In 1922, Cartan introduced into differential geometry, as well as the Riemannian curvature, the new concept of torsion. He visualized an homogeneous and isotropic distribution of torsion in three dimensions as depicted by a so-called helical staircase which was constructed by starting from a three-dimensional Euclidean space and by defining a new connection via helical motions. This geometrical procedure was described in detail and the corresponding connection and torsion were defined. The interdisciplinary nature of this subject was already evident from Cartan’s discussion, since he argued—but never proved—that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. Cases in physics where the helical staircase was realized were described: in the continuum mechanics of Cosserat media, in speculative three-dimensional theories of gravity (Einstein-Cartan gravity with constant pressure and constant intrinsic torque, and Poincaré gauge theory with the Mielke-Baekler Lagrangian) and in the gauge field dislocation theory of Lazar et al., which first proved it for a suitable distribution of screw dislocations. However the main emphasis here was placed on the case of dislocation field theory.

Cartan’s Spiral Staircase in Physics and, in Particular, in the Gauge Theory of Dislocations. M.Lazar, F.W.Hehl: Foundations of Physics, 2010, 40[9-10], 1298-325