La paradoja de la costa ROMPE la REALIDAD | Fractales - YouTube

Do you know the good Lewis Fry Richardson?

Well, he was a little bit obsessed with ending the war in the world.

But not with tanks and bullets, but with mathematical relationships.

He was certain that there was some relationship between borders and wars.

If the border between two countries was small, chances are they would get along.

And if the border was large, they would end up in a fist fight sooner or later.

The theory was interesting and to test it he took the case of Spain and Portugal.

And something strange happened to him: according to the Portuguese, the border measured more than 1,200 kilometers;

and according to the Spanish, less than 1000.

And it was not an isolated case: in most countries the figures did not fit.

In other words, they did not agree on the size of the border.

And it wasn't that they'd left a decimal, no, no. It was a big difference.

Richardson still didn't know, but he had just discovered a flaw in mathematics and

in 20th century physics. A failure that was about to start a revolution.

Mathematicians have an ancient and infallible technique to measure lengths:

to use a ruler.

Yes, measuring the length of a straight segment is very easy:

you use a ruler and write down the Outcome. Easy-peasy.

With some figures you have to use it several times, but you put it here, you put it there, you measure,

You add the results and you're done. That's why the ruler is such a good method.

Although, well, it has a drawback, and that is that the figures which are not straight are more difficult to measure.

And fixes can be made, but almost all fail.

For example, we cannot measure it directly because the ruler could slide and we would lose count.

to move the ruler, not a good idea.

And we couldn't use a flexible ruler either, because sometimes the marks would be closer together

or more separated, and the results wouldn't match.

In the end with so many restrictions there are not many things we can do.

The only way would be to be patient and go little by little.

We draw a few straight lines on the circumference, just like that.

Okay, the figure we get doesn't look much like the original curve,

but its sides are straight and we can measure them with a ruler.

We add the pieces and we get a result.

Meh, it doesn't look much like the original length, but hey, it's an approximation.

We can use smaller rules.

Like this the drawing will be more adjusted to the actual circumference, and when adding the length

from all the pieces we will obtain a more accurate result than before.

I think the idea is simple:

if we use smaller rulers, we can get as close to the original curve as we want.

The smaller the strokes are, the more accurate the result will be.

For you to see that there is no trap or cardboard here, I want to show the numbers of the previous circumference.

First of all, if we draw ten strokes we get a result with a correct number.

(Okay, 6!)

If we double the strokes to 20, we get another correct figure.

And if we are very motivated and we keep doubling the number of strokes

we will get more and more correct figures.

Oh, and we could have seen it in graph form:

As we increase the number of strokes, the length we measure approaches a fixed value.

That is the actual length of the figure.

And I emphasize this, but this method is the Holy Grail of mathematics,

because it allows to calculate the length of almost any figure.

For example ... Oh, let me see which one I choose ... All right, yes, the lemniscate curve.

The procedure is the same: we draw a few straight lines and we increase the quantity.

You don't have to do this yourself, you can make the computer do it.

The point is that the more strokes, the more precision.

In the end, the length stabilizes and then you have found the result you were looking for.

And here comes the good thing, because if you have understood how to measure curved figures with a straight ruler,

then you have understood the essence of calculation.

To convert a difficult problem that we don't know how to solve

into many easy problems that we do know how to solve.

And when we have solved the simple problems, we put them back together

to get the solution.

Moreover, you have surely heard the names:

studying the small strokes is a topic related to derivatives.

And putting it back together is what young people call integrals.

Afterwards everything gets complicated with formulas, nuances, this and that... But this is the idea.

That's why they say that the calculation is the language with which the universe is written.

And we will dedicate some videos, but the important thing now is to know

that calculation simplifies the universe so we can study it.

Without it we would be lost.

Luckily, we have been using calculus for centuries and it has always worked.

Or it worked, until Richardson arrived.

Ok, so we go back to the original problem.

Actually, the fact that the size of the border doesn't fit isn't such a mystery.

Each country was using a different rule, that's why the results did not match.

In this case, the Portuguese had a more approximate result.

But Richardson was not very happy,

and he knew that precision could be gained by using smaller strokes in the drawing.

The idea that had occurred to him was to use the official measures, the one from Spain and the one from Portugal,

and then to make more and more measurements until gradually approach the real value of the border.

Come on, the way it had always been done.

The problem was that the next measurement was much larger than expected. And the next, much more.

According to the map, the length of the border kept growing without limit towards infinity.

The smaller the strokes, the larger the border.

This was the flaw Richarson had found, which is currently known as the coast paradox.

Natural borders, whether between countries or with the sea, seem to be infinite in length.

The calculation does not work with them, and this is due to the shape they have:

no matter how much you zoom, more and more details appear every time.

I mean, classic figures can be measured well because when you zoom they end up

looking like straight lines.

But with the natural figures the zoom does not eliminate the details.

No matter how small you make the rules, there will always be details that you have not measured before

and they increase the length over and over again.

These types of figures are the famous fractals:

figures that have details at any scale.

So it looks like they are broken, or fractured, or have holes everywhere.

According to Richardson's studies, fractals existed beyond the mathematical world.

In fact, you have to be a little blind not to see them, because they are everywhere in nature:

the trees, the veins of the leaves, the vegetables in general, the human body,

rivers, lightning, ice ...

Objects full of detail that challenge the mathematical models in front of us.

Figures with infinite lengths, with fractional dimensions, and other crazy things

related to each other in very strange ways.

So we are going to dedicate a series of videos to explain all these characteristics.

(And we could talk about this theme for a while, because the story begins three centuries before Richardson.)

But before we get into trouble, we have to solve the paradox of the coast.

Is the border really infinite or does it have a catch?

Well, at first we thought that nature was described by the laws of calculus.

And then the fractals appeared.

But the truth is that nature is between the two theories.

Zooming to the coastline makes details and details appear and the length grows endlessly.

This is the same thing that Richarson found when he made the graph of the border,

that it grew and grew without stopping.

But if you keep insisting, the details disappear, and in the end the length stabilizes.

In other words, nature does not have infinite precision, it does not have infinite details.

At some point they disappear.

Therefore, sometimes it is convenient to study it as if it was a fractal and sometimes with the classical theory.

Of course, there are other nuances that make coasts difficult to measure:

Does not the length of the coast vary with the waves and the tides?

Or what grain of sand do we consider land and which one not?

In the end, seeking so much precision doesn't have much sense.

Nowadays, if you want to know how much a border measures, look it up in wikipedia and leave the ruler at home.