The Perfect Road for a Square Wheel and How to Design It

There seem to be a lot of comments questioning the practicality/usefulness of square wheels, particularly whether you can turn side-to-side with them. The short answer is there's likely not much practical use for them and you can't turn side-to-side. To be clear, this video was mainly meant to be an interesting application of math and geometry to a fun problem and was not meant to be practical in the slightest.

I was going to mention that a second requirement for a smooth ride is that when the rotational speed of the wheel is constant, then horizontal speed of the axle is also constant. Otherwise, you could have a 'smooth ride' where the car constantly speeds up and stops short even when the wheels are not accelerating. However, the equation dx = r*d(theta) very simply shows that the only shape that could satisfy this new condition is a circle.

We live in an astoundingly amazing age. One person is able to singlehandedly write, animate, narrate and publish such a polished, professional, easy to understand, and intriguing video. not to mention doing all the math and even providing a formal proof they crafted themselves. Such incredible talent has existed in past ages (rare tho it may be), but never before has the common man been able to so easily and readily benefit from it. I am astounded and humbled and grateful.

I feel like I gained brain cells despite not understanding a word

22:12 super hype to see my favorite curve show up in this video!! A Catenary Curve is also very commonly used in architecture for its even distribution of weight/pressure.
The most famous catenary curve is the St Louis Arch which is over 600ft tall! It differs from the identity curve by having 0.01 in each exponent of e, as well as multiplying the entire equation by -68.8, resulting in a curve almost exactly as wide as it is tall!

Awesome video! It's been just a few days since I have fallen in the rabbit hole of differential equations. I must say that I love your videos and that they inspire me to keep on improving and learning. Thank you!

I'm late: but there's a single also important point to make a "square wheel" work. The very point that needs to stay at the height also needs to be the center of mass. Otherwise a wheel would give a force rolling back/forward during part of it's movement.

The flaw in this is a vehicle with a continuous force applied through is engine to the axle wouldn't experience bumps in the x axis, but it would experience lurches and lags in it's movement on the y axis. Therefore it still would not be a comfortable drive unless the wheels rotational speed was constantly adjusted.

When I watched this video, I just realised that my intuition is strong that without even a mathematical description I can jump to the right conclusion, but at the same time I realised I lacked the ability to articulate since I didn't understand it mathematically or completely realising the fact that how this is so or 'How come?' in simple terms.
I need to strengthen my mathematical comprehension of data into equations and other methods.
Thanks 👍

Great tutorial. Good didactic structure. Instructive, helpful and optically "super nice" to look at.

This video made me love catenaries even more, and I already considered them one of my favorite curves of all time! 🤩
I like catenaries bc they appear everywhere, from the Brachistochrone problem to architecture. For instance, Catalan architect Antoni Gaudi took pictures of carefully arranged sets of hanging chains and turned them upside down to model the structure of the most famous church he designed, bc upside-down catenaries make EXTREMELY stable arches. Isn't that beautiful? 🥰

Great video! You took an idea that seemed complicated at first and explained it so well that it seemed almost obvious in hindsight.

I feel like I've stumbled onto a video about a question that I never had in mind, and, along with an amazing explanation of the entire problem, has given me a solution that I am really satisfied by and solves that problem?
plus the explanation is amazing so like, mad props

I never expected it to be that intuitive! Thanks for the really really great video.

The dopamine hit i got when i successfully calculated the equation of the road was something else. I thank you for presenting this problem.

Maybe you get to this later, but the "stationary rim property" also follows from the pivot principle. When the point in the wheel is the contact point itself, then any line through that point can do for the reference line in the orthogonal motion property. Only one possible velocity could be orthogonal to every line: 0.

I deeply agree with your channel description and the Poincaré quote, i'm in for what you do, keep the good work !

great tutorial. good didactic structure. instructive, helpful and optically "super nice" to look at.

Incredible video. I just took a dynamics course at university and I learned so much. This is an incredible application of maths. Bravo 👏

You explained that beautifully! I am definitely looking forward to your future videos.

19:45 It seems like the road shape depends on how you parameterize the wheel's rotation then -- the function I always instinctively reach for when parameterizing straight lines in polar coordinates is the secant function, and I'd have written that line as { r(t) = sec(t), θ(t) = t }

Do you mind if I ask what programs/language/code you used to make this video? I'm attempting to learn this sort of simulation, but I'm not sure where to start.
Thank you for making these videos. I've been trying to figure out this topic in my head for several years and this is the first meaningful insight I've come across in a good long while.

Great video! Love how you started by making the equations and then deriving the shape from them! Can't wait for the next video.
also, wouldn't the wine glasses in the thumbnail be knocked forward/backward due the second law of road-wheel motion?

4:15
This is the most insane wheel I've ever seen, and I'm here for it

Nice video ! I would be interested to see how you would present the optimal road shape taking into account a specific mass for the wheel, the gravitationnal force.

