar axes of spin to generate the 25 great circles of the vector equilibrium. The 25 great circles of the vector equilibrium are the equators of spin of the 25 axes of the 50 unique symmetrically positioned topological aspects of the vector equilibrium (see also drawings section ).

450.12 Six of the faces of the vector equilibrium are square, and they are only corner-joined and symmetrically arrayed around the vector equilibrium in respect to one another. We can pair the six opposite square faces so that there are three pairs, and we can interconnect their opposite centers of area to provide three axes, corresponding to the XYZ coordinates of Cartesian geometry. We can spin the vector equilibrium on each of these three intersymmetrically positioned axes of square symmetry to produce three equators of spin. These axes generate the set of three intersymmetrical great-circle equators of the vector equilibrium. Together the three great circles subdivide the vector equilibrium into eight octants (see also drawings section).

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Fig. 450.10 The 12 Great Circles of the Vector Equilibrium Constructed from 12 Folded Units (Shown as Shaded).

The 12 Great Circles of the Vector Equilibrium

Fig. 450.11A Axes of Rotation of Vector Equilibrium:

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1. Rotation of vector equilibrium on axes through centers of opposite triangular faces defines four equatorial great-circle planes.

2. Rotation of the vector equilibrium on axes through centers of opposite square faces defines three equatorial great-circle planes.

3. Rotation of vector equilibrium on axes through opposite vertexes defines six equatorial great-circle planes.

4. Rotation of the vector equilibrium defining twelve equatorial great-circle planes, each of which passes through two opposite vertexes and two midpoints of the edges of two opposite triangular faces. The axes of rotation pass through opposite square faces.

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Fig. 450.1 1B Projection of 25 Great-Circle Planes in Vector Equilibrium System: The complete vector equilibrium system of 25 great-circle planes, shown as both a plane faced-figure and as the complete sphere (3 + 4 + 6 + 12 = 25). The heavy lines show the edges of the original 14-faced vector equilibrium.

1. There are also eight symmetrically arrayed triangular faces of the vector equilibrium. We can pair the symmetrically opposite triangular faces so that there are four pairs, and we can interconnect their opposite centers of area to provide four intersymmetrically positioned axes. We can spin the vector equilibrium on each of these four axes of symmetry to produce four intersymmetrical equators of spin. These axes generate the set four intersymmetrical great-circle equators of the vector equilibrium.

2. When the 12 intersymmetrically positioned vertexes of the vector equilibrium are polarly interconnected, the lines of most economical interconnection provide six symmetrically interpositioned axes of spin. These six axes generate the set of six intersymmetrical great-circle equators of the vector equilibrium.

3. We may also most economically interconnect the 24 polarly opposed midpoints of the 24 intersymmetrically arrayed edges of the vector equilibrium to provide 12 sets of intersymmetrically positioned axes of spin. These axes generate the set of twelve intersymmetrical great-circle equators of the vector equilibrium.

4. As described, we now have sum-totally three square-face-centered axes, plus four triangular-face-centered axes, plus six vertex-centered axes, plus 12 edge-centered axes (3 + 4 + 6 + 12 = 25). There are a total of 25 com- plexedly intersymmetrical great circles of the vector equilibrium.