How to Design the Perfect Shaped Wheel for Any Given Road

Last video, we looked at finding the ideal road for a square wheel to roll smoothly on, but what about other wheel shapes like polygons and ellipses? And what about the inverse problem: finding the ideal wheel to roll on any given road, such as a triangle wave?

=Chapters=
0:00 - Intro & Review
1:48 - Polygon Wheels
3:49 - Elliptical Wheel
5:30 - Focus-centered Ellipse
8:50 - Wheels From Roads
11:24 - How to Get a Closed Wheel
14:10 - The Many Wheels for a Sinewave
16:24 - The Wheel for a Triangle Wave Road
19:16 - The Wheel(s) for a Cycloid Road
20:24 - The Wheel for a Parabolic Road
20:58 - A Look Ahead and a Challenge

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Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive, I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:

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CREDITS

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Thank you for your support!

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The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:

[morphocular]

As some of you have noted, the shape I've been calling a "sawtooth" in this video is actually what's usually called a "triangle wave". Sorry about that! Clearly I am not an engineer.

EDIT: Also, I had no idea the pronunciation of "foci" was so contentious! My pronunciation is what I was taught growing up in the US, but evidently it's different elsewhere. Obviously the correct pronunciation is as follows:





"GIF"

[AbiGail-okfc]

I'd say that a "smooth ride" also implies that a constant rotation frequency of the axle leads to a constant speed forward.

[sarthaksharma]

I've always felt stupid with Maths since high school days. Still do. But things like these keeps the flame of curiosity going and help me study more. Thank You.

[wmbtngn]

Apple making proprietary roads for apple car

[sf_board]

never have i been so interested in geometry in my life. this guy taught me more in one video about shapes than i have ever known

[LeoStaley]

Hey, the next video is gonna be about gears! Yeah your videos are absolutely on par with 3b1b. I say this as an educational YouTube junkie.

[IONATVS]

The final challenge, two wheels rolling against each other, is the classic problem of Gear design—except you can change the shapes of BOTH curves to make the “ride” as “smooth” (in this case, where there is always a contact point applying constant torque throughout the rotation, so not exactly the same problem, but a related one) as possible, with the most popular solution for in-plane gearing is the spur gear, with repeating teeth made from 3 sections: the walls which actually perform the contact are made from involute curves (the curve traced by the end of a string as it held taught and unwound from a circular spool), connected piecewise with other short curves (the tips of the teeth and trenches between them), generally computed numerically like the elliptical integrals, to avoid the problem you mentioned with triangular and cardioid wheels—two involute curves too close to each other would not be able to roll over each other in practice without help, but by having a trench and a point in a different road-wheel pair of shapes you can ensure there is always a point of contact on the involute curves and any contacts through the other parts transfer no torque.

[nate]

I'm so glad we finally have a method for making the driving experience bearable in Oklahoma, thank you.

[realcygnus]

Priceless content ! Many years ago I was playing around with this(mostly just the regular polygon cases anyway), & there was VERY little information available about it, especially in one place. If there was, I never found it. I did eventually manage to solve those rather simple cases. But you took this lightyears beyond anything I ever even imagined, which is so cool. Really great work ! 👍

[thefullestcircle]

I'm now wondering if you could take a shape with rotational symmetry, find its road (using the point of symmetry as an axis), then adjust the depth of the road until you get a wheel that doesn't have rotational symmetry around its axis anymore to find a "prime" version of the shape.

Great video!

[ethanos]

Anyone else watching this because it showed up on recommended even though it’s not anything to do with your normal content recommendations?

[RinKin]

One note though - they are THEORETICALLY ideal wheels in IDEAL experimental environment.
Also it assumes that vehicle is PULLED by something along the road, while wheels just keep the vehicle horizontally stable.
If you calculate what road will be ideal for DRIVING wheel, the shape of road will be different.

[jmiki]

As for the questions at the end of the video: I didn't do the math, but I have a hunch, that the main difference would be that instead of the "vertical alignment property" it would be a "radial alignment property" meaning the axle and the contact point are collinear with the center of the road wheel. The other big change is, I think, that for the coordinate system of the road an other polar system would be useful instead of a cartesian one.

Great video, btw.

[NEKRATOS]

Omg, this video deserves millions of views, the maths and visuals are amazing! I wish you all the best and hope you’ll get the recognition you deserve! Less than 800 views right now is a crime! And when I started watching the video a few days ago it was less than half.

Someone from the future please leave a comment when this video reaches 100.000 at least!

Keep up the good work, I think we’ll see you among the big educational channels one day!

[rpyrat]

11:33 makes for a pretty fun screenshot when taken out of context

[_bman]

i love the style here! honestly, it gets kind of repetitive seeing the same 3b1b visual style on tons of math videos, you're putting effort into giving it a cozier feeling more fit to your own style of teaching and i am noticing and appreciating that effort!

[duncanrobertson]

I'd be interested to see the equivalent for shapes of constant width, where the definition of a 'smooth ride' is having the top of the shape, rather than some fixed point, travel horizontally.

[zxuiji]

2:31, Never thought I'd say this but I never wanna see a pair of testicles roll again

[droro]

Dude, that video was so cool. I never stop being amazed by the beauty of math and how complicated structures can arise from a very simple set of rules! Thank you for this content 🙏🏾

[mathematicalmachinery]

That 6:04 animation between “foci” and “focuses” got you a subscriber. That was cool.