If you had a shape large enough, then theoretically, you could.

you just dropped a new mathematical challenge

I would pay for that

You could use their algorithm to make a shape that does that, assuming you had the material

Cursive text with a leap?

Ah yes lemme roll the Euler dice

Roll Them Bones!!!

A bakugan-type game with just these

L pfp

And somehow it reminds me of spinors and how Clifford Algebra describes them.

I think you're right: it could be a way to measure the geometry (the convex hull, to be precise). Somehow it didn't occur to me. But I think your idea will work. However, any object rolls in the same way as its convex hull does, so it will be impossible to distinguish a true nonconvex shape from its convex hull just by inspecting its slipless rolling. The trajectoid algorithm calculates the convex hull needed to do the job. In the paper, the shape of each trajectoid is simply this convex hull. And the object must be rolling sliplessly on a slope under action of gravity alone (or some other constant force). I'm afraid, slipless rolling driven by gravity alone is not very common. Planets and nanoparticles (let alone molecules) don't normally roll on a slope, and if they do it's not a slipless solid-body roll. In the case of nanoparticles, for example, electrostatic and Van der Waals forces come into play, as well as diffusion and fluid flows: nanoparticles stick to surfaces, or don't touch them at all.

Do you have an English translation of this?

​@@ivanvanogre-nd1sw Translation: those things don't roll the same way. Gravity + surface to roll on not included here.

Not really because here, only slope and gravity are used while in natural environment, you have wind, currents, tidal and plate action, etc.

The butterfly effect exists to disprove this theory.

I had an infestation of Gömböcs in my basement last year. I didn't spot any shaoes though.

Yup.

I also feel this way but in a way I cant articulate

@@banann_ducc maybe how a polypeptide wraps around a metal ion?

@@Komadaki maybe?? I havent gotten that far into chem yet. (freshman biochem major with a vague idea of what protein folding is from youtube videos)

Could you roll 100 balls that leave an imprint and be left with a piece of art

been updated tons of times recently, got no notifs?

Well at least annoying, if not harmful.

I see what you mean, but it this specific case this claim is not unfounded. It's shown in the paper that it's so easy to make a two-period trajectoid because, it turns out, almost any finite sequence of 3D rotation matrices whose axes are coplanar can yield the identity matrix when applied twice in a row, if all rotation angles are multiplied by appropriate shared constant. This peculiar property of 3D rotations is directly applicable to the Bloch sphere representation of a qubit. In the context of Bloch sphere, this property means that almost any planar field pulse, once scaled by an appropriate factor and applied twice in a row - will return the quantum system exactly to its original state. You may ask what's the point of performing an action that brings the system back to precisely the same state it was in before this action -- but it's actually one of important operations in pulse sequences used for rotary echo, it's also found in widely-used Wimperis sequences -- see the classical paper at DOI: 10.1006/JMRA.1994.1159 In Wimperis sequences, this operation is done as a single 360-degree rotation. The property found in trajectoids can be directly applied to construct an infinite variety of such identity-matrix-equivalent pulse sequences. It's just a new tool in the pulse sequence designer's toolbox, as I see it.

@@yaroslavsobolev9514 thank you for pointing this out

Look, everyone inflates the "applications" section of their paper/grant proposal.

@@ShankarSivarajan That doesn't make it right.

Each path is infinite, translationally-periodic. And there are infinitely many paths for which a trajectoid exists. But some parts of the associated math are surprizingly deep, who knows when and what will be found once the bottom is reached? Mathematicians should have considered this problem in 23 B.C., not in 2023 A.D.

Since a single path repeats it can roll forever. Since the surface of a sphere has infinitely many points, an infinite non-repeating path should also be possible. Not practical of course, but in theory there should even be infinitely many of those infinite paths.

​@@FHBStudioHaving an infinite number of points isn't sufficient for an infinite path. Any (nondegenerate) path already has infinitely many points, but of course not all paths have infinite length. It's still super easy to find examples of infinite curves. Spirals are the easiest to construct for this purpose since hyperbolic and euler spirals can be cut off at a point and the remaining piece have infinite length but occupying a bounded finite-area region. The simplest example for the sphere, however, is a rhumb line, which has a wikipedia entry if you're interested

Thinking about finite vs infinite 1.As time increases the Objects would wear i wonder can you create complete paths that will always wear into other complete paths. 2. Are there complete paths where reversing the slope changes the path en.wikipedia.org/wiki/Sisyphus?wprov=sfla1

not everything has to be useful tbh

There's a possibility it can be used to solve the protein folding problem, a solution essential in finding the cures to cancer.

I think i can use this in my research Will comeback to this comment after i publish it