Abstract
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.
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Data availability
All data in support of the findings of this study, including 3D shape files of demonstrated trajectoids, are available from a public repository: https://doi.org/10.5281/zenodo.8116412.
Code availability
An online open-source demonstration on Google Colab allows users to design trajectoids for a path of their choice: https://colab.research.google.com/drive/1XZ7Lf6pZu6nzEuqt_dUCHormeSbCCMlP. All Python code used to obtain results that are reported in the paper is available from a public repository: https://doi.org/10.5281/zenodo.8116412.
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Acknowledgements
This work was supported by the Institute for Basic Science, Republic of Korea, project code IBS-R020-D1 and by Swissmap (to J.-P.E.).
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Y.I.S. designed the algorithm that solves for trajectoid’s shapes, implemented it in the software, fabricated the trajectoids, performed experiments, trajectory tracking and analysis. R.D. helped with experiments. T.T. and J.-P.E. developed the mathematical treatment. T.T. and Y.I.S. developed quantum-mechanical and classical analogies. S.G. and B.A.G. conceived and supervised research. Y.I.S., S.G., J.-P.E., T.T. and B.A.G. wrote the paper.
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Extended data figures and tables
Extended Data Fig. 1 Additional illustrations.
a, Potential energy curve corresponding to one half of a period of a sideways rolling of a capped-cylinder trajectoid (orange, yellow) from Fig. 1d. The potential energy curve (green) starts at a minimum energy \({mgr}\), follows a cycloid until reaching maximum energy \({mgR}\) and remains constant afterwards until the edge E contacts the plane. In other words, spherical side caps of radius \(R\) define the maximum of the potential surface. Increasing \(R\) increases the \({mg}(R-r)\) difference and deepens the “trench” in the gravitational potential surface, making the adherence to the target path \({{\rm{T}}}_{\infty }\) more reliable. b, Relative positions of objects defined in the numerical implementation of the algorithm. See Methods for details. Sphere G (the initial “shell” mesh, radius \(R\)) is concentric with \(r\)-radius core \(W\) and intersects with the cube \(C\), whose top face is tangent to \(W\) and lies in the plane \(Q\) of rolling. c,d, Illustration of the MPT Theorem (see Methods for details) with the same path as in Fig. 3d for \(n=3\) (c) and \(n=5\) (d).
Extended Data Fig. 2 Deforming the input path non-uniformly to make it have a one-period trajectoid.
a, Angle \(\theta \) between starting and final orientations of a sphere rolled along a target path (same as in Fig. 2a and Fig. 4d) that was deformed non-uniformly in two directions by scaling factors \({k}_{x}\) and \({k}_{y}\) (see Methods for details). Examples of deformed path on the left correspond to \({k}_{y}=1,2,3\) and fixed \({k}_{x}=1\); examples below correspond to \({k}_{x}=0.5,2\), and \({k}_{y}=1\). Fulfilling the \({R}_{{\rm{A}}\Omega }=1\) condition requires that \(\theta \) be zero (blue regions). b,c, Curves traced over the sphere by sphere-plane contact point when using paths corresponding to the local minima marked as b and c in the \(\theta ({k}_{z},{k}_{y})\) map in panel (a). d–f, Deformations additional to non-uniform scaling affect the minimal mismatch on the \(\theta ({k}_{z},{k}_{y})\) map: a path (blue, same as in Fig. 2a and Fig. 4d) is partially modified by adjusting its Bezier curve nodes; the modified section is shown in orange in e. Maps (d,f) of mismatch angle \(\theta \) for both the original path (d) and its modified version (f) are plotted for the vicinity of the useful minimum of the original path (see full map in panel a). Note the increase of mismatch in f compared to d.
Extended Data Fig. 3 “Bridging” the input path to make it have a period-one trajectoid.
Example of input path (blue) with “bridges” (red and orange) appended to it. Bridge consists of three arcs (red) and two straight “arms” (orange). b, Family of bridges constructed for the path in a at a fixed scale.
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).
Extended Data Fig. 5 Search for paths not having Property π.
