| buckyball - the graph of truncated icosahedron is also the graph of truncated
aligned cube shown on the right. This representation of the new form of
carbon is a result of my work on a conjecture of computer
program Graffiti that in stable
n-atom fullerenes, the independence number is greater or equal to
n/2 - 6. The uniqueness of the construction among IP-fullerenes
(dodecahedron is another example) is based on the structure of very
special 24-element "pentagonal" independent sets shown below.
Picture by: Craig Larson |
Drawing by: Tomaz Pisanski and Matjaz Zawersnik |
Pentagonal
independent set determine the structure of bromofullerene C_60Br_24 and
very likely also of the fullerol C_60(OH)_24, which was reported to have
taxol-like effects. Taxol is a rare drug for lungs, breast and ovarian
cancers. This is intriguing in our context. Many viruses have icosahedral
symmetry studied by Coxeter along with Fuller's constructions. The
pentagonal independent sets of the buckyball may be related to Golay code;
the soccer ball with pentagonal independent set seems to be naturally
suited for a geometric and hopefully a physical model of the Miracle Octad
Generator - a method to compute with Golay code. The latter is the best
device to transmit information to minimize the chance of unrecoverable
errors. The new representation of the buckyball is a special case of a
general construction: alignments are special involutions of
line-graphs and
the possible relation to Golay code may be an indication of structural
representations of some sporadic simple groups as automorphisms of certain
geometric and related to platonic, cubic graphs, probably including
generalized Petersen graphs. Some of these ideas are discussed
in greater details in my
"C60"
paper published in DIMACS volume "Graphs and
Discovery"