The Brachistochrone

[Music] hey Vsauce Michael here if every single one of us held hands together in a chain of unity around Earth would there be enough of us to go all the way around the planet there are about seven and a half billion of us and that's a lot but remember that that many human bodies thrown together into one big pile would barely fill the Grand Canyon this is all of us in one place the physical bulk of all living human flesh on earth today would only make a cone about 7,000 people tall and 2,000 across that's it but that's a three-dimensional shape what if we made a one-dimensional single-file line of people each with say 1 meter of room and we stretched that around the planet well we would make it all the way around and still have 99.5 percent of the human population left if we made a circle that included everyone the ring of people would be more than 2.4 million kilometers in diameter dwarfing the orbit of our own moon now that's not just a circle that is a sir cool let's talk about circles today specifically something that they do roll but what is rolling well rolling occurs whenever something moves with respect to something else and is always in contact with that something else and the contact points have instantaneous velocities of zero that is there's no sliding in mathematics the path traced by a point on a rolling object is called a roulette French four little wheel the centre of a disk produces a roulette that's just a smooth straight line while rolling on one and this is why disks are good wheels a square on the other hand would be bumpy but square centers can make straight smooth Roulettes across the right surfaces this is the principle behind square wheels which I recently had the pleasure of enjoying at the Museum of mathematics in New York City Stan wagon wrote a fantastic and famous article about wheels which I've linked down in the description below it's a great read he also contributed a fantastic interactive tool to the Wolfram demonstration project that allows you to build a wheel and then find the corresponding road shape that allows it to roll smoothly the roulette traced by a point on a disk as it rolls on a straight line has a special name it's called a trochoidal Greek word for real TRO Coase now trochoidal prolate depending on whether the tracing point is inside or outside the circumference if the point is on the circumference the resulting curve is called a cycloid cycloids are very special and they are the star of this episode now I've been working with Adam Savage a lot lately as we gear up for brain candy live our 40 city tour that I hope to see you at recently I asked Adam for some help with rillettes Adam give a favorite polygon you can't they're like your children you got a favorite eye I don't have a favorite either I do have a favorite thing that you can do with the polygon okay make a cyclic on a cyclic cyclic on is an actual actual thing it's an actual thing do have a polygon around here here this this here's a squid right right say this is a square yeah if I take one of the vertexes of this squid and I start rolling this square and you follow where that vertex oh shoot let me roll this better are you ready yeah all right the actual path it describes a curve described here yeah that curve is called a cyclic on and as you pick polygons with more and more sides you get closer and closer to what I want from you today okay a cycloid a cycloid now I I could have just done this in Photoshop and done like an animation like I normally do but you have a shop we can actually build and you can build the bit okay why did I want to build a cycloid well let me ask you this with only gravity to move you what's the fastest way to roll or slide from a point to some other point below but not directly below would it be a straight line well that would certainly be the shortest path but when you fall gravity accelerates you and falling vertically a lot right away would mean having a higher top speed during more of the journey and that can more than make up for the fact that a path like this is much longer than a straight line but of these two considerations accelerate quickly and don't have too long of a path what's the optimal combination finding the answer the path of least time is called the brick east a chrome problem and it's been around for a while Galileo thought the answer was a path that's just a piece of a circle but he was wrong there's a better one and in 1697 Johann Bernoulli came up with the answer using a very clever approach to see how he solved it let's start with a similar problem you are standing in some mud and you want to run to a ball in the street as quickly as possible now a straight line would be the path of shortest distance but if you can run on pavement much faster than you can run in mud the path of shortest time would be one in which you spend less time moving slowly in mud and more time on the surface your bast on the angle you should enter the pavement at depends on how fast you move on both surfaces as it turns out when the ratio between the signs of these two angles equals the ratio between your speeds across both surfaces the resulting path will be the optimal quickest route this is called Snell's law light always obeys Snell's law when it changes speeds like when it leaves a material in which it moves more slowly like water and enters one in which it moves more quickly like air it always refract according to Snell's law in other words light always follows the route that is the fastest for it to take Bernoulli used this fact to tackle the brick ista chrome problem light changing speed is analogous to a falling object changing speed but of course a falling object doesn't just speed up once its speed is always increasing its accelerating to mimic this using light which Bernoulli knew would always show him the fastest possible route all he had to do was add more and more thinner and thinner layers in which the speed of light was faster and faster and faster and well what do you know there it is the brick East a chrome curve the path of least time roll down a track like this and you will beat anything rolling down any other path every time Bernoulli was clever enough to realize that this curve can be described in another