I have a physics PhD and I never heard of the tractrix. I’m glad you made this video because I learned something. Very well done and please keep it up!
Isn't this an Irodov problem? This feels like an Irodov problem! This is like bringing back some massive nostalgia from my college entrance prep days ❤
I love Calculus. Dog will be infinitely close to owner path but never touches. Your visual method of explaining is very appealing, I imagine it would be interesting for young audience.
Wow, that's a question! Since the first dog isn't moving in a straight line, the second one can't be moving in a tractrix. The angle of the second leash would lag behind the first, so the super-tractrix must be steeper than the original. Whatever the case, the area between the two dogs must be the same as the tractrix. But now I want to know what happens as you approach many dogs with infinitesimal leashes!
@@physicsforthebirds you should probably know that dogs do not do a tractrix they go where they want whether it's nearby, ahead of you or even using the leash to make a tangled mess
I’ve been slowly going through your videos this past week, and I absolutely love all of them. Please keep up your content, I love how you focus on explaining through the theoretical/contextual lens rather than force feeding all of the practical information and hard numbers
I can totally see Matt Parker calculating Pi from the trails of a bicycle for his next Pi-day video. Interesting video, and very nice visual explanations!
Love it! You'd think after years of roaming around the maths youtube you wouldn't find any more thingy that's both simple and full of secrets but here we are... And oh such a lovely explanation too, congrats!
this video is incredibly easy to follow even though its describing more advanced mathmatical concepts, and its fun to look at! definitely dererves more subs
my favorite fun fact about spheres and pseudospheres is the right angles in a "square." on a plain (surface with constant 0 curvature) a square is shape made entire of right angles and it has 4 sides. on a sphere, a shape made of only right angles would be 3 sides (equator, up to north pole, turn, back down to equator, turn, back to where you started), but on a hypersphere, an only-right-angled shape would have 5 sides! pretty cool
We can view any closed shape polygon is just a special form of a circle, a circle which has edge length > 0, thus made of finite number of line segments, unlike a circle which can be thought of being made of infinite line segments, thus the smooth shape. Once you realise this, all paths that end up at the start, traced by a bike, are just forming concentric circles in the end. Just special forms of circles.
I remember that old youtube video about turning a sphere inside out... IIRC it said that turning a 2d circle inside out is not possible. Well, well, well...
Your videos are great but you should really do something about your microphone or audio processing. Those *cs* sounds (not sure how else to describe them) are really annoying and make it difficult to watch for me.
i wanted to know what shape you would walk in if you're walking at the same speed as another person who is walking in a straight line and you're always walking towards them, and i think this is the same problem, so yea!
It does depend on *some* physics. If you did this experiment in zero g vacuum the dog will end up rotating around you. This is obvious if you view the problem from the human's (inertial) frame
Yeah, if the front tire turns around in a small enough radius then the back tire will make a sharp stop and then start moving backwards. At 7:34, the back tire (green) was moving backwards between those two sharp points and the front tire was always moving forwards.
In the original question it doesn't state the dog needs to be facing the owner, so couldn't the dog just walk to the right at the same pace as the human and draw a straight line? lol
I don't like that you chop off all the weird angles of the rearranged triangles to make a circle. That feels to me like a step is missing and it makes it harder to be convinced by the explanation. Where do all those bits go?? How is that a circle!
Visual calculus wasn't Mamikon's, it is literally what Newton was using that lead to the development of calculus itself. That is where the idea of a limit on approach to infinity itself came from. 100% the same thing. The visual part came before the math. Did other calculus classes not cover this?
I’m just flabbergasted at how good your videos are. Please keep going. First, your casual style really minimizes completely in a beautiful way. It’s just so simple!! Meaning that you introduce complexity in such a natural way that it doesn’t feel complex at all. Second, and following from the first, I just feel so happily, jubilantly surprised when you use a simple example to describe a complex result. The bike path metaphor leading to 2pi just shocked me in the best way. I love it. As a casual math geek, I just love how you present complex ideas without getting too deep in the details. I don’t care about the details, I care about the joy of the relationships, and you do that just so wonderfully.
The factoid about area traced out while riding a bike is delightful. And your illustrations to support the idea were perfect! It really made it click for me, and I think the idea will stick with me every time I ride a bike
I'm in SHOCK that you don't have as much subscribers as i thought you would. Your visuals, audio, script and overall production quality are incredible. Keep up the work!!!! Love it. 💕
The whole analogy with the bike is actually used in safety measures in cars. We always suggest you have your best tires in the back; even if you have a front wheel drive car. The reason is if you hydroplane in the read, but not the front, the rear tires new angle (total toe/ thrust angle) will dictate where you go. On the other hand if your front tires hydroplane your steer ahead will change and this is much easier to recover. Along with that, if you step on the brakes and your rear tires hydroplane, your rear end will be pushed forward making your thrust angle exponentially greater. If your front tires hydroplane and you brake, the front end will pull on the vehicle which will not put you into a uncontrolled drift of over 45 degrees. A 45 degree spin out is much easier than a 360 spin out. These both are extremes, but we’ve seen you guys drive… I’m amazed anyone is alive to be honest. If you want more details, I’d love to share them. I’m alignment certified by Hunter Engineering, and that’s why I may use fancy terms like “total toe, steer ahead, thrust angle, etc.” and if you need me to explain, I’m happy to. I just will say that it’s easier to show than it is to tell. After all, it is geometry… and physics. Stuff is hard to explain with visuals, but is much harder using only words.
That's actually a very relevant example; some of this same bike track math has applications in parking control systems for autonomous vehicles with trailers.
Just stumbled upon this channel in my feed. Really like the way you explain stuff! Another example I found it useful about this tractrix curve is on CNC drag knife. It works just like the bike example! One question though, when you mention that differential equation (DE), I don't think that is a DE? It looks like an ordinary integration problem because from what I know it has to have a "y" Variable in the equation for it to be DE. I hope someone can confirm or tell why if it is otherwise. I'm kinda new to DE stuff... Thanks!
The author has an unfortunate habit of lying in their video titles. They made another video the title of which claimed that the universe has negative curvature, but the video actually concluded that the universe is flat.
I'm leaving a comment here to let you know that I am shocked by how well you transitioned between seemingly irrelevant yet logically relevant concepts across applications in different fields... you've beaten SciShow as my favourite science channel 👍🏻
Hmm, isn't this the shape that makes balls roll down the fastest? Why are these curves connected? I feel like there should be an underlying logic here.
The brachistochrone is a cycloid, a curve drawn by a point on the edge of a rolling circle. The tractrix can be drawn as the curve that always perpendicularly intersects a rolling circle, so they're somewhat related!
"so next time you're riding your bike or walking your dog, you can tell your friend all about the tractrix" GODDAMNIT i already have too much to talk about! thanks tho
Area using visual calculus can’t be exact. The tip of every triangle is curved no matter how many slices. I saw an explanation like this on why pi is used to determine area of a circle. No matter how many times you slice the circle the short end is curved. Even if it’s not visible to the eye. It’s wholly unsatisfying.
The rule to get area between tracks only applies to Euclidean geometry or if the plane of the paths is flat. If you make a trip on a bicycle around the Earth wiggling around the equator - the area is positive but you made no turns cause you end up in the beginning in the same orientation
