In everyday life, rolling motion is typically associated with cylindrical
(for example, car wheels) or spherical (for example, billiard balls) bodies
tracing linear paths. However, mathematicians have, for decades, been interested
in more exotically shaped solids such as the famous oloids1, sphericons2, polycons3,
platonicons4 and two-circle rollers5 that roll downhill in curvilinear paths
(in contrast to cylinders or spheres) yet indefinitely (in contrast to cones,
Supplementary Video 1). The trajectories traced by such bodies have been studied
in detail6,7,8,9, and can be useful in the context of efficient mixing10,11 and
robotics, for example, in magnetically actuated, millimetre-sized sphericon-shaped
robots12,13, or larger sphericon- and oloid-shaped robots translocating by shifting
their centre of mass14,15. However, the rolling paths of these shapes are all
sinusoid-like and their diversity ends there. Accordingly, we were intrigued
whether a more general problem is solvable: given an infinite periodic trajectory,
find the shape that would trace this trajectory when rolling down a slope. Here,
we develop an algorithm to design such bodies—which we call ‘trajectoids’—and
then validate these designs experimentally by three-dimensionally printing the
computed shapes and tracking their rolling paths, including those that close onto
themselves such that the body’s centre of mass moves intermittently uphill
(Supplementary Video 2). Our study is motivated largely by fundamental curiosity,
but the existence of trajectoids for most paths has unexpected implications for
quantum and classical optics, as the dynamics of qubits, spins and light polarization
can be exactly mapped to trajectoids and their paths16.