way as a roulette specifically he noticed that it was a cycloid the path traced by a point on a circle rolling along a line a cycloid satisfies Snell's law everywhere to see why I highly recommend watching this video on the brick East a chrome problem this channel is fantastic by the way I'm a huge fan the visuals and explanations are top-notch anyway a cycloid provides the perfect balance between keeping the journey distance short and picking up speed early now I told Adam all of this and I told him that it would be really fun to have a cycloid curve we could roll things down and he said clearly we should start building we should just start building it what we need is a circle okay that's a the height that we want mm-hmm and then we're going to use that circle to trace a night cycloid curve I want to do a race right so yeah yeah look if we're gonna do like if you have point a here and point B here and you say that there's some kind of curve that's better than a straight line in terms of something traveling between those two then I want to also make a straight line from A to B yeah and maybe also a really extreme curve like that right okay got some protractors there you have a compass extension I do I have all sorts of I've never seen such thing is that it yeah that connects to that give or take I almost am I feel guilty that this is like a dream of mine coming true oh really but it's such a nerdy dream too it's not like I want this you know Red Ryder BB gun it's like I just want to curve that things roll down okay so then clearly when we're done with this this is this is my Christmas present to you I think I'm currently like working out a way to do this in my head that actually makes it fairly compact and not super not super complicated yeah take a blade and cut out like an inch around the whole thing [Music] how are you gonna do the finishing I'm going to cut this unless you would like to on the band's you can start okay [Music] okay now that's a little bit rough so we're gonna finish it on my disc sander oh wow that's close enough it's pretty good yeah so now you want to use this to draw a psych psych Lloyd yeah so you'll need a little hole that's what yeah so that the point of our drawing implement is on the rim not above it I'm doing this right inside I don't think so it is that's even better than having the rim hole because we want the line to be right on the edge okay so what we're gonna do is we're gonna create a pattern for the cycloid curve that I'm then going to transfer to acrylic to clear acrylic regular right allow us to see things really clearly yeah this allows you right roller sight because we're just doing it it's literally like that right oh yeah see this is perfect yeah pushed against here I shouldn't slip this is like a Ouija board for geometry nerds it is a Ouija board for geometry nerds okay there we go that's smooth out Ricci stack rone that's that's that's the curve that we're talking about yeah that's the beginning that's the end and this is our pattern yeah cool that's cool okay so I'm gonna end up with let's say 3/4 of an inch thick but it's gonna have channels table saw it out of it mm-hmm travel its length and in those channels will sit my clear acrylic acrylic curves and it'll also have an upright that also will have the channels milled out of it on the table soft and that'll allow the curve to sit and be supported yeah a little backstop here easily at the bottom that will allow us to hear that they all hit at the same time there's a couple things going on one is is that we've got cycloid straight line and then we've got extreme curve right and are these you had mentioned bending the acrylic so we could adjust the no magically I have this is you like this the acrylic will just be a thin of acrylic just a thin sheet of it traveling on that will be and I have material for this some Delrin or acetyl rollers that look like this so from the side they'll look like this they'll look like an H ah huh in which the acrylic sits in there and the roller is self-supporting on the acrylic but rolls down ah that is tolerance love it I like those curves while Adam and I build a real-life cycloid track let's take some time to appreciate other kinds of Roulettes as mentioned before trochoidal anon straight lines but an epitome oi is made when a disk rolls around the outside of a circle roll a disk inside a circle and what you've made is a hypo trochoidal names for the curves you make when using a spirograph toy notice that the holes don't lie on the circumference of the disks though some do come close there's a special name for epic rock woods and hypo trochoidal an algis to disks rolling on straight lines they are epicycles and hypo cycloids now if two circles have the same radius a point on the rolling one will touch the stationary one exactly once always in the same spot creating a cusp this cute heart-shaped epicycle ID is also known as a cardioid if the rolling circle has half the largers radius you'll get a 2 cusped epicyclic the shape of which is called a nephrite because it apparently looks like a kidney I guess 1/3 the radius gives you three cusps 1/4 four cusps and so on as for hypo cycloids if the inner circles radius is 1/4 of the larger x' the resulting Roulette curve is called an asteroid because it looks like a star which the ancients also thought about asteroids 1/3 the radius and you've got a deltoid named after its resemblance to the Greek letter Delta 1/2 the radius and well you get a straight line this fun relationship is called a 2 C couple rotational motion turned into linear motion follow a number of points on the rolling circle and you'll get the famous illusion where every individual point moves in a straight line but the whole thing describes a rolling circle put a handle on it and you've built a trammel of Archimedes aka an ellipse a graph when used to ellipses AKA a hillbilly entertainment center when bought in Osceola Missouri anyway let's get back to Adam and eyes curve comparison build okay okay there's your finish line you ready I'm ready all right I'll count us down three two one go okay here we go 3 2 1 go 1 2 3 with the asleep was second straight was last straight was last the shortest distance between two points was last slowest way to