When Pi is Not 3.14

[MUSIC PLAYING] We're going to learn how to change the value of pi by changing the way we measure distance.

Before we learn how to change the value of pi, let's review the standard value, why you've been told your whole life that pi is 3.14.

First we need some high school geometry.

Well, pi is defined to be the ratio of the circumference to the diameter of a circle.

So to determine pi, we just need to measure a circle.

But what precisely is a circle?

A circle, by definition, is all the points that are a fixed distance away from a central point.

This is why distance matters.

Since distance is in the definition of a circle, the way we measure distance affects the shape and circumference of the circle.

And the circumference of the circle affects pi.

The usual way to measure the distance between these two points is given by this formula.

Using variables, the formula looks like this.

This is the familiar Euclidean metric.

Metric is just a fancy word for the way we measure distance.

Now we can draw a unit circle centered at the origin, which, by definition, is all the points that are distance 1 from the point 0, 0.

Finally, we define pi to be the ratio of the circumference to the diameter.

Since the circumference is roughly 6.28 and the diameter is 2, that makes pi roughly equal to 3.14.

But there are plenty of other ways to measure distance, ways you use in your everyday life.

Let's say we're in Manhattan, and I'm trying to give you directions from MoMath, the Museum of Math, to MoMa, the Museum of Modern Art.

I'd probably say, it's 18 blocks away, one block east and 17 blocks north.

Mathematicians call this the taxicab metric because a taxicab traveling through a city grid can only move up and down or left and right.

It can't move diagonally.

So the taxi would have to drive a distance of 18 blocks to get from MoMath to MoMa.

In other words, the distance in the taxicab metric from MoMath to MoMa is 18 units, not the square root of 290 units that a bird would fly using the Euclidean metric.

Here's the general formula for the distance between two points using the taxicab metric.

So now let's try drawing a circle of radius 3 in this metric.

That's all the points that are distance 3 from the origin.

For starters, we have these 12 points.

Because a taxi can reach these points by driving three blocks without backtracking, they're all distance 3 from the origin.

But here's where I'm going to break from reality and ruin the metaphor of a taxi.

Instead of focusing on a physical taxi, let's just look at the formula for the taxicab metric.

What if I wanted to measure the distance between 0, 0 and 1/2 comma 2 and 1/2?

This is like asking a taxi to drive half a block east and then 2 and 1/2 blocks north.

Well, that's not possible, since it would drive through buildings.

But our formula still works.

Math isn't restricted to reality.

The taxicab metric says that the distance between those two points is 1/2 minus 0 plus 2 and 1/2 minus 0, which is 3.

Since the taxicab distance between the origin and the point 1/2 comma 2 and 1/2 is 3, we need to put that point on our circle of radius 3.

In fact, now that we know we're allowed to drive through buildings and the only restriction is that we drive left, right, and up, down, not diagonally, we need to add a bunch of points, like negative 2.3 comma 0.7, 1 and 1/2 comma negative 1 and 1/2, and negative 1.4 comma 1.6.

All these points are taxicab distance 3 from the origin and belong on our taxicab circle of radius 3.

If we keep filling in the points, we eventually get a shape that looks like a diamond.

That's the taxicab circle.

Now, let's compute pi, the ratio of the circumference to the diameter.

To figure out the circumference, we need to figure out how long each of the four sides is in this diamond-looking shape.

Let's measure the length of this side.

It extends from the point 3, 0 to the point 0, 3.

We need to measure the distance between these points using the taxicab metric.

To get from 3, 0 to 0, 3 we have to drive three blocks west and three blocks north, which is six blocks total.

So the points are distance six away from each other.

That means the line has length 6.

That's kind of weird.

Your Euclidean intuition might tell you to use the Pythagorean theorem and compute that the line has length square root of 18.

But your intuition, and the Pythagorean theorem, don't work in taxicab land.

The line has length 6.

In fact, by similar reasoning, all four lines have length 6, which means that the circumference has a total length of 24.

Now, the diameter of our taxicab circle is 6.

That means that pi is 24 divided by 6, which is 4, not 3.14.

