The Brachistochrone

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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 for "little wheel," the center 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, a Greek word for "wheel." Trochoidal can be prolate or 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 roulettes.

"Adam, give me a favorite polygon!"

"You can't! They're like your children! You got a favorite?"

"I don't have a favorite either. I do have a favorite thing that you can do with the polygon."

"Okay, make a cycloid."

"A cycloid is an actual thing! It’s an actual thing!"

"Do you have a polygon around here?"

"Here, this is a squid."

"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, the curve described here..."

"Yeah, that curve is called a cycloid. 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!"

"Now 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 Brachistochrone 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 path 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 sines 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 refracts 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 Brachistochrone problem. Light changing speed is analogous to a falling object changing speed, but of course, a falling object doesn't just speed up; its speed is always increasing—it's 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 Brachistochrone 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 Brachistochrone 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 it!'"

"We should just start building it. What we need is a circle."

"Okay, that's the height that we want."

"Mm-hmm."

"And then we're going to use that circle to trace a nice 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?"

"I have a compass extension."

"I do! I have all sorts of—I've never seen such a 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 a curve that things roll down!"

"Okay, so then clearly when we're done with this, 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 complicated."

"Yeah, take a blade and cut out like an inch around the whole thing."

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"How are you gonna do the finishing?"

"I'm going to cut this unless you would like to on the bands; 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 cycloid?"

"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?"

"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—clear acrylic."

"Regular, right?"

"Allow us to see things really clearly."

"Yeah, this allows you right roller site 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 smooth out."

"Ricci stack rone, 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-sawed out of it, travel its length, and in those channels will sit my clear acrylic curves, and it'll also have an upright that will also 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'll hit at the same time. There's a couple things going on: one is that we've got cycloid, straight line, and then we've got an extreme curve, right? And are these—you had mentioned bending the acrylic, so we could adjust the..."

"No. Magically, I have this! The acrylic will just be 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."

"Aha?"

"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 on straight lines, but an epitrochoid 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 hypotrochoid.

"There are 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 epitrochoids and hypotrochoids, and that’s epicycles and hypocycloids. 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 epitrochoid is also known as a cardioid. If the rolling circle has half the larger's radius, you'll get a 2-cusped epicyclic, the shape of which is called a nephroid 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 hypocycloids, if the inner circle's radius is 1/4 of the larger's, 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 two-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 my 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; the cycloid was second; straight was last! The straight was last!"

"The shortest distance between two points was last—the slowest way to get there!"

"Yeah, it certainly was. Let's try it one more time!"

"Yep, because it was super close! The Brachistochrone curve was by far the winner."

"Yeah, what a mouthful of a name, by the way—Brachistochrone. But Keith's Brachistochrone is not related to the Brachiosaur. I once looked up the difference between genius 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 Brachistochrone curve is the fastest way to get there. Well, a Brachistochrone curve is also known as a tautochrone curve. That has another property that we should test."

"What's it?"

"That is—hold up; before you get to this, we've established between these three curves that the cycloid-made Brachistochrone 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, what you want is that perfect balance of gravity's acceleration but also moving to where you need to be—and that is 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, the straight line would probably be the fastest."

"Right, okay; what is the other quality of the tautochrone curve?"

"You just said it! Tautochrone means same time! So as the geometry and math tell us, no matter where you start an object—"

"Whoa! Whoops! The clamps hit—it was my fault! But also, come on, clamp no—it's sector!"

"You know how to do these tables? They are a pain in the ass because they have these lips on them, and they actually—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 Brachistochrone 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!"

"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!"

"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 get one to work from here like that. There might be a lot of friction involved, but look!"

"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 temper the edge as I did on this one."

"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 sayings is, '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!"

"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 my...this one?"

"Yeah, go first! Ready?"

"Yeah!"

"Good! Pretty good! That was awesome! That was so—they all ended up lined right up!"

"All right, let's switch it up."

"Okay, even I think you're gonna be the one to release them. Otherwise, it won't be easy to time. Here we go!"

"Three, two, one!"

"I'd be garnishing them lining up!"

"Yeah, they lined up like they're waiting for each other! I'm gonna stand further away because I want to see the full path."

"Okay, ready?"

"Yeah! Three, two, one!"

"How cool is that? They totally lined up to hit at about the same periodicity!"

"They went for here! So given the vagaries of some extra frictions here and there, they actually hit 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 tautochrone curve demonstration ring of your dreams?"

"It really is!"

"It is! It's also the Brachistochrone 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 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 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!"

"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 tautochrone—the solution that brings you and others together, even if you started in different places!"

"And as always, thanks for watching!"

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