get there yeah it certainly was let's try it one more time yep because it was super close the brick ista Crone curve was by far the winner yeah what a mouthful of a name by the way barista drunk East uh Crone but Keith's brick east across not related to the brachiosaur looked at our that's I once looked up the difference between ingenious and ingenuity and they don't have them and they're not related no all right now ready I'm ready 3 2 1 go yeah same results same result 1 2 these are so close it's hard to tell [Music] well we're answering the main question which is that the barista chrome curve is the fastest way to get there well a brick easter chrome curve is also known as a tata chrome curve that has another property that's that we should test what's it and that is it's all you know hold up before you get to this we've established between these three curves that the cycloid made brick easter chrome curve is by far the fastest it's by far the fast okay great you get steep and unfortunately you pick up a lot of speed right away but then you've got a lot of this with little acceleration yep you go straight down and you know funny enough you what you want is that perfect balance of gravities acceleration but also moving to where you need to be and that is asked that's fascinating to me that a geometry which is the cycloid curve would yield such an efficient exploitation of the forces involved yeah exactly because if there was no acceleration if there was just one force in the beginning straight-line would probably be the fastest right okay what is the other quality of the Tata chrome curve you just said it Tata chrome means same time so as the geometry and math tells us no matter where you start an object yeah what whoops the clamp city it was my fault but also come on clamp no it sector you know how to do these tables are a pain in the ass because they have these lips on them and they actually that's like it's it's actually my fault so I'm gonna remove the straight line guy as well so we're left with a cycloid which is called a brick East a Crone curve but it also has a bizarre property where no matter where I start an object on it when I let go they always reach the bottom in the same amount of time wait a second so if I started here the amount of time it takes for this to get to the end is the same amount of time it takes for it to get to the end from here yeah and the same from right here well Wow okay so starting from here it's gonna be tough because of friction yeah now if you do this in software it's perfect but that's boring yeah this is the real world and maybe we won't start the network I mean if we do one two and three yeah are we actually I feel like we could probably we could probably get one to work from here like that there might be a lot of friction involved but look at always see we can always see we've got enough of them that's why we made three curves so we can test this property and these were all cut and sanded clamped together so they are incredibly similar I will have to tamper the edge as I did on this one yeah a little known fact that once you've torn paper off of Plexiglas it crumples into a really nice wall to be thrown long distances a Plexiglas in the first place I've never seen it happen because it doesn't really feel glued I think this is I hope that's cool because in theory and practice theory in theory theory and practice are the same thing but in practice I went to the University of Chicago where one of their sings well that sounds good in practice how is it in theory hahahaha life of the mind there so let us see if you do if I do one up here and you do one there and okay yes so this has got to be without this I think the same person should release them because then okay I'm it better so here put that one here okay all right so three different positions three this one has longer to go yep this one has the shortest path so who's gonna reach the bottom first okay here we go let's see would you wish it would say the one in might this one yeah go first ready yeah good pretty good that was awesome that was so they all ended up they lined right up alright let's do let's let's switch it up okay even I think you're gonna be the one to release them yeah otherwise it won't be easy to time here we go three two one I'd be garnish them lining up yeah they line up like they're waiting for each other I'm gonna stand further away cuz I want to see the full path okay ready yeah three two one how cool is that they totally line up to hit at about the same periodicity they went for here so given the vagaries of some extra frictions here and there there they're actually hitting this at the same sort of periodicity that they are when they start at the same point so there's that and then [Music] is this some is this the Tata chrome curved demonstration ring of your dreams it really is it is it's also the brick Easter Kron rig of my dreams and the cycloid rig of my dreams so check that I keep doing that yeah that is really cool another one yeah that was yeah exactly on point Wow raesha see in this game no matter where you start no matter who you are you're always a winner there's always a tie so this is brain candy for me Adam toasties this is something that was previously abstract and only seen in animations and in text made real now I can put these wherever I want I'm not stuck with what someone else did I can physically hold it and that's that's what makes brain candy exciting for this it is like that I love taking the theoretical and making it physical and actually honestly we've always had what seemed to me like sister enterprises yeah and it's nice to join them together yeah this is this is a little child of ours isn't it maybe maybe one of us wasn't so good at sanding and maybe the other of us kind of saved the day but it's real now yeah and it's alive and I think it's loving its life that was fun it was really fun true Adam thank you so much for your help working with you is always amazing I hope to see all of you out there watching at brain candy live it's going to be incredible and in your daily lives may you always find the Tata chrome the solution that brings you and others together even if you started in different places and as always thanks for watching [Music]

