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

Abstract

The four-bar linkage is a basic arrangement of mechanical engineering and represents the simplest movable system formed by a closed sequence of bar-shaped bodies. Although the mechanism can have in general a spatial arrangement, we focus here on the prototypical planar case, starting however from a spatial viewpoint. The classification of the mechanism relies on the angular range spanned by the rotational motion of the bars allowed by the ratios among their lengths and is established by conditions for the existence of either one or more bars allowed to move as cranks, namely to be permitted to rotate the full 360 degrees range (Grashof cases), or as rockers with limited angular ranges (non-Grashof cases). In this paper, we provide a view on the connections between the "classic" four-bar problem and the theory of 6j symbols of quantum mechanical angular momentum theory, occurring in a variety of contexts in pure and applied quantum mechanics. The general case and a series of symmetric configurations are illustrated, by representing the range of existence of the related quadrilaterals on a square "screen" (namely as a function of their diagonals) and by discussing their behavior according both to the Grashof conditions and to the Regge symmetries, concertedly considering the classification of the two mechanisms and that of the corresponding objects of the quantum mechanical theory of angular momentum. An interesting topological difference is demonstrated between mechanisms belonging to the two Regge symmetric configurations: the movements in the Grashof cases span chirality preserving configurations with a 2 pi-cycle of a rotating bar, while by contrast the non-Grashof cases span both enantiomeric configurations with a 4 pi-cycle.