The Brachistochrone

published on July 13, 2020

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 995 percent of the human population left if we made a circle that included everyone the ring of people would be more than 24 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 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 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

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

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

Related Videos

hey guys techrax here coming at you guys with another giveaway this is gonna be an iPhone 5s two of these brand new phones not the one in this video guys this i...
hey guy there's a popular game in the app store called flappy bird right now and it's supposedly really annoying believe it or not I haven't played ...
hey guys tech Rex here so right here with me I have an Apple iPhone 5s this is the gold one I also have with me some liquid nitrogen yes this is the real deal a...
hey guys techrax here right here I have the iPhone 5s with me this is the champagne or the gold color I still can't really figure out if it's champagne ...
hey guys tear cracks here so this video really exciting video I have the new Samsung Galaxy s5 for you guys this is the long-awaited cellular devices releases i...
everyone techrax here in this video I've got the latest Samsung Galaxy s5 right here as well as the Apple iPhone 5s and we're going to be doing a simple...
everyone techrax here in this video guys have a really exciting device this is the Samsung Galaxy s5 charcoal black and I'm really liking I after seeing the...
everyone techrax here here with me off the Samsung Galaxy s5 this is a perfectly fine s5 it is cracked from the drop test that I had with also one minor neck as...
hey guys tetrax here so in this video I'm going to try and burn the newly released Samsung Galaxy s5 this is the shimmering white 16 gigabyte model and if y...
everyone techrax here so I got my burn Samsung Galaxy s5 and I wanted to see whether the heartbeat sensor would still work the heart monitor on your galaxy s5 a...
hey guys Tech Rex here so I'm really excited to bring you guys a giveaway for my channel but this time I'm actually teaming up with a buddy of mine your...
hey guys techrax here so in this video I'm going to be hopefully instructing you guys how to make your very own a tech sandwich slash burger slash meal so y...
hey guys techrax here so I've got a galaxy s5 here this is the copper gold hopefully you guys can see pretty well it is sunset so it's getting a little ...
hey guys techrax here so right here with me I have a professional deep fryer in here is already some canola or corn or whatever oil I don't know vegetable o...
hey guys techrax here so just trying to make this video short and quick i'm having recently i got five hundred thousand subscribers and yeah most of you guy...
hey guys texture so in this video I have a drop test on the latest LG g3 device now this is actually the gold-coloured LG g3 this has not been released in the U...
hey guys techrax here so in this video we'll teach you guys how to make your iPhone indestructible this is essentially a case that's been around for yea...
hey guys techrax here soon in this video we had a train run over the iPhone 5s so we actually did this in two different instances initially we had a Space Gray ...
hey guys tech cracks here so in this video we're going to be dropped testing the newly released Amazon fire phone this is exclusive for AT&T and I belie...
hey guys techrax here so right here with me I have the Amazon fire phone this is the one I dropped and you know I thought what better what else do I do with thi...