Channel: Vsauce

Views: 8,962,392

Rating: 4.9548736 out of 5

Keywords: michael stevens, adam savage, brachistochrone, curve, math, geometry, cycloid, roulette, rolling, brain candy live, tautochrone, maths, science, snell's law, light, refraction

Id: skvnj67YGmw

Length: 25min 57sec (1557 seconds)

Published: Sat Jan 21 2017

Reddit Comments

[u/WPattempter]

I watched the whole video and I can honestly say it's a god damn interesting content.

[u/CylonBunny]

Adam seemed a little less enthused and felt like he was reading from a script up until the test where the three disks started at different points but ended together. I loved how excited he got at that point.

[u/macnicool]

Good double act potential, although I would like to see a more casual relationship sort of atmosphere, they both were very polite together

[u/iamthepkn]

In your daily life may you always find the tautochrone.

[u/kennyD97]

Interesting video but that Michael stare in slow-mo at the end.

[u/Tszemix]

https://www.youtube.com/watch?v=5F1noPMJmAc

[u/Tractor_Pete]

That's a cir-cool! intense grinning

Let's talk about circles today.

You simply cannot write stronger hooks than this.

[u/Plasma_000]

Wow, Vsauce plugged 3blue1brown - It's a really small maths channel that I discovered on reddit just a few weeks ago. Nice one Michael!

[u/CottonBalls26]

The tauto chrone thing is amazing.

At first I thought no way - they're violating some law of physics having 3 objects complete different distances in the same time on the same curve then I thought oh right - they're experiencing different accelerations