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.
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!
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
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 ❤
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.
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. 💕
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!
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 👍🏻
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
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
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.
"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
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
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!
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...
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.
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.
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
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!
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.
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!!
That's exactly how it works! As long as you keep track of the angles that went in the other direction the area will still be proportional to the total angle. At 4:11 the red triangles are ones that went the other way, so you can see how I subtracted them instead of added.
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.
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.
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!
The brachistochrone is drawn by a point on the edge of a rolling circle, but the tractrix can be drawn as the curve that always perpendicularly intersects a rolling circle, so they're somewhat related!
your presentation is absolutely adorable, your explanation is satisfying but accessible, please please please keep making content!!
Will do!
Thought the same thing
If you like it that much, it means... you're a bird! 😝 (jk I love it too)
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.
Exactlyyyyy
Although I do like details, i think it shows the relationships even more clearly
same as mihail
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!
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
This was beautifully done. As a math teacher myself I cannot help but thank you for the brilliant perspective and detailed presentation.
I'm glad you liked it! I think visual calculus fits well in a classroom since it's so hands on.
@@physicsforthebirds Do you use a wacom tablet or iPad to make the animations?
@@katarixy I make the animations on iPad with Procreate
This channel better blow up soon. You deserve to be so much bigger with this kind of quality.
Frrrrr
When you first posed the question, I imagined the dog making a perfect circular orbit around the owner
That trick with the area between a bicycle's tracks is also (with some coordinate substitutions) the way a planimeter works.
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 ❤
Well done. I was enthralled through the entire presentation. Please, continue to make videos.
Yup yup
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.
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. 💕
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 got a couple of recent videos recommended and now im in a deepdive of all the other videos. I love this channel
Love your videos, so entertaining and still super informative.
Hopefully you get a bit more attention in the future!
Me too, Fable, me too...
Give this man more subscribers.
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.
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.
What if you leash another dog to the first one? Is that just another tractrix, or do you get a super-tractrix?
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
Top tier content. This is genuinely enjoyable.
Thank you very much for these very nice explanations and demonstration.
You are my new favorite UA-cam channel! I feel lucky that UA-cam reccomended you!
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!
already my new favorite youtuber :))
Don't know if I needed to know this but I am highly intrigued, will be staying around for a while
bro this is incredible
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 👍🏻
I really wish you had been my Algorithms teacher back in college.
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
Pretty sweet! That's how I was able to make the 5-sided and 3-sided origami paper for my non-Euclidean cranes
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
This was brilliant
The area between bike tire tracks is easily the best fun math fact I learned in years!
You are awesome your videos are so much fun!
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 usually go home and watch UA-cam to get away from geometry (haven't learned claculus yet) but damn this is mad interesting
You are one of the best math youtubers out there! Shame that you are so underappreciated!
Broh, just found out your channel.
I just fucking love it. Please, keep doing videos like this!
UNBELIEVABLY UNDERRATED. Thank you so much for this video, so interesting!
You Sir Bird have earnt yourself a sub for a great video.
This is fantastic!
This is an amazing video. Keep up the great work!
"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
careful you're pinching it infinitely tight
Wrong dimension buddy
4:35 is a perfect visual for integration by substitution
This is really excellent! I am amazed and learned something cool!
I just love this videos
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
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!
It's similar to when slices of fruit lean on each other and make a curve on top
this feels like an April fool's day prank
dog drawing is underrated
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...
As I watch this, I thought of another tractrix. An 18 wheeler truck.
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.
i love this channel, somehow you explain physics in a way that’s kinda similar to computer science
great video! hope to see more from this channel
This was really insightful and interesting to listen to, after just taking trig and going into calc1 lol
great video! thanks!!
Making these shapes with string, nails and a board when i was a kid might be part of why i like maths.
Superb video!!
Awesome video!
I can see this outro jingle being as culturally significant as the Vsauce one in a few years.
Great Content!
Can you make a video on how to turn a sphere inside out?
Why are the comments on this video so kind?
so fire dude
Let it be known I was your 484th subscriber once your famous. Great job!
4:39 Oh my god this example of your is painfullll! xD
But yes, I accept that, keep going, don't stop now! :)
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 can't imagine how these triangles make up the circle or it's quarter.
3:45 aperture science refrence lol
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
Good video
thanks!
Hey !Really cool video ! y the way, what is the veeery cool music in the background at 8:04 ? Really nice music !
The music is my own, made for the video. I'm glad you like it!
@@physicsforthebirds Please release it!
I have that same Yoda backpack
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!
Quick question: is that race happening in San Fierro?
It's a map of San Francisco, which I guess San Fierro is based on 🤣 ...wasn't meant as a reference!
Isn't that also the shape of gabriel's horn? The one with finite volume but infinite area
this is cool
Cool!
What are the intro/outro songs. They are really good!
Where do you turn the církve inside out?
sorry autocorrect
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.
Nice
Imagine if they made a bike called the Tractrix
It's Appa ! =D
Love it
I wonder how shapes could vary if you were to introduce another variable for if the dog was to walk closer to the owner than the leash is long
Okay. Maybe you're right, and I hear you, 100%. But how do you turn a circle happy?
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
so a sphere in 3d creates a funnel in 4d...?
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!!
That's exactly how it works! As long as you keep track of the angles that went in the other direction the area will still be proportional to the total angle. At 4:11 the red triangles are ones that went the other way, so you can see how I subtracted them instead of added.
@@physicsforthebirds awesome! Thank you so much for letting me know - very very cool! Keep it up!
Visual Calculus [Challenging: Success]
7:32 what is happening here, is the bike moving backwards
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.
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.
Is it the braquistochrome curve?
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!
Isn't that called a brachistochrone?
The brachistochrone is drawn by a point on the edge of a rolling circle, but the tractrix can be drawn as the curve that always perpendicularly intersects a rolling circle, so they're somewhat related!