This channel is a hidden gem of maths KRplus

First video ive seen on this channel.
Wondering why youtube took so long to recommend me stuff from here.
This channel is amazing!

fantastic video… cant even express how impressive this is to me, I try to do math recreationally after getting my masters in applied math…

What the hell is this? It's awesome. I think it would be more complete/satisfying to state that the vertical alignment property relies on shapes being convex, but honestly this is one of the best math(s) videos i've seen for a while

I was hoping to make a video on this exact topic, but I guess it has already been beautifully covered by this channel. While checking for that, I came across this channel and I love the animations and their interactivity. Already subbed. Expect a video soon covering more stuff, cus I'm not leaving the idea :)

Very interesting video and analysis. I'll be watching more of your videos. This one reminds me of an old comic strip. It was called BC and based in prehistoric times. Their only form of transportation (other than walking) was what they called the wheel. It was a circular wheel with an axle through the centre and they stood on the axle to ride the wheel. (How they propelled it - especially uphill - is beyond me.) In one of the strip's comics (presumably before they thought of using circular wheels and hence only had square wheels) one character declares to another he has derived an improvement to the square wheel and produces a triangular wheel. "Improvement?", the second character says, confused. The first character replied "It eliminates one bump". But of course if they designed their roads as you specified they could actually have square or triangular wheels with no bumps. (Somehow I think it would be easier to come up with a circular wheel.)

I'm safe to say, that this is the most engaging video that I've ever watched.

Watching the shape of that ellipse move around makes me wonder if that visual perspective doesn’t unlock a thought on how to attack the unknown equation of the perimeter. For those of us who are visual, this was absolutely gorgeous to watch.

This is an amazing video! Thank you.

This is a really good video. The math is fascinating, and you present it clearly with exceptional visuals, and I greatly appreciate it

very nice video, i really enjoyed the small steps taken each time to get to the answer. and even then, there's so much more to discover! well presented and paced, didn't feel like half an hour. your consistent use of both visual and verbal explanations for each new idea is great.

Great video, I like to wonder what this would look like in practise, if someone were to try this in the real world, but of course there would be a great deal of other things to consider

I love the production quality

I want an entire video, or at least a short, dedicated to the orthogonal movement principle. It’s a mess and I want to dive in with full understanding! Great video about the wheels too. I feel like many of the wheels shown would slip a lot on their roads, so I guess the dream of bumpy square wheeled roads is a long shot lol.

that “Pivotal Role” pun at 11:14 was painful, well done

This is the perfect example of "I have no idea what this man is talking about, but I like it"

I can't really point to what it is in your videos that makes them one of the best I discovered through 3B1B's SoME. Whatever it is, you are grokking it, man.

it always amazes me how e manages shows up everywhere even when the problem looks like it has nothing to do with it

While a flat ride is certainly an important thing for a smooth ride, I'm not convinced it's sufficient. It seems reasonable to describe a jerky ride as also a non-smooth ride. That is to say, given a constant torque applied to the wheel, the third derivative (the jerk) of the forward motion produced by the wheel spinning should be precisely equal to zero.
Put another way, a linear acceleration of the rotation of the wheel should produce linear forward acceleration for the whole system.
Now, I think the stationary rim principle should be sufficient to ensure that this is the case because it ensures that the rim speed and the axle speed are equal, but I think it'd be insufficient to consider only the flatness of a ride to determine if it's properly a smooth ride.

I reeeally love your content. Thank you for all your videos :-)

I am pretty sure you can easily derive the pivot principle from the fact the contact point is stationary:
observation 1: the wheel is a 2D rigid body, so its motion is fully described by horizontal speed, vertical speed, and rotational speed, so it has 3 degrees of freedom.
observation 2: the constraint that the contact point is stationary restricts 2 degrees of freedom, thus leaving 1 degree of freedom.
observation 3: pivoting motion satisfies the stationary contact point constraints and has 1 degree of freedom.
therefore pivoting motion is the only possible way to satisfy the stationary contact point constraint.

I'm curious what would happen if you impose the additional restriction of making the axle's horizontal speed (and, hence, velocity) constant (given constant rotation speed). I noticed the speed seemed to vary a lot with that particularly arbitrary-shaped wheel example at 4:18, which would probably be a disconcerting experience as a driver. Still I imagine the answer is that you can't have a road that does both - to prevent a change in horizontal speed you'd probably need a different road that causes vertical changes. What if we just say "constant velocity", allowing the vertical position of the axle to change as long as it feels like a smooth slope would for a circle-wheeled driver. I don't know how that would go, but it feels more likely to be possible.

This video is amazing, and all of his videos, ngl are basically 3b1b on light mode

Awesome 👍 I tried to solve this problem on my own once. I'm glad I watched this video, because now I know that I would never have been able to do it 😂

amaizing! just amazing video! Thanks

thats so sick, finally some applied mathemathics!!!

This problem (or rather a simpler version of the problem) ended up in an italian high school final exam, in 2017. It is to this day one of the most iconic problems to ever appear on the test.

it's honestly fascinating how many titles and thumbnails this exact video's had. i've heard about this but never gotten to see it firsthand

The animations are so satisfying

The way you use - I assume - MAnim is absolutely outstanding. I bet you come to understand every concept you explain in an incredible depth as you code these. Really impressive!

So good explanation!

