Extended Data Fig. 4: Existence of trajectoids completing two periods in one revolution: paths having Property π.

a,d,g, Two periods of the input planar, complex paths. Color represents progression within a single period (from blue to yellow, see color scale in a). Orange circle shows diameter of the sphere from c,f,i relative to the path. b,e,h, Correspond to paths in a,d,g respectively. Top plot is the mismatch angle (degrees) between initial and final orientations of the sphere after completing two periods of the scaled path – plotted against the path scale \(\sigma \). This angle is obtained by Euler’s axis-angle representation of the matrix of net rotation accumulated by the sphere formula (6) in Methods: \(\theta =\arccos (({\rm{t}}{\rm{r}}{R}_{{\rm{A}}\Omega }-1)/2)\). Bottom plot in each panel shows oriented spherical area \(S(\sigma )\) enclosed by the spherical trace of scaled first period, also plotted against the scale \(\sigma \). Scale corresponding to a two-period trajectoid is marked by red dots. c,f,i Spherical trace of contact point of a unit-radius sphere rolling along scaled paths in a,d,g corresponds to value of \(\sigma \) indicated by red dots in respective plots b,e,h. a, Archimedes spiral with random noise added. d,g, Path obtained by a 2D random walk (making equal steps in random directions) – in piecewise linear version (d) and smoothed version (g).