  New analyses of the organization of the genetic code system together with
their relation to the two classes of aminoacyl-tRNA synthetases are reported in
this work. A closer inspection revealed how the enzymes and the 20 amino acids
of the genetic code are intertwined on a polyhedron model. Complimentarily and
cooperative symmetry between class I and class II aminoacyl-tRNA synthetases
displayed by a 28-gon model are discussed, and we found that the two previously
suggested evolutionary axes of the genetic code overlap the three two-fold
symmetry axes within the two classes of aminoacyl-tRNA synthetases. Moreover,
it is identified that the amino-acid side-chain carbon-atom numbers (1, 3, 4
and 7) in the overwhelming majority of the amino acids recognized by each of
the two classes of aminoacyl-tRNA synthetases meet a mathematical relationship,
the Lucas series.
