Modelling equiangular spiral arc kinematics of the human finger: exact conditions, viable parameter regions, and the role of phalangeal proportions

The equiangular spiral has long been proposed as the natural trajectory of fingertip motion during finger closure, and specific phalangeal length ratios — in particular those approximating the golden ratio — are advocated in reconstructive hand surgery as necessary for spiral arc kinematics. This paper analyses both claims quantitatively. An exact equiangular spiral is unattainable for any such chain, for any link lengths, coupling coefficients or pole position — not merely over an unbounded flexion range, but on any interval of positive length, including the degenerate case of a circle. That incompatibility, established previously, leaves open how closely the fingertip path approximates a spiral arc over the physiological range, and for which parameter values. For incommensurate coupling coefficients, the generic case, the instantaneous spiral cotangent is quasi-periodic in the proximal joint angle, oscillating within bounds about a mean set by the coupling and pole position. The viable approximation region — the coupling pairs (k1, k2) for which the fingertip traces an excellent spiral arc (fidelity R2 ≥ 0.999) over the physiological range — occupies approximately 80% of the displayed parameter space. At anatomical coupling, fidelity is essentially uniform across the full range of phalangeal length ratios examined. Spiral arc kinematics is therefore a robust property of the coupled finger across a wide range of coupling states and anatomical proportions, and the surgical prescription of golden-ratio or other length targets as necessary for it is not supported by the analysis.