Representation analogous to Extended Data Fig. 4, but for different input paths. a, Acute-angle isosceles V-path does not have Property \({\boldsymbol{\pi }}\): for this path, \(\left|S(\sigma )\right| < {\rm{\pi }}\) for all \(\sigma \). The trace c corresponds to a one-period trajectoid as shown by the green dot in b. d, V-like path whose arms have “kinks” parameterized by angle \(\alpha \). It does not have Property \({\boldsymbol{\pi }}\), and a two-period trajectoid does not exist for this path. The trace f corresponds to a green dot in e showing a near miss (i.e., not a trajectoid). g, V-like path similar to the one in panel a, but here the acute corners are tapered (“dulled”) by straight segments \({A}_{1}{T}_{1}\), \({T}_{2}{T}_{3}\), \({T}_{4}{M}_{1}\). Trajectoid for this path exists (red points in h). k, Asymmetric V-like path with kinks – as in panel d, but here kinks in two arms are unequal: angle of the left kink is \(\alpha \), but angle of right kink is \(\beta =\alpha (1+\Xi )\). Trajectoid for this path exists (red points in l). In panels k,l,m, we used asymmetry \(\Xi =0.37\). See Supplementary Video 4 for dependence of these plots on angle \(\alpha \) in d, taper ratio in g, and asymmetry \(\Xi \) in k.
Extended Data Fig. 6 Trajectoid tumbling at sharp turns.
In Fig. 4d, g, h, the trajectories feature some sharp turns (marked therein by green circles) at which trajectoids tumbled/recoiled. These features of dynamics can be understood by looking at the trench made by the trajectoid in the gravitational potential surface. The representation here resembles Fig. 1b except that the trench makes a sharp turn. In this dynamic analogy, resulting motion can be illustrated by a small particle (colored here in blue) rolling down such a trench: prior to encountering a sharp turn, the particle may have a zig-zag trajectory bouncing between the trench walls (cf. Fig. 4d, h yellow arrows), and then might recoil back at the sharp turn instead of making the turn immediately. It may take a few bounces before the particle finally turns the corner. For the actual trajectoids we fabricated, these bounces and zig-zagging correspond to precessions that may result in net rotation of the object around the axis normal to the sloped plane. In the case of the second corner in Fig. 4d (bottom orange circle), the unintended net rotation due to recoils causes a turn of the entire subsequent trajectory with respect to the intended path \({\rm{T}}\). We believe that these dynamic defects may be minimized by engineering the trench profile (presently we keep it cycloid-like, which might be overly sharp) or making smoother turns of path \({\rm{T}}\).
Extended Data Fig. 7 Energetic considerations for intermittent uphill rolling.
Energy landscapes illustrating design considerations for traversal of uphill excursions (as in the path in the main-text Fig. 4n, o). The gravitational acceleration vector \({\bf{g}}\) (vertical arrow) points down. The trajectoid’s total energy (blue surface) decreases with descent due to losses such as friction. The potential energy surface (green) is defined by trajectoid’s shape and the angle of the slope. Parts of the potential surface that are below the total energy surface are accessible to the rolling trajectoid. Proper balance between losses, shape, and slope are shown in a: the total energy suffices to overcome the potential barrier along the target path (uphill excursion, short arrows), yet is insufficient to escape walls of the potential trench that follow the target path. b, When losses are too high, the net energy is insufficient for an uphill excursion. When losses are too low, as in c, net energy decreases more slowly than the potential energy and becomes sufficient to escape the potential trench as indicated by the dashed arrow. Note that strictly speaking, the net energy is not a function of only the 2D location on the plane and, furthermore, the potential energy depends on the trajectoid’s 3D orientation, which is not an unambiguous function of the 2D planar location even for slipless rolling. Still, the concept of potential landscape is useful as a first approximation.
