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Hi Vladimir,
You asked "whether a buckyball constructed of dipole "rods" has a weak spot due to Brouwer's theorem".
My guess is that IF the buckyball has a weak spot due to Brouwer's theorem, then this weak spot would be part of a hexagon. Clearly, the pentagons have the wrong symmetry for this type of instability, but if a Buckyball is orientated similarly to Figure 2 of The Nature of Dimensions, then it might have a weak spot (and this weak spot might be partially responsible for inducing a triality of generations). If a sphere collapses due to this instability, then the natural new shape would be toroidal. Two nested Buckyballs are homotopic to a torus, so I anticipate that this new toroidal "lattice" should have the equivalent of 120 Carbon sites (although this "lattice" is comprised of the very fabric of Spacetime near the Black Hole "singularity", and not actual Carbon atoms - the same concept as Subir Sachdev's graphene analogy [Reference 12 of the above linked paper] to the Holographic Principle). If we place a spinning tetrahedron at each of those 120 sites, then we may have as many as 480 degrees of freedom on the surface of our torus (plus the frame degrees of freedom), which may be related to an E8xE8*~SO(32) TOE of order 496.
I especially like the pentagon symmetries of the Buckyball (and my TOE), because these pentagon/pentagram symmetries lead to the possible application of the Golden Ratio, as experimentally determined by Coldea et al [Reference 6 of the above linked paper], and as pictorially represented by the appendix figure in my essay.
Have Fun & may your eyes continue to heal!
Dr. Cosmic Ray