[MUSIC PLAYING] Singularities are ubiquitous in math and physics, but making sense of them is often a serious challenge.

In general, mathematicians use the word singularity to describe a problematic point.

In its most common form, a singularity is a point where a quantity becomes infinite.

When your teacher told you that something divided by zero was undefined, they were sweeping under the rug how complicated this type of singularity can be.

Let's look at the function 1 divided by x.

When x equals 1, it's 1.

When x equals 1/2, it's 2.

When x equals 1 over 100, it's 100.

When x equals 1 over a million, it's a million.

See what's happening?

The closer x gets to zero, the bigger one divided by x gets.

There's no limit to its growth.

It just keeps getting bigger and bigger and bigger, which is why it's awfully tempting to say the function equals infinity at the point x equals zero, even though that's a bit confusing.

This kind of infinity tends not to bother mathematicians.

They just say the point x equals 0 is undefined, which is shorthand for bad, off limits, or just don't talk about that point.

And in mathematics, that's fine because math is abstract.

We can choose to ignore the difficult points, like when a quantity becomes infinite.

But things get weird when that mathematical quantity is supposed to represent something in the physical world.

It's often the case that a mathematical theory describes some aspect of reality extremely well except for one pesky singularity.

Then you're left with a choice, admit that the math might not accurately describe reality if you look too closely, or claim that reality actually has infinities.

In some cases, it's not clear which one is the right choice.

But to start with, here's a case of a total mathematical fail.

In a 1960 article in Science magazine, Heinz Von Forrester tried to formulate equations to predict human population growth, but they had one big problem-- a singularity.

The mathematical theory aptly titled "The Doomsday Equation," predicted that population would become infinite on Friday, November 13, 2026.

This type of singularity is called a finite time blow up because it predicts that some variable, in this case population, will blow up in a finite amount of time.

In the tension between mathematical theory and reality, the doomsday equation seems like a simple case.

It's almost certain that his mathematical model is wrong, as in it doesn't actually predict human population growth, rather than saying that the math is correct and the population will explode in 2026, especially since there are physical restrictions like a nine month gestation period and the Earth's finite capacity.

But not all finite time blow ups are as obvious as the doomsday equation.

In fact, a million dollar prize in mathematics hinges on a finite time blow up in water.

The Navier-Stokes equations describe fluid motion, and one of the seven Clay Millennium problems in mathematics is to show whether or not these equations allow fluid to move in wild and unexpected ways.

Recently, mathematician Terence Tao showed that a slight variant of these equations, a sort of physics in an alternative reality, do allow water to blow up in finite time, to explode in an infinite spike.

If mathematicians prove that the real Navier-Stokes equations also allow for this possibility, it'll expose a disconnect between the mathematics and the physical reality where water can't actually explode infinitely.

But there might be situations where the math is correct, where the infinity is actually exhibited in our world.

One of the best examples of a mathematical singularity having profound implications for its guiding physical theory are black holes.

Hmm, who do I know that knows about black holes and also has a PBS digital series?

Oh, hey, Matt from Space Time.

Hey, Kelsey.

Thanks for having me on.

Glad you could make it.

There are lots of types of singularities in math.

So why do astrophysicists so often talk about black holes as singularities?

Well even Newton's good old universal law of gravitation has a mathematical singularity, gravity increases as the square of distance as you approach a massive object.

From the math, you'd expect the gravitational force to become infinitely strong at zero distance, but that zero distance to the center of mass of an object, for most things, planets, stars, etc.

that means you're inside the object, and most of the mass is above your head no longer contributing to gravity.

But what if the massive object itself has zero size?

That's the singularity of the black hole, where density and gravity do go to infinity.

Really, we need Einstein's general relativity to describe the weirdness of the black hole, but even in Einstein's theory you get a singularity.

In fact, you get more than one.

Many physicists believe that this is reason to think that general relativity is not the complete picture.

Come to think of it, I'm going to head out and start making an episode on possible ways out of the mathematical singularities of the black hole.

See you, Kelsey.

Thanks, Matt.

You can watch Matt discuss more about the implications of a black hole singularity at Space Time.

Here on Infinite Series, we're going to bring it back down to earth and talk about some of the singularities that you might interact with on a daily basis.

So Matt mentioned that the force of gravity is proportional to 1 over r squared, where r is the distance between the center of the massive object, like the center of earth, and the object being pulled toward it, like you.

A similar situation holds for vortices.

The velocity of water spinning in a whirlpool is proportional to 1 over r, where r is the distance of the water to the center of the whirlpool.

This looks a lot like our graph of 1 over x from the beginning.

The closer the water gets to the center, the faster the velocity is approaching a singularity in the center.

But a water molecule has a fixed size.

It can't get arbitrarily close to the center of a vortex.

Unlike with a black hole, physicists have been able to resolve the vortex singularity explaining the difference between this idealized model of a vortex and its actual physical nature.

We can even find an infinity hidden in the North Pole.

It's the spot where all the time zones meet, so it doesn't really have one.

The Earth's northernmost point is a singularity.

First, imagine you're walking along the equator all the way around the earth.

Since you're a pretty fast walker, it only takes 24 hours to walk through all 24 times zones.

Now, you walk due north to the 60 degree line of latitude.

At the same speed you traversed the equator, you walk for 24 hours around the 60 degree line of latitude, only this time, you make it all the way around twice.

That is, you've cycled through all the time zones twice.

Now, you trek way up north to the 89.99 degree line of latitude.

Again, with the same speed you followed the line of latitude for 24 hours.

This time, you've made it through all the time zones 100 times.

The further north you make this journey as your loop closes in on the North Pole, you'll cycle through the time zones quicker and quicker.

Once you reach the North Pole, you'll cycle through infinitely many time zones during your 24 hour walk, whatever that means.

In some sense, the North Pole singularity comes from the weirdness of our coordinates.

On a flat, two dimensional map, the time zones are perfect rectangles.

There's no singularity there.

But flat maps are an inaccurate representation of our curved globe, and in real life, these rectangles are smushed at the ends into two points-- the infinities on our globe.

Can you think of any earthly singularities?

Let us know in the comments, and we'll see you next week on Infinite Series.

Hello.

Thanks to Matt O'Dowd for joining me to talk about singularities within black holes.

To learn more about this, you can watch his episode on black hole singularities right here.

I highly recommend it.

Now, I want to respond to some of the comments from the "Random Night" episode.

First one from Matthew Wilson, who says, "Can anyone explain the process of finding the stationary distribution of the second radio station example?"

This is a great question, and it points to the math that underlies Markov chains.

So the basic tool that we use to study finite Markov chains are called matrices.

And so, you study matrices to learn things like the stationary distribution.

So that's how I did it, and it's a good reason to learn more about linear algebra and matrices.

All right, the second question is from Veene Michael Eco, who says, "What about 3D chess?

How long would it take a knight to get back to its starting square on a three dimensional, 8 by 8 by 8 chessboard?"

What a cool question.

So, they answered their own question in the comments, and you can go ahead and read that, or you could think about it on your own and try to apply the method that we came up in the episode to three dimensional chess.

Finally our last comment is from IamGrimalkin, who says-- who wants to clarify that the average is the mean, not the mode or the median.

This is a reasonable clarification.

I also wanted to point out that a lot of people use the word expectation or expected value to mean the same thing, to mean average or mean.

All right.

See you next week.