Extended Data Fig. 8 Centroid of a visible shape as an estimator of the center of mass.
a–c, Comparisons between the target path of a trajectoid (solid black curves) and the theoretical trace of the centroid of trajectoid’s shape projection, simulated assuming perfect performance (orange curves). The difference between curves in a is barely visible. Note that the discrepancy between orange and black curves in a,b,c is smaller than or comparable to the discrepancies between the experimental trajectories and the respective target paths (Fig. 4o, n, l). d,f, For every possible orientation of the trajectoid (parameterized by latitude and longitude of contact point on the trajectoid), color on these maps shows planar distance between the projection of the center of mass and the centroid of the shape projection. Maps in d and f are computed for the trajectoids constructed for paths b and c, respectively. Note that the distance in these maps never exceeds of the trajectoid’s minimal radius \(r=1\). e,g, Illustration of 6D pose (full orientation and position) tracking from video frames (see Supplementary Video 7): e is the raw video frame of the trajectoid (same as in Fig. 4k) whose surface has black dots (markers) painted on it; g shows the same raw frame overlaid with the trajectoid’s green 3D mesh (the one used for 3D-printing the trajectoid) matching the location and orientation evaluated by 6D pose tracking algorithm applied to apparent trajectories of black markers on the video (reconstructed 3D locations of markers are shown in orange). h, Comparison of theoretical (intended) path of the 2D projection of the center of mass (black curve) and the experimental path evaluated by two methods: by the centroid of the visible shape (blue curve) or by the 6D pose tracking algorithm (orange curve). Both methods were applied to the same video. i, Difference in the Y coordinate between the experimental and the intended path for the two methods of evaluating the center-of-mass location: here, the black curve in h has been subtracted from the blue and orange curves in h, respectively. The standard deviations for the two methods are: 0.845 mm when using centroid of visible shape, 0.765 mm when using the 6D pose tracking.
Extended Data Fig. 9 Optical and quantum-mechanical analogies to the existence of two-period trajectoids.
a, Illustration of the Bloch sphere representation of a single qubit. The state represented by a red circle on the sphere rotates (orange arrow) around an instantaneous axis \({\bf{n}}(t)\), which is defined by the driving field. b,c, Field pulse equivalent to a single period \(T\) (b, solid curves), its envelope (b dashed curves) and phase shift (c) as functions of time. Shown are two possible analogies to varying the radius \(r\) of the rolling sphere in case of a given field pulse (black): either scaling the applied pulse’s magnitude (green) or stretching the pulse’s functions (envelope and phase shift) in time (blue). d, Illustration of the Poincaré sphere representation of polarization state of light. Squares show polarization states corresponding to respective black points: two circular polarizations at the poles and four linear polarizations at the equatorial plane. e–g, Given almost any (i.e. those having Property \({\boldsymbol{\pi }}\)) sequence of waveplates (dark blue in e), their thicknesses can be scaled (f) by such a factor \(1/r\) that the doubled sequence (g) has no net effect on polarization state of light (yellow helices) passing through it. In this example, curved green arrow shows left-handed circular polarization, curved orange arrows show right-handed elliptic (f) and right-handed circular (g) polarization. See also Supplementary Videos 5, 6.
Extended Data Fig. 10 Small-radius (“adiabatic”) limit of trajectoids.
a–c, Effect of ball radius \(r\) (or path’s scale \(\sigma =L/(2{\rm{\pi }}r)\)) on the trace of the contact point upon rolling along a finite smooth path (two periods of path from Fig. 4k). Value of \(r\) decreases from a to b to c. See Supplementary Video 8 for more details. d,e–g, Effect of ball radius \(r\) (or path’s scale \(\sigma =L/(2{\rm{\pi }}r)\)) on the trace of the contact point upon rolling along a polygonal path (path shown in d). Value of \(r\) decreases from e to f to g. See Supplementary Video 8 for more details.
Supplementary information
Supplementary Video 1
Narrated presentation of the main concepts and results of this work. Note that this video contains an audio track. (00:00–00:48) Problem formulation. (00:48–01:11) Demonstration of solutions. (01:11–05:12) Conditions of trajectoid existence are discussed. (05:12–06:42) Heuristics of computing the shape of a trajectoid. (06:42–07:16) Experimental validation of trajectoids. (06:16–08:26) Performance of trajectoids with uphill excursions and self-intersections. (08:26–09:49) Applications of TPT theorem in quantum mechanics.
