--- In synergeo@yahoogroups.com, Rybo6 <rybo6@...> wrote:

On May 21, 2008, at 12:44 AM, Alan Michelson wrote:

> AM] But wouldn't the 6 ico great circles be furthest away from the
> 12 vertexes of the reg icosahedron?

Yes indeed they are Alan. But your "but" comment appears to me to be
irrelevant, to those 6 GrC relationships to the VE's 12 vertexes, as
was previously stated.

Oh yes, the six icosahedral Great Circle Planes vs. the six cuboctahedral Great Circle Planes. If you look at Equilateral Zonohedron -- from Wolfram MathWorld, then you will notice that:

> AM] After all, those 6 ico great circles are perpendicular to the
> body diagonals of the reg icosahedron.

"body diagonals" aka volumetric/diametric diagonals aka an axis ,
however Alan, that these 6 GrC are perpendicular to the 6 GrC's is
most likely well known to all synergeticists here at Synergeo. I
dunno.

> The Td Great Circles are perpendicular to the body diagonals
> through the 12 vertexes of the VE/cuboctahedron.

That may true Alan, as it would explain some of the 14 redundant
congruency, however, I was mistaken, i.e. 3 of those congruences
*ARE* involve with the Ico's 15 GrC's( probably the octahedral three,

Each 3 of those congruences are mutually perpendicular to each other, and there are 5 sets of those mutually-perpendicular Great Circles. See Jim Lehman's L9834. Of course, I am talking about George Hart's compound of five cubes.

and the four VE's that define the cubo-octahedron are congruent with
the Icoss 10 GrC's.

You might also want to look at five tetrahedra inscribed in a dodecahedron and five tetrahedra in five different colors and a compound of ten tetrahedra inscribed in a dodecahedron. They are at George Hart's Compound Polyhedra.

http://www.rwgrayprojects.com/synergetics/s11/figs/f3201b.html
...."Four of the VE great circles are congruent with four of the
icosa's 10-great circle set. Three of the VE great circles are
congruent with three of the icosa's 15-great- circle set. Thus seven
positive are redundant and seven negative are redundant. (87__14=73.)
There are 73 great circles in the composite set.

--- End forwarded message ---