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 7.5 1,000,000,000 of us.

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 7000 people tall and 2000 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, one meter of room and we stretched that around the planet?

Well, we would make it all the way around and still have 99.5% 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 role.

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 disc produces a roulette that's just a smooth straight line while rolling on one.

And this is why discs are good wheels.

A square, on the other hand, would be bumpy, but square centers can make straight smooth.

Let's 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 tro coin from the Greek word for real tro cose.

Now TRO coins could be curtained or pro late, 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 psych Lloyd Psych Lloyds 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 rule ETS Adam, you have a favorite polygon.

Um, the whole group.

I just literally, actually, you can't like your Children.

Do you have a favorite?

I don't have a favorite either.

I do have a favorite thing that you could do with.

Okay, make a cycle ago.

A cycling cycle.

Gone is an actual thing.

It's an actual thing.

Um, do have a polygon around here.

Here, This this here's a square, right?

Right.

Say this is a square.

If I take one of the Vertex is of this square and I start rolling this square and you follow where that Vertex Oh, shoot.

Let me roll.

This better get ready.

All right.

The actual path it describes occur described here.

Yeah, that curve is called a cycle ago.

And as you pick polygons with more and more sides, you get closer and closer to what I want from you today.

Okay?

A psych Lloyd.

A cyclone.

Now, I could have just done this in photo shop and done looking animation like I normally do.

But you have a shop we can actually build, and you could build things.

Why did I want to build a psych oy?

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 crone 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 16 97 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 and mud and more time on the surface.

You're fast 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 brutally used this fact to tackle the Burke East.

A crone problem like 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, it's accelerating to mimic this using light, which Bernoulli knew would always show him.

The fastest possible route I only had to do was ADM or and more thinner and thinner, laters, 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 crone curved path of least time rolled 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 psych Lloyd, the path traced by a point on a circle rolling along a line.

A cycling satisfies Snell's law everywhere.

To see why I highly recommend watching this video on the brick East.

A crone problem.

This channel is fantastic, by the way, on the huge fan, the visuals and explanations are top notch.

Anyway, a psych Lloyd 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 cycle lead curve.

We could roll things down and he said, Clearly, we should start building.

Would you just start building it?

What we need is a circle.

Okay, that's the height that we want.

And then we're going to use that circle to trace amount cycle occur.

I want to do a race, right.

So you're gonna look, if we're gonna do like, if you have a 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 curved like that.

Right?

Okay.

Got some pro tractors there.

Who?

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.

That connects to that.

That's gonna be 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 Rider, BB gun.

It's like I just want a curve, that thing's roll down.

Okay, so then clearly, when we're done with this, this is This is my Christmas present, E, um, I think I'm I'm currently like working out a way to do this in my head.

That actually makes it fairly compact and not super, uh, not super complicated.

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

And how are you gonna do?

Finishing?

I am going to cut this.

Unless you would like to on the bands you can start.

Okay.

Okay.

Now, uh, that's a little bit rough, so we're gonna finish it on my disk.

Sander about that's close enough.

It's pretty good.

Yeah.

So now you want to use this to draw us like a psych Lloyd?

Yeah.

So you'll need a little hole, That's what.

Yes.

So that the point of our drawing implement is on the rim, not above it.

I'm doing this right inside.

I messed that up.

I don't think so.

It is.

That's even better than having the rim hold 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 psych light curve that I'm then going to transfer two acrylic to clear acrylic.

Allow us to see things really clear.

Yeah, um, so here just allows you, right?

Roll your cycling because we're just doing it is literally like that, right?

Yeah.

See this?

Perfect.

Yeah, pushed against here.

I shouldn't slip.

This is like a wee gee board for geometry nerds.

It is a wee gee board for geometry nerds.

Okay, there we go.

That's spilled out.

Ricky sta crone.

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.

Um, okay.

Show.

I'm gonna end up with, uh, a piece of applied, Let's, let's say, 3/4 of an inch thick, but it's gonna have channels table sought out of it.

Travel its length and in those channels will sit my clear acrylic acrylic curves.

And it will also have an upright that also will have the channels milled out of it on the table soft and that will allow the curve to sit and be supported.

A little backstop here easily at the bottom metal.

Allow us to hear that they all hit at the same time.

There's a couple things going on.

One is that we've got, uh, Psych Lloyd, straight line.

And then we've got extreme curve.

Right?

And are these you had mentioned bending the acrylic so we could adjust The known, Actually, I have.

This is you'll like this 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 Cem, Delron or a Siegel rollers that look like this.

So from the side they'll look like this.

They'll look like an H er in which the acrylic sits in there and the roller is self supporting on the acrylic but rolls down.

Oh, that is Torrance for love.

Like those curves.

While Adam and I build a real life cycle track, let's take some time to appreciate other kinds of rule ETS as mentioned before tro coid czar curves made by discs rolling on straight lines.

But an EPA TRO coid is made when a disc rolls around the outside of a circle.

Roll a disc inside a circle, and what you've made is a high pop tro coid.

These are the mathematical names for the curves you make when using a spiral graph 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 DePetro coins, and Hypo TRO coins, traced by points on circumference, is analogous to discs rolling on straight lines.

They are episodic, Lloyd's and Hypo Psych Lloyd's.

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, episodic Lloyd is also known as a cardio.

If the rolling circle has half the larger Sze radius, you'll get a two cussed, episodic Lloyd, the shape of which is called a Neff roid, because it apparently looks like a kidney, I guess 1/3 The Radius gives you three cusp CE 1/4 4 Custis and so on.

As for Hypo Psych Lloyd's If the Inner Circle's Radius is 1/4 of the larger Tze, 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 to 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.