your presentation is absolutely adorable, your explanation is satisfying but accessible, please please please keep making content!!

I'm floored by how good your videos are, please keep making more!

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 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!

This channel better blow up soon. You deserve to be so much bigger with this kind of quality.

This was beautifully done. As a math teacher myself I cannot help but thank you for the brilliant perspective and detailed presentation.

That trick with the area between a bicycle's tracks is also (with some coordinate substitutions) the way a planimeter works.

Well done. I was enthralled through the entire presentation. Please, continue to make videos.

I didn’t realise this channel only had 9k subs until a couple videos in. With this level of quality you’ll definitely have a bunch more in no time.

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. 💕

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 ❤

When you first posed the question, I imagined the dog making a perfect circular orbit around the owner

Love your videos, so entertaining and still super informative. Hopefully you get a bit more attention in the future!

Top tier content. This is genuinely enjoyable.

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

This is an amazing video. Keep up the great work!

You are my new favorite KZfaq channel! I feel lucky that KZfaq reccomended you!

You are one of the best math youtubers out there! Shame that you are so underappreciated!

already my new favorite youtuber :))

UNBELIEVABLY UNDERRATED. Thank you so much for this video, so interesting!

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

Thank you very much for these very nice explanations and demonstration.

Give this man more subscribers.

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!

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.

loved the video. i used to watch a lot of cliff stoll's lecture on topology back in my freshman. it reminded me of that.

You are awesome your videos are so much fun!

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 👍🏻

bro this is incredible

great video! hope to see more from this channel

This is really excellent! I am amazed and learned something cool!

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.

I just love this videos

Don't know if I needed to know this but I am highly intrigued, will be staying around for a while

great video! thanks!!

Superb video!!

What if you leash another dog to the first one? Is that just another tractrix, or do you get a super-tractrix?

This is fantastic!

This was brilliant

The area between bike tire tracks is easily the best fun math fact I learned in years!

Great Content!

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!

i love this channel, somehow you explain physics in a way that’s kinda similar to computer science

careful you're pinching it infinitely tight

Awesome video!

This was really insightful and interesting to listen to, after just taking trig and going into calc1 lol

You Sir Bird have earnt yourself a sub for a great video.

so fire dude

I usually go home and watch KZfaq to get away from geometry (haven't learned claculus yet) but damn this is mad interesting

Broh, just found out your channel. I just fucking love it. Please, keep doing videos like this!

Good video

"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

this feels like an April fool's day prank

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

thanks!

I really wish you had been my Algorithms teacher back in college.

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!

this is cool

Making these shapes with string, nails and a board when i was a kid might be part of why i like maths.

Cool!

4:35 is a perfect visual for integration by substitution

Let it be known I was your 484th subscriber once your famous. Great job!

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.

Love it

As I watch this, I thought of another tractrix. An 18 wheeler truck.

4:39 Oh my god this example of your is painfullll! xD But yes, I accept that, keep going, don't stop now! :)

Nice

I can see this outro jingle being as culturally significant as the Vsauce one in a few years.

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

Can you make a video on how to turn a sphere inside out?

What are the intro/outro songs. They are really good!

I have that same Yoda backpack

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...

Why are the comments on this video so kind?

Hey !Really cool video ! y the way, what is the veeery cool music in the background at 8:04 ? Really nice music !

dog drawing is underrated

I can't imagine how these triangles make up the circle or it's quarter.

Your videos are absolutely incredible! Just watched the jazz one and now this! I do have but one question however: if the bike changes direction, i.e starts turning left then turns right, does the area kind of cancel itself out a but like being above or below the x axis in an integral, in which case the degree it turns through doesn't represent the area? I'm not sure if this is right but if anybody knows I would appreciate it! Keep up the absolutely perfect videos!!

It's Appa ! =D

It's similar to when slices of fruit lean on each other and make a curve on top

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!

Imagine if they made a bike called the Tractrix

3:45 aperture science refrence lol

Isn't that also the shape of gabriel's horn? The one with finite volume but infinite area

Quick question: is that race happening in San Fierro?

so a sphere in 3d creates a funnel in 4d...?

Okay. Maybe you're right, and I hear you, 100%. But how do you turn a circle happy?

Visual Calculus [Challenging: Success]

Where do you turn the církve inside out?

Wow

Is it the braquistochrome curve?

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

This was a phenomenal video, but if I may make a suggestion, please try to filter out the wet sounds from your mouth when you're talking

7:32 what is happening here, is the bike moving backwards

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

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.

Dogs are the best

Isn't that called a brachistochrone?

Bro used Roblox to visualize falling 💀