Combinatorial and geometrical origins of regge symmetries: Their manifestations from spin-networks to classical mechanisms, and beyond

Tullio Regge discovered new symmetries in 1958, hidden in formulas for calculations of the coupling and recoupling coefficients of quantum angular momentum theory, as developed principally by Wigner and Racah: the only known (limited) application appeared computational. Ten years later, in a paper with Ponzano, Regge provided a semiclassical interpretation showing relevance to the basic geometry of quadrilaterals and tetrahedra, and opening also a promising road to quantum gravity, still currently being explored. New facets are here indicated, continuing a sequence of papers in this Lecture Notes series and elsewhere. We emphasize how an integrated combinatorial and geometrical interpretation is emerging, and also examples from the quantum mechanics of atoms and molecules are briefly documented. Attention is dedicated to the recently pointed out connection between the quantum mechanics of spin recouplings and the Grashof analysis of four-bar linkages, with perspective implications at the molecular level.