Videos like this are why I love math

The horizontal motion of the axle is necessary for a smooth ride, but not sufficient. It needs to be *smooth* horizontal motion, not jerking forward and back. That, in turn, requires the wheels to rotate at highly variable speed. But that's not how driven axles tend to work.
Additionally, when the wheel is moving up the slope, the wheel will be moving too fast at any given moment. Combined with the uphill configuration, you basically guarantee slippage. You face a similar problem on the downhill side, but inverse. Those novelty tourist attractions tend to reduce both these effects by having the front & back wheels be exactly a half wave out of phase. That way the slippage either way hopefully cancels out, and one axle can be speeding up while the other is slowing down.

This is as good as mathematics vidéos get. The pinacle.

6:10 I love that the first and last terms cancelled happily :D
Loved the proof too, I must remember to check through if complex numbers might help when I come across a problem.

I'd be interested in a sister video where "smooth" was defined as "constant velocity" rather than "constant axel height", ie changing the axel height in the wheel as it rolls to keep it moving horizontally at constant speed

this was a very enjoyable video

KEEP IT UP BRO U ARE DOING GREAT WORK ❤❤❤❤❤

This is a beautiful video.

This is fantastic ! 2 + 2 = 5 for large values of 2
But would a square wheel do good in snow or maybe even ice ?

What I love about this is, it has a simple answer. Think gears, and a gear rack. But is far more complicated to preform

this video is so good. its criminal that you don't have hundreds of thousands of subs

This video is pure gold

If you want a non circular wheel that moves with constant speed, you can give the wheel a non uniform mass density such that when the wheel would slow down, the part on the bottom that is moving slower is made more massive. It's momentum is transfered to the entire wheel body, maintaining a constant velocity.
Most likely the mass distribution would be such that every dTheta slice around the axle has the same mass regardless of radius.
(Constant moment of inertia)

It is a nice video, even though I think some properties have different names in here. Instant center of rotation is the center (no pun intended) of all this procedure, and wasn’t mentioned. The animations were very good!

this should have way more views

I know I'm late, but this is really good!

I have a way I like to think about it, if you take the path that the axle takes when the shape is rolled continuously over a flat surface, and use that for the road surface, the shape will roll smoothly. It's cool to see the algebraic representation of that though.
Very cool video! ^^

Exited for the next videos!

I'm kinda proud of myself I grasped the first analytical definition more easily than the second visual one

Awesome!!! You used math and physics so cool

this is an amazing video, it went much more in depth that i thought it would and im so glad for that, 10/10

I'm a sophomore in high school so I have no clue what this video is talking about but it's still interesting

Its so much more understandable than PHysics I and Technischemechanik at ETH together eith explaining all the concepts

Great video. You've earned a subscriber

Thanks for the video. I learned a lot. Also, I have a question: If the axle moves at a constant velocity, does the wheel rotate with a constant angular velocity?

Is there another way to relate the coordinate x from the road with the r an theta ecuations beside the derivatives?
I think it would be interesting to see it in other way, but I can't do it

When describing the small timesteps in the visual prrof at the end, maybe it would be more specific using a Δt->dt for smaller and smaller time steps :) But great video! really liked it!

I have absolutely no idea what any of this means but I find it interesting

This was a nice brain teaser before I go off to do maths at uni. GL everyone off to uni in Septemember!

These are the type of videos I watch at 3 AM

Why isn’t x equal to the hyperbolic sin plus a constant because we integrated
Where’s the +c

While I don’t understand 80% of what I’ve been told here, I did finally understand the purpose of imaginary numbers here.
I’ve struggled through so many math classes which could never just explain it so effectively.

When you try to stop the wheel, wouldnt it be unable to stop at some certain points though? For example, for the square wheel, if you trying to stop just a bit further forward from the x value of top of a "bump", wouldnt the wheel roll toward the next "ditch"?

Very nice and interesting video

This is awesome

Assuming lateral motion in respect to the axle is constant, wouldn't that mean the rotation of the 'square wheel' varies depending on which part of the catenary is in contact with it? The radius of the corners being √2 (1.414) more than on the middle of its sides. If rotation was constant, wouldn't lateral motion undulate?

Nice to meet you Grant Anderson Junior!

Great video! I just have a question though, during the proof for the orthogonal motion property, why must the road and wheel path functions be complex?

I'm more interested in how this transfers over to 3 dimensions and how turning affects how the shape of the road is made.

watch more and learn more your are great thanks.

Does the axle move without accelerating if no kinetic energy is put into or out of the system? And if the general answer is no, for which shapes does it behave in that way?

Can you explain stationary rim property because if it doesn't move when it touches ground it will never move.

great video, good job :D

Can the pivot principle also be explained as being necessary from the no-slip principle and the fact that the shape never deforms ie all distances stay the same?
If all distances stay the same, then the rate of change of the distance between a particular point and the contact point must be 0.
Therefore, the displacement can only change direction, not magnitude.
Therefore the rate of change of displacement (aka relative velocity) must be perpendicular to the displacement vector as that's the condition for circular motion
Therefore because of the no-slip principle, the absolute velocity of that point must be perpendicular to the displacement vector.