We've looked at two different ways to measure distance, the Euclidean and taxicab metrics and the circles created by those metrics.

But, as we mentioned in the beginning of this episode, there are a lot of different ways to measure distance.

Let's go back to the Euclidean metric where the distance between two points is measured this way.

Let's rewrite it this way.

The Euclidean metric is also called the L2 metric because of all the 2's in the formula.

What happens if we replace all the 2's by 3's?

Well, we get the L3 metric.

What does a circle look like under the L3 metric?

This.

It's all the points x, y, such that x cubed plus y cubed, all to the 1/3, is equal to 1.

The circumference of this circle, measured using the L3 metric, is roughly 6.52, which makes the value of pi under the L3 metric roughly 3.26.

There's nothing really special about the number 3.

We can actually replace it with any number, p, that's greater than or equal to 1.

Mathematicians call these Lp metrics, and they're used all the time.

So what do circles look like under these Lp metrics?

When p equals 1, you get the taxicab metric we saw before, so the circle looks like a diamond.

As p gets a little bigger, closer to 2, the circle begins to look more like the circle you know and love.

When p equals 2, it's the Euclidean metric, and so we get a normal circle.

Then as p keeps getting bigger, the circle starts to look more like a square.

In the limit as p goes toward infinity, it turns into a square.

But here's the amazingly cool part about how that changes pi.

When p equals 1, the value of pi is 4.

Then as p gets a little bigger, closer to 2, the value of pi gets smaller.

When p equals 2, pi is the normal 3.14.

Then as p keeps getting bigger, the value of pi goes back up toward 4.

So among the Lp metrics, 4 is the maximum value of the ratio of a circle's circumference to diameter.

And 3.14159 and so on is the minimum.

That is, pi is the minimum value of pi.

There are actually metrics different from the Lp ones which gives smaller values of pi.

It's a little complicated to write down.

The details are in the description.

But there's a metric whose circle looks like a hexagon.

And the value of pi, it's 3, exactly 3.

In fact, it's a theorem that for metrics given by a linear norm, which is a technical term for the kind of metrics we've been looking at, the value of pi is always between 3 and 4, inclusive.

So while pi is not always 3.14, it's always kind of close.

What's your favorite metric?

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

Hello.

This week I'm doing comment responses from the Joint Mathematics meetings, which is a huge math conference.

And I have the awesome privilege of working for their press team, which means that I'll spend the weekend interviewing awesome mathematicians.

But right now I'm going to respond to your comments about our video, "Can You Hear the Shape of a Drum?"

A lot of you wanted to know, does this question make sense in other dimensions?

It does.

And in fact, it makes sense in other contexts.

The famous topologist John Milner pointed out that there are two 16-dimensional tori that sound the same.

A torus is just the fancy math word for a donut.

So he's really saying that there are two 16-dimensional donuts whose eigenvalues of the Laplacian are the same.

So in that sense, they sound the same.

But they have a different shape.

Huh.

I got a good chuckle out of people like Chris Morong who were really frustrated that I used the triangle, which is the Laplacian symbol, instead of the upside down triangle squared.

The upside down triangle is sometimes called the nabla or del symbol.

These are the same things, mathematically identical.

But I really like that people are debating about which symbol to use because it really shows that mathematics has an aesthetic to it, that there's qualities besides the literal meaning of the symbols that matter.

So keep arguing about those things.

Calvin Jones asked, "Would you expect those two drums to actually sound the same?"

211 00:10:33,940 --> 00:10:36,790 I don't know.

The real world isn't exactly my thing.

I work in the math world.

And I know that those two drums have different shapes but produce the same eigenvalues.

So in that sense, they sound the same.

There's a more detailed sense in which something could sound the same, which is called homophonic.

And I included a link to that.

But that's sort of a technical thing.

I think he's really asking about the physical question because waves interact with space as they come out of the drums.

They interact with your ear as you hear it.

And I have no idea what the answer to those questions are.

And I would love it if one of you experts in the real world could answer those questions.

All right.

That's it for this week.

See you guys next week.

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