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The four-bar linkage is a basic arrangement of mechanical engineering andrepresents the simplest movable system formed by a closed sequence ofbar-shaped bodies. Although the mechanism can have in general a spatialarrangement, we focus here on the prototypical planar case, starting howeverfrom a spatial viewpoint. The classification of the mechanism relies on theangular range spanned by the rotational motion of the bars allowed by theratios among their lengths and is established by conditions for the existenceof either one or more bars allowed to move as cranks, namely to be permitted torotate the full 360 degrees range (Grashof cases), or as rockers with limitedangular ranges (non-Grashof cases). In this paper, we provide a view on theconnections between the "classic" four-bar problem and the theory of 6j symbolsof quantum mechanical angular momentum theory, occurring in a variety ofcontexts in pure and applied quantum mechanics. The general case and a seriesof symmetric configurations are illustrated, by representing the range ofexistence of the related quadrilaterals on a square "screen" (namely as afunction of their diagonals) and by discussing their behavior according both tothe Grashof conditions and to the Regge symmetries, concertedly considering theclassification of the two mechanisms and that of the corresponding objects ofthe quantum mechanical theory of angular momentum. An interesting topologicaldifference is demonstrated between mechanisms belonging to the two Reggesymmetric configurations: the movements in the Grashof cases span chiralitypreserving configurations with a 2 pi-cycle of a rotating bar, while bycontrast the non-Grashof cases span both enantiomeric configurations with a 4pi-cycle.

rendiconti lincei. scienze fisiche e naturali

Quadrilaterals on the square screen of their diagonals: Regge symmetries  of quantum-mechanical spin-networks and Grashof classical mechanisms of  four-bar linkages