> I parameterized the wheel as a twenty sided polygon with a variable radius for each vertex, ri.

Could this way of representing the problem (radius) introduce bias to the solution? It could probably make many kind of shapes, but it may need to overcome some huge local maxima in order to drastically change it to something else by tweaking multiple of the radii.

When writing my thesis on spoke patterns for bicycle wheels, one of the representations outperformed the others, as it had symmetry baked in. That could have made it not find some esoteric solutions (which is why I had multiple representations).

I totally agree! It would have been better to avoid bias by using a pixel/voxel representation (1=material present, 0=no material present), like Airbus did with wing structure optimization: https://backend.orbit.dtu.dk/ws/portalfiles/portal/138121106...

(Not claiming that OP's work isn't absolutely cool, rather suggesting paths for future improvements. :D)

On/off representations are cool, but also a bit harder to work with. One can end up having disjoint groups that don't really make sense, so lots of permutations has to be thrown out. So finding the correct representation can actually be very hard.

The problem you're describing is kind of dual to the problem with parametrized representations.

It's easy to construct parametrized boundary representations (as OP used) that maintain topological invariants, but it's very difficult to use them to explore the space of possible topologies (which would be relevant if OP wanted to say, design a lighter wheel).

@matsemann Would you mind sharing your work?

It was discussed here[0] at the time :)

[0]: https://news.ycombinator.com/item?id=10410813

Responding broadly to this and other sibling comments about different representations for optimization...

This can be considered a generalized example of the bias-variance tradeoff. Imposing symmetry or a parametric form can be considered forms of regularization/bias that help converge to a well-formed solutions.

> Could this way of representing the problem (radius) introduce bias to the solution?

Yes, but you're always going to have a prior.

You could model it as, let's say, a shape in a square grid with a certain area. Or a collection of line segments that are closed.

This is an issue common to most optimization problems, you want the simplest model that will be efficient, otherwise you might risk overfitting (or just taking too long to optimize)

I think actually ri doesn’t introduce bias, maybe N=20 sides can be problem for bias, sufficiently large N should not cause problem.

The issue is the points are all equally spaced. Working back from a circle that’s not an issue, but suppose the optimal solution had varying side lengths. You could approximate that with very large n, but that’s about it.

The Reuleaux triangle is for some measurements (for example material used), more optimal than a circle, while still able to roll smoothly.

https://en.wikipedia.org/wiki/Reuleaux_triangle

Only in theory. A Reuleaux triangle would need to be connected to an axle which is connected to the thing being moved. If it's on vehicle or trailer you have one on each side. The axle needs to go up an down 3 times per circle revolution.

If the wheels on each end of an axle aren't perfectly aligned the up and down motion won't be the same on each side. The speed of the up and down motion while moving down a modern road would be very fast. This would wear our parts far more quickly.

Theory and practical implications don't always line up. In practice it's not more optimal when applying to every day real world application.

In addition, that up and down motion is annoying to passengers, and a waste of energy. The suspension has dampers, and the tires have hysteresis (it takes more energy to squish a tire down than you get from letting it spring back up), so more energy is put into the "up" parts of the cycle than is recovered from the "down" parts.

Which they won't be perfectly aligned as soon as you start turning. Because as you turn the wheels rotate differently as they travel a different distance.

As soon as i saw that, i recognized the shape as the rotor from a wankel engine, but apparently its subtley different. According to that wiki article.

The problem is that you can't put an axle on this.

You can, however, put it between two planes. (say for needle bearings?)

It’s interesting to note that the final result, though essentially circular, has an offset axle. This certainly seems suboptimal, and it would’ve been nice to have some analysis of why this differs from the expected result.

I feel like the resulting shape is more of a fluke here.

> The performance of the wheel was based on the final speed achieved by an accelerating imaginary car, vf. The wheel was driven with a constant torque, τ, and no slipping.

There is nothing in this fitness value that defines the shape of the wheel. As described here, there's no clear reason why the shape of the wheel would affect the speed of a car (no slip, no taking into account up/down movement of the car, etc.) Rather this algorithm seems to optimize for a certain radius and the circular shape is the result of the fact that all points converge to the same optimum.

I also find the fact that it converges at a finite radius using this model suspicious. If you look at this optimization problem analytically, the optimum is obviously a wheel with a zero radius: Force on car is torque over wheel radius. If radius goes to zero, then force goes to infinity and acceleration goes to infinity and car speed goes to infinity.

It doesn't take the vertical movement of the car into account directly, but that cosine term in the force does mean that having to raise the car (or lower the car) saps it of effective torque.

Also, the wheel spoke lengths are normalized, so the wheel can't grow or shrink overall.

I feel the same thing. It's basically optimizing the "gear ratio" to get maximum speed. The wheel isn't centered because it doesn't complete an integer number of revolutions during the simulation.

I'm still scratching my head while trying to figure out what's the mechanism that prevents the solution blowing up to infinity by setting the radius to zero and therefore achieving huge accelerations from a constant torque.

I couldn't figure this out at the time, but a potential explanation seems obvious to me now.

I think this may have been because the wheel starts accelerating from a particular point. To optimize distance traveled, it's better to have more acceleration up front. Shorter spokes means more acceleration. For the overall wheel, the spoke length is normalized, but it makes sense that the optimization would set the starting spoke shorter.

I should randomize the starting angle of the wheel and see if that fixes it.

I'm surprised to see that I didn't discuss this in the blog post. You're right, I should have.

I think the author's speed calculation model doesn't care about the axle position with respect to the wheel's centre. Could be accounted for, but I'm guessing that's why the model doesn't fit for it.

But axle position and constant radius are the only reasons why a circle is the best wheel shape, instead of Reuleaux triangles or something entirely different. If axle position isn't a factor that makes it seem like the code is just searching for an object with some constant radius X. Which is a fine task, but it's not really about the "Optimal Rolling Shape"

Maybe it does not matter much for the rolling property, but for sure an offset axle will not improve comfort.

It could just be a local maximum, with respect to the model.

I cannot be a local optimum: moving the axle just sligthly towards the center of the disk will produce a slightly rounder, and better, shape. It seems that the optimization is botched here.

Spoiler alert: It's still round :(

That's cool. I think though that framing the overall problem as starting from pushing and pulling an irregularly shaped object over irregular ground would be much more interesting. "Discovering" sliding, and "discovering" tipping and then rounding would be a nice hack. That seems like a reachable problem in today's RL if you score "the game" in terms of net exerted force, and if you represent both the shape of the ground and shape of the object as mutable ...

I intended to extend this to non-flat surfaces, but never got around to it. Wanted to input that rolling-square cycloidish shape and see if I find a square wheel as optimal. I should give it a go.

"How could ancient Sumerian wheel builders have calculated the ideal shape without modern computers and software? We may never know." I don't think figuring out the shape of the wheel was ever the problem. People have been moving huge stones by rolling them over logs since the dawn of time. Figuring out that the round thingy is of more use if you pierce it in the middle and let it rotate around an axle was, I think, the tougher nut to crack.

Hey hey, this is my blog! Thanks for posting it, @sillysaurusx.

Give this man his Ig Nobel Prize!

A better wheel is no wheel. Use superconductor based magnetic levitation instead!

Now do it in 3D to 1-up Shark Wheels.

Would be interesting to see the same in a 3D space

Assume a spherical cow ...

QED

... in a vacuum

...on a frictionless plane that goes to infinity.

Leave it to hacker news to reinvent the wheel without making it round.