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A unified vision of the symmetric coupling of angular momenta and of thequantum mechanical volume operator is illustrated. The focus is on the quantummechanical angular momentum theory of Wigner's 6j symbols and on the volumeoperator of the symmetric coupling in spin network approaches: here, crucial toour presentation are an appreciation of the role of the Racah sum rule and thesimplification arising from the use of Regge symmetry. The projective geometryapproach permits the introduction of a symmetric representation of a network ofseven spins or angular momenta. Results of extensive computationalinvestigations are summarized, presented and briefly discussed.

Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective