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

[@morphocular]

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.

[@nyxalexandra-io]

yea

[@bopcity5785]

I hope you've seen the Cody dock rolling bridge which now applies this math

[@lukcurious]

7ýyy ÿt5vg 22:25

[@lukcurious]

Ÿtyfus y😅ÿyÿy y yt 24:55

[@lukcurious]

24:58

[@kindoflame]

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.

[@kfawell]

I thought of the same thing as I watched. And I imagined what it would be like to ride in such a car that's constantly jerking you forward and backwards. It made me laugh out loud. I think we would be used to a bumpy road going up and down. It would be somewhat tolerable at least. We experienced that walking and jogging for example. On the other hand, having our head jerked back and forth would be hilariously unpleasant or at least irritating. For example, as though somebody has grabbed our collar and is shaking us back and forth. I don't want to detract from the video. It was very enjoyable and solves the smoothness problem as defined.

[@fghsgh]

@@kfawell I've tried out one of those square wheel cars in a museum before. It was exactly like that.

[@kfawell]

@@fghsghI am laughing again thinking about that. Were you able to watch others doing that before you rode? If yes, I suppose you had to find out first hand. I just realized you had that memory while you watched the video. I wonder how you reacted when you saw the word smooth. I really appreciate that the creator specifically defined smooth. Thank you for telling me.

[@fghsgh]

@@kfawell I mean, you had to pedal yourself forward, and it was pretty slow so not too bad. It also mostly felt like variable resistance, not so much speed (because that's how inertia works). But yeah it seemed like it would not be entirely smooth from seeing others too. This was also at least 8 years ago so although my memory is pretty good, I can't give an exact description of the scene ;). But anyway I thought the lack-of-smoothness was just from the physical thing being imperfect, until this comment said otherwise.

[@Zildawolf]

Well now I’m wondering what’s the shape that’d make the most speed inconsistency possible lol

[@cambridgehathaway3367]

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.

[@AsiccAP]

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

[@aartvb9443]

You didn't gain brain cells, you gained connections between brain cells ;)

[@catcritical]

*Brain.exe has stopped working.*

[@wildcard_772]

Same

[@EverythingLvl]

It's an illusion, still super dum

[@BloonMan137]

@@aartvb9443 🤓

[@juanroldan529]

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!

[@morphocular]

That's great! I'm so glad you found these videos so valuable. One of my hopes for this channel was to inspire others to learn and love math, so it pleases me deeply to be succeeding in that. I wish you the best on your continuing studies :)

[@redtortoise]

@@morphocular first

[@maxwellhavoc6996]

@@redtortoise I am confused by what you are trying to say.

[@mohre401]

Hi, is this even a differential equation? To me a differential equation is an equation in which the function itself and one or more derivatives appear which isn't the case in this example

[@juanroldan529]

@@mohre401 have you even watched the video?

[@thomasrosebrough9062]

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!

[@ethansmith876]

Saarinen my beloved

[@csar07.]

You ascend to a new level when you get your own favourite mathematical curve

[@pulli23]

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.

[@mujtabaalam5907]

We can assume a powerful motor is spinning the wheel on a fixed gear system so the wheel's mass doesn't effect the motion

[@whoisgliese]

@@mujtabaalam5907 epic lateral thinking thanks

[@gcewing]

You can always achieve that by weighting the wheel appropriately, so it's not a constraint on the wheel's shape.

[@johnmount5487]

That “force” exists even if the axle is at the center of mass. If the wheel is rotating at a constant angular speed the horizontal speed is by definition not constant (changing by a factor or r).the effect is exaggerated as the axle is moved away from the axle as the extremes of the bounds of the radius get larger. The wheels horizontal speed, speeds up and slows down constantly throughout its travel for any shape other than a circle

[@aaaab384]

its*

[@alriktimo644]

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 👍

[@meade6291]

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.

[@eventhisidistaken]

Sure, but 'continuous force' was not specified. Yes, I'm an engineer.

[@ob_stacle]

and if there's any wheelspin at all you'll be on the worst road in existance

[@meade6291]

@@ob_stacle holy shite I hadn't thought about that

[@afoxwithahat7846]

I think you switched the axis, the axles aren't moving vertically at all.

[@meade6291]

@@afoxwithahat7846 yep, and I teach coordinate plane. Shame on me

[@sozo8537]

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

[@RFVisionary]

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

[@ineedtogetoutmore1848]

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

[@mullactalk]

This channel is a hidden gem of maths KZbin

[@drmathochist06]

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.

[@enbyarchmage]

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? 🥰

[@AJMansfield1]

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 }

[@AJMansfield1]

(In fact, you can choose *any* θ(t) parameterization you want, and just use r(t) = sec(θ(t)) to get a straight line for whatever speed you rotate the wheel at.)

[@Chariotuber]

This is interesting. I suppose you can get from your parameterization to his by the change of variables t → tan(t'). I wonder if this freedom of parameterization has any physical meaning.

[@AJMansfield1]

@@Chariotuber I went and simulated it, and the resulting road curves *are* actually different from each other.

[@Chariotuber]

@@AJMansfield1 Oh, how did you simulate it? On my end, starting with your parameterization, I ended up doing the standard integral of sec(t) which is ln(|sec(t)+tan(t)|). I then plotted this parametrically on Desmos (typing in "(ln(|sec(t)+tan(t)|),-sec(t))" on the first line) with the domain [-π/2,π/2] for t. It already looked close to the catenary shape. But to make sure, on the 2nd line I put in his solution of y=-cosh(x), and the curves stack on top of each other rather exactly.

[@dyld921]

The parametrization of the road would change, but the shape (x-y relationship) wouldn't.

[@dj_laundry_list]

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

[@AlmondAxis987]

I actually watched this last year when I was in 8th grade. I didnt understand anything, obviously. And now, after learning basic calculus from youtube, it makes SO much more sense! Also, I want to say that the visual building of the road section is BEAUTIFUL.

[@philosophymikebill]

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.

[@morphocular]

I actually used my own homemade software to make the animations in this video. You can find the software here if you want to play with it: github.com/morpho-matters/morpholib However, it's still largely just a personal project and the documentation is rather sparse. A more well-established and popular tool for making similar animations is called Manim, which you can find here: www.manim.community/ Hope this helps :)

[@philosophymikebill]

@@morphocular I really appreciate the advice and even sharing your program! Thank you for getting back to me

[@alexv1129]

@@morphocular Math is interesting and fun - but I am subbing because of this right here. Amazing of you to be so kind and helpful. Good luck, creator!

[@ianhickey3423]

@@morphocular This is so unbelievably cool

[@danelyn.1374]

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

[@LunaAlphaKretin]

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.

[@WaluigiisthekingASmith]

The second equation says dx/dx =rdtheta/dt. Differentiating a second time d^2x/dt^2= dr/dt dtheta/dt +r dtheta^2/dt^2. Given your restriction dr/dt dtheta/dt = -r dtheta^2/dt^2. Thus r'/r =u'/u. Doing what any good physicist would do and pretending we can just cancel our differentials like fractions, we get ln(r *dtheta)= c and thus dtheta/dt =c/r

[@joaogiorgini1326]

Make velocity constant with constant rotational speed? In other words, dx/dt=cte and d0/dt=cte. Meaning, in the second equation, r must also be a constant. In other words, the only shape that satisfies a truly smooth ride is a circle.

[@bears7777777]

I think the only way this would be possible would be to allow wheel slip. The amount of slip would be the fastest angular speed - slowest angular speed. The slip would have to occur when the point of contact is farther than the minimum. For the square, this would be when the point of contact tends towards the corners as they are farther from the center then the center of a side. I’m not sure that’s even solvable though

[@scifiordie]

Nobody cares bro get a life

[@_Mike57_]

it's all fun and games until you have to turn

[@bloomp7999]

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

[@LoganCralle]

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

[@gergonagy846]

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

[@officiallyaninja]

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

[@Happy_Abe]

In time we’ll get this channel there

[@SophiaBrouchoud-se1ht]

Who needs to spend thousands of dollars on therapy when you have this guy and his wheels? This genuenly sooths my brain and I love to learn things like this so yippy!

[@tsar_asterov17]

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

[@bifidoc]

Thanks!

[@morphocular]

Thanks for watching!

[@saintgermain6694]

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

[@amaarquadri]

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

[@RFVisionary]

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

[@DitieBun]

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

[@TheCynicalOne]

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.

[@DonkoXI]

The proof he gave is actually pretty clean all things considered. If you are interested in understanding it, I highly recommend looking through it and trying to understand his reasoning one step at a time. You can ignore the algebraic details at first, but try to understand the concepts in the argument. If you understand the way complex numbers work well enough, it should all be pretty intuitive with some time. If you don't feel very comfortable with how complex numbers work, then stopping and thinking about each detail of this proof will actually be a pretty good way to get a better understanding of how they work. What feels clean to me is of course subjective though.

[@vikn331]

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

[@小绿苦力怕]

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"?

[@geckoanims]

True

[@the25thdoctor]

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

[@stuartl7761]

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.

[@morphocular]

A good hint that complex numbers might help is if your problem involves 2D rotation or 2D rotational symmetry. That's where complex numbers often come in handy!

[@TerrifyingBird]

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.

[@lenskihe]

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 😂

[@deathpigeon2]

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.

[@klikkolee]

We are used to vehicles which are propelled by the wheels. However, if the vehicle is moved by means unrelated to its wheels, then the criterion in the video is sufficient. For vehicles which are wheel-propelled, unless a fanciful control system regulates the wheel speed, your additional criterion is required to make the vehicle feel subjectively smooth to a real human occupant. The no-slip condition (stationary rim principle in this video) does *not* guarantee your criterion. The r in the no-slip equation is a function of t. Your criterion is only consistent with the no-slip condition if the radius is constant -- meaning a circular wheel.

[@deathpigeon2]

@@klikkolee ...Right. I was thinking it'd ensure 0 jerk because it ensures that the rotational velocity at the touching point and the forward velocity at the axel are the same, but, for constant torque, the velocity at the touching point would be in part a function of the distance from the axel so you *need* at least some slipping to ensure a smooth ride unless you have a constant distance from the axel (ie being a circle as you said).

[@Nuclear868]

What if, in case of a car, we make the distance between the front and the rear wheels such that front and rear wheels are offset - when the front wheels have the highest angular speed, the rear wheels have the lowest angular speed? Yes, they will not cancel out completely, but will reduce the 'jerk' feeling.

[@eventhisidistaken]

Who said the torque had to be constant? Stop trying to impose your roundism on the rest of us.

[@klikkolee]

@@eventhisidistaken It would be a substantial engineering challenge to create a vehicle where the torque applied by the wheels varies in perfect concert with the road shape. Without that perfection, a wheel-propelled vehicle can't have a smooth ride on an extreme road without slipping.

[@ДаниилРабинович-б9п]

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.

[@kindlin]

When he said it was _really hard to prove_ I was confused, as this is the only motion available due to the no-slip-condition and the rigid body motion. But honestly, the statement of the question itself is almost the proof of the question. You want to figure out how to prove that all points on the wheel move periductular around the contact point, well, proof by exhaustion, there are no other ways it could move around the contact point but to pivot, and the definition of pivoting, as noted in this video, is perpendicular motion about a point.

[@k7iq]

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

[@brucea9871]

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.)

[@H3xx1st]

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

[@mateuszbaginski5075]

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.

[@pastadcasta]

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! ^^

[@steffahn]

A square wheel rolled over a flat surface will actually just pivot around each of the 4 corners. Thus, the axle would take a path composed of a series of arcs (i.e. sections of the perimeter of a circle), which is definitely *different* from the series of catenaries that are shown in this video to be the shape of road that you need.

[@iskallman5706]

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

[@tracy449]

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?

[@morphocular]

Thanks for watching! To answer your question: Not necessarily! The second Road-Wheel equation says the axle's velocity is dx/dt = r dθ/dt, where dθ/dt is the angular velocity. So the only way both the axle velocity and the angular velocity can be constant is if the wheel has a constant radius, meaning this will only happen for the case of a circular wheel.

[@TheHuesSciTech]

@@morphocular Fascinating -- I *believe* an involute rack and pinion has the property of dx/dt = k dθ/dt, where k is a fixed property of a given gear (the radius of the gear's "pitch circle", or half the "pitch diameter", perhaps?). This would appear to contradict the statement you made above, but I believe that might be because you're assuming no slippage between the wheel and road in your video, whereas an involute rack and pinion does have slippage?

[@cheshire1]

@@TheHuesSciTech The equation you gave is approximately true, since a gear is pretty close to a circle.

[@TheHuesSciTech]

@@cheshire1 It's approximately true for all gears, yes. But I believe it's *precisely* true for an involute gear. (Neglecting real-world clearances and manufacturing tolerances, of course.)

[@cheshire1]

​@@TheHuesSciTech You may be right, involute gears do have slippage (and the contact point jumps around instead of staying on a vertical line), so the argument from the video doesn't work in their case.

[@adrianmisak07]

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

[@phlapjakz]

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

[@joeeeee8738]

At 11:04 you put the curve down, as if it's the road, and that's what get you the minus sign for y in the differential. But if this road was also the "ceiling", then we would have a plus sign for y. Wouldn't this affect significantly the curve if we had a + ?

[@sriramn1809]

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

[@blackboxlearning]

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 :)

[@KOZMOuvBORG]

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?

[@TheGiuse45]

This was a question on the high school finals in italy a few years back

[@duncanhw]

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?

[@jursamaj]

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.

[@blaze8862]

I haven't learned trig yet and I've only slightly touched on graphing, yet I watched a 30 minute video on the topic, and I loved every last minute of it

[@lamenwatch1877]

Let's make an ATV that dynamically morphs it's wheel shape for the terrain. EDIT: 27:13 But what is the factorial of negative "ish"?

[@ultimagnum3511]

Is it just me, or is the beginning cut off?

[@MelonKingdomOfficial]

I noticed that also

[@Happy_Abe]

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

[@morphocular]

Man! Nothing gets past you guys! :) Yes, technically a +C belongs there, but all it will do is shift the road forward or backward; it won't affect the actual shape of the road, which is what we were after. So I picked C = 0 in order to get the simplest and cleanest possible final answer.

[@Happy_Abe]

@@morphocular Thanks! Yeah I didn’t think it was a problem for the solution, I just thought it should have been addressed. Amazing video, can’t wait for your channel to explode in popularity!😊

[@manifaridi9200]

I hate mathematics but man... look how beautiful it is.

[@WAMTAT]

You don't hate math, you just hate how it was taught to you.

[@blackbeast9268]

Trust me nobody hates math. I used to have E and was on my way to F but then i moved school and my teacher was amazing and i got A because he explained everything so well and got me motivated. Math is a language with rules and if you're teacher doesn't explain the rules in details it will be boring because you rely on common methods and formulas instead of understanding why they work. It's very fun and i would argue chemistry or physics are much harder then any math expect super high level .

[@Adam-pj2qh]

thats so sick, finally some applied mathemathics!!!

[@OhGoodGreef55]

i know this is completely off topic but whats the song at 9:01, its not said in the discription

[@UrielDávilaOrtega]

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

[@Error-xl3ty]

Videos like this are why I love math

[@vitorguilhermecoutinhodeba3253]

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!

[@miguelcabaero5843]

I love the production quality

[@thirockerr]

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.

[@user-hd9oh9bk8b]

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

[@iwersonsch5131]

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?

[@argolake8623]

I find myself wanting dx/dθ to be constant. So if I am a driver and pressing the gas so my wheel rotates at a constant rate, then the horizontal speed of the car does not oscillate like it does at 16:00. The equations shown didn’t leave any free parameters for creativity, so is there a class of shapes that have this property? Is the only member of that class a circle?

[@bogiokas7426]

Regarding the proof at 6:00: Fixing a unit parametrization for f and defining w(s) as position of a point on the wheel at the time s, when f(s) is exactly the contact point, already parametrizes w with a parametrization that may not be of unit speed. So, I don't think you can always satisfy all three constraints at the same time.

[@diribigal]

Minor point: around 13:55, I think the distance would be -y(t), not y(t). Edit: Though you clarify this about a minute later. Also, around 26:14, was your step size dt or actually dθ?

[@morphocular]

I think in that particular animation, to make my life easier, I actually used t = θ, meaning dt = dθ, but that's not required. The main point is that as dt approaches 0, so will dθ, and so the segment sequence will converge to the true curve even if the wheel is parametrized differently.

[@vicslav4030]

aye Buddy thats aboot the moost accurate film aye seen in ayewhile friend. oh Canadaa a great and noble land oh Canada we stand on your glory

[@archie1490]

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

[@agy3256]

This video is pure gold

[@johnnyvishnevskiy8090]

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

[@josiphanak9130]

noone cares...youtube is in 2D

[@arkdotgif]

13:30 WELCOME TO WHEEL SPACE. YOU HAVE BEEN LIVING HERE FOR AS LONG AS YOU CAN REMEMBER.

[@Josephi_Krakowski]

These are the type of videos I watch at 3 AM

[@jorgec98]

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

[@ZotyLisu]

this should have way more views

[@debmalyalodh1]

1:35 IDC what anyone says but THAT INTRO SONG IS FIRE!!!!!!!!!!!!!!!!!!111111

[@g10royalle]

The animations are so satisfying

[@Smitology]

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.

[@WeeIrishLaddie1]

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

[@ANZEMusic]

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

[@pietandersen6120]

Quick question, but wouldn’t y(t)

[@nathansos8480]

so the road would have gaps, right?

[@rebeccastevens2903]

Wheels are paradise. The wheel groups form paradise!

[@badpriestess_]

this video gave me calculus PTSD flashbacks. loved it

[@MF-dz7cp]

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

[@negi6071]

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

[@HelPfeffer]

Some tips if I cannot integrate to obtain x? I'm trying to get the ideal road for a Cubic Bézier curve, so I can make more complicated shapes, and to integrate I try using python and the library SymPy, but it just doesn't integrate finish integrating Update: I achieved it by taking little steps and computing

[@learningFalcon108]

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

[@kinkinawesome]

Exited for the next videos!

[@bitroix_]

This is an amazing video! Thank you.

[@elitestryker5709]

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

[@arlyu606]

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

[@VagueHandWaving]

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)