Supplementary Video 2
Top view of validation tests showing a comparison between the experimental trajectories (bottom) and the intended ones designed numerically (top). The specific fragments of the video correspond to Fig. 4 as indicated in the top right corner. A green or blue circle overlaid in video frames marks the centroid of the trajectoid (Methods and Extended Data Fig. 8). The trace of this point gradually appears behind the rolling trajectoid. Colours of some trajectoids differ from their photos in Fig. 4 because they were three-dimensionally printed in different colours (for no specific reason). This video shows true colours but with for Fig. 4 their photos were recoloured blue (except the Fig. 4j) to avoid confusion.
Supplementary Video 3
Proof of the TPT Theorem illustrated for the path of Fig. 1c. This animation illustrates the proof steps: using property π to scale the first period as in Fig. 3a–c, then rotating it by 180° with respect to axis OC to arrive at Fig. 3e. For details, see the Methods.
Supplementary Video 4
Investigation of the existence of TPTs. This animation shows dependence of the plots described in Extended Data Fig. 5 on the angle \({\rm{\alpha }}\) in Extended Data Fig. 5d, the taper ratio in Extended Data Fig. 5g, and the asymmetry \(\Xi \) in Extended Data Fig. 5k. Taper ratio is defined as \(2({\rm{A}}{{\rm{T}}}_{1}/{\rm{AC}})\) in the scheme on Extended Data Fig. 5d.
Supplementary Video 5
Scaling the driving field pulses in quantum-mechanical analogy to the existence of TPTs. In this analogy, modulating the radius r of the rolling ball can take two possible forms (Extended Data Fig. 9b,c): either scaling the applied pulse’s magnitude \((E(t)\to E(t)/r,{B}_{xy}(t)\to {B}_{xy}(t)/r)\) or stretching the pulse’s functions in time (envelope \(E(t)\to E(rt)\) and phase shift \(\Phi (t)\to \Phi (rt)\) for NMR and electric dipole transitions; planar magnetic field \({B}_{xy}(t)\to {B}_{xy}(rt)\) for magnetic dipole in magnetic field). These forms are shown on left and right sides of this video.
Supplementary Video 6
Scaling the waveplates in the optical analogy to the existence of TPTs. This video corresponds to Extended Data Fig. 9e–g. It starts with a given sequence of waveplates (dark blue) with some thicknesses and orientations of optic axis (shown by dark arrows). Thicknesses are then scaled by such a factor \({\rm{\sigma }}\) that doubled sequence has no net effect on polarization state of light (yellow) passing through it.
Supplementary Video 7
Tracking the CM by 6D pose tracking. Illustration of 6D pose (full orientation and position) tracking from video frames. The first part of the video shows the trajectoid (same as in Fig. 4k) with black dots on its surface rolling down a plane. Footage is shot at 120 frames per second and shown at 30 frames per second at 25% of the true speed. The trajectory of the centroid of projections is superimposed on top of the raw footage. The second half of the video shows the results of 6D pose tracking in Blender software: trajectoid’s true 3D model and its reference frame are superimposed on the raw footage. Detected markers are shown in orange. The green circle is the true CM estimated by 6D pose tracking and white curve is its trajectory as viewed from the perspective of the camera. Note that the respective plot in Fig. 8h is the trajectory’s orthogonal projection onto the plane of rolling and is therefore free of the perspective distortion of the camera.
Supplementary Video 8
Small-radius limit of trajectoids. The right side shows the spherical trace of the contact point of an r radius sphere that has completed a roll from start to end along the path shown on the left side. Colour corresponds to arc length along the path, measured from the starting point. The value of r decreases over the course of the video, as reflected by the ratio of path period length to the sphere’s circumference 2πr indicated in the top left. The orange circle shows the sphere’s circumference relative to the path. The first half of the video (0:00–0:13) shows the case of a differentiable planar path (two periods of path from Fig. 4k). The second half of the video (0:13–0:26) shows the case of a polygonal path.
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Sobolev, Y.I., Dong, R., Tlusty, T. et al. Solid-body trajectoids shaped to roll along desired pathways. Nature 620, 310–315 (2023). https://doi.org/10.1038/s41586-023-06306-y
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DOI: https://doi.org/10.1038/s41586-023-06306-y
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