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Increasing interest is being dedicated in the last few years to the issues ofexact computations and asymptotics of spin networks. The large-entries regimes(semiclassical limits) occur in many areas of physics and chemistry, and inparticular in discretization algorithms of applied quantum mechanics. Here weextend recent work on the basic building block of spin networks, namely theWigner 6j symbol or Racah coefficient, enlightening the insight gained byexploiting its self-dual properties and studying it as a function of two(discrete) variables. This arises from its original definition as an(orthogonal) angular momentum recoupling matrix. Progress also derives fromrecognizing its role in the foundation of the modern theory of classicalorthogonal polynomials, as extended to include discrete variables. Features ofthe imaging of various regimes of these orthonormal matrices are made explicitby computational advances -based on traditional and new recurrence relations-which allow an interpretation of the observed behaviors in terms of anunderlying Hamiltonian formulation as well. This paper provides a contributionto the understanding of the transition between two extreme modes of the 6j,corresponding to the nearly classical and the fully quantum regimes, bystudying the boundary lines (caustics) in the plane of the two matrix labels.This analysis marks the evolution of the turning points of relevance for thesemiclassical regimes and puts on stage an unexpected key role of the Reggesymmetries of the 6j.

Exact and asymptotic computations of elementary spin networks: classification of the quantum-classical boundaries