3nj-symbols and harmonic superposition coefficients: an icosahedral abacus

Angular momentum recoupling coefficients of angular momentum theory and matrix elements for basis set transformation of hyperspherical harmonics enjoy properties and sum rules crucial for applications but complicated without the guidance of graphical techniques. These coefficients being related to Racahʹs polynomials, the graphs also apply to polynomials of the hypergeometric family, their q-analogues and their `ellipticʹ extensions. A useful `abacusʹ exploiting the connections with presentations of icosahedral and related symmetries is introduced. Particular and limiting cases, such as those of the semiclassical type, allow a unified view of properties of angular and hyperangular momentum algebra, including relationships among vector coupling coefficients and rotation matrix elements.