What is more terrifying than the non-existence of physics is pi丨Happy Pi Day

In The Three-Body Problem, before committing suicide, Yang Dong asked himself fearfully:

“Nature, is it really natural?”

What do you think?

Think again

This question

you

Are you scared?

...

1

Proposal Day

This is the most unique proposal I have ever seen.

Today, March 14th.

A boy from the mathematics department suddenly knelt on one knee, looked at his girlfriend affectionately, and took out a...

...Apple pie?

Copyright image, no permission to reprint

I took a closer look and found that this apple pie was a perfect triangular slice, and its top view perfectly matched the outline of the following formula:

It suddenly dawned on me that this was what his proposal meant.

If this formula is unfamiliar to you, let's reset the slice:

Copyright image, no permission to reprint

Now, doesn't its shape resemble a circle?

If you are baking a mini pie with a diameter of only 1 cm, then its circumference will become the main focus today:

π.

Do you remember the pi formula that your math teacher asked you to memorize when you were a child?

3.141592653..

Today.

March 14, International Pi Day.

You will soon realize:

Behind these numbers,

Included,

The entire universe.

2

Specializes in treating all kinds of dissatisfaction

As I recall, in May when the college entrance examination was approaching, everyone became restless.

One day, the math teacher wrote this formula on the blackboard:

"How many times have I emphasized that the big questions must have a complete calculation process, but what about the result? You like to just write the answer, right? OK, who can tell me what the answer to this formula is?"

The classroom was silent.

The teacher tapped the blackboard with a ruler and said, "You can't even recognize π, what are you so proud of?"

Later, in college calculus class, I learned that this was the Wallis product discovered by John Wallis in 1655, and was the second infinite-term formula for pi discovered in Europe :

Simple proof process of Wallis product

No wonder we couldn’t answer it back then!

It’s a dimensionality reduction attack, a naked dimensionality reduction attack.

However, I think this trick of the math teacher is worth learning!

Next time, when you want to test the other person's depth, you can also ask the value of the following formula:

If the other person remains silent, you can say earnestly: " You don't even know π, you should read more books. "

After all, this is Leibniz’s formula for π , and given enough terms, the sum will slowly approach π.

Although, the convergence rate of this series is very slow, and it takes 500,000 items before it can be accurate to the fifth decimal place of π...

But no matter what, π is the cure for all kinds of dissatisfaction!

3

Infinite non-loop

Let's review an elementary school math problem:

Please move a matchstick so that the equation below becomes another approximate equation.

View Answer

The reason why "approximate equation" is emphasized in the question is because π is an irrational number and cannot be expressed as the ratio of two integers . Although we often use fractions such as 22/7 to approximate π, in fact π is an infinite non-repeating decimal .

However, every irrational number can be expressed in the form of a continued fraction , and π is no exception, for example:

If we cut off at any point, we can get an approximate value of π. If we cut off at the second line, we get 22/7; if we cut off at the fourth line, we get 355/113.

These two values ​​are pointed out because they have historically shined as approximations of pi.

In 250 BC, Archimedes proposed in his treatise "On the Measurement of the Circle":

He used the circle cutting method:

Schematic diagram of the circle cutting method, source [1]

The circumference of a circle is between its circumscribed polygon and its inscribed polygon . As we continue to increase the number of sides of the polygon, we can continue to reduce the difference in circumference. Therefore, by calculating the circumference of the polygon, we can obtain the upper and lower limits of the π value with a certain accuracy.

In ancient China, our exploration of pi also has a long history.

In the Zhoubi Suanjing, the oldest astronomy and mathematics book in my country, there is a sentence: "The method of numbers comes from the square of the circle." Zhao Shuang, a mathematician in the Three Kingdoms period, commented on it as: "The diameter of a circle is one and the circumference is three." This means that the circumference of a circle with a diameter of 1 is approximately 3.

It can be seen that at that time, the rough estimate of pi we used was 3 .

In 462 AD, Zu Chongzhi recorded in his book "Zhuishu" the approximate value of pi he calculated, which is 355/113. The value expanded into a decimal is 3.1415929203...

For nearly 800 years, this remained the most accurate estimate of π.

In fact, the estimation of pi had very direct practical significance in ancient times.

For example, at that time, both ordinary people and royal nobles were very concerned about one thing: when would it rain and how much would it rain.

For this reason, court officials needed to revise the calendar, which involved calculations of circles. If the approximate value of π had a large error, it would be impossible to accurately predict the four seasons of the year, which would ultimately directly affect the people's livelihood of the entire country .

The accuracy of 355/113 can be specifically felt by taking an example:

Assuming that the diameter of a circle is 10,000 meters, the circumference calculated using it is only less than 3 millimeters more than the true value!

Therefore, Zu Chongzhi's achievement is of great significance both to the common people at that time and to the research progress of later generations.

4

Randomly throw a needle to get π

Since today is International Pi Day, why not play a little game while eating pie?

Draw parallel lines 4 cm apart on a piece of paper, find n 2-cm-long toothpicks, randomly throw them on the paper, and finally count the number of times k that the toothpicks intersect the parallel lines, and calculate the value of n/k.

Random Toss

After statistics, it was found that the value of n/k is very close to pi!

This is actually the famous Buffon experiment.

Assuming there is a set of parallel lines with a distance a, and the length of the thrown toothpick is l, the probability of the toothpick intersecting the straight line can be simply calculated as follows:

Simple schematic diagram

Assume that the toothpick AD intersects the straight line MN, B is the midpoint of the toothpick, the angle between the toothpick and the straight line is θ, and the vertical distance from point B to the straight line MN is s. Then s≤lsinθ/2 must be satisfied for the toothpick to intersect the straight line.

The angle θ where the toothpick intersects the straight line MN varies from 0 to π, and the range of s varies from 0 to a/2. A simple diagram is drawn as follows:

The curve in the diagram is s = lsinθ/2, and the shaded part represents the intersection of the toothpick and the straight line. The area of ​​this rectangle represents the total number of throws , so the intersection probability can be calculated as follows:

In the above game, we chose the parameter a=2l, so we got n/k=π.

Theoretically, as the number of tosses increases, you can get a more and more accurate value of π. Many people in history have conducted this experiment :

Table of some historical experimental data, source [2]

If you observe carefully, you can find that the accuracy of the value of π does not seem to be proportional to the number of tosses.

Rudolph threw the dice 5,000 times, while Lazzlini only threw it 3,408 times, but the value of π he obtained was much more accurate than that of Rudolph.

In this regard, many scholars have suspected that Lazlini's data is falsified.

But in fact, this throwing experiment also involves the optimal stopping problem : how many times to throw to stop to get a better solution.

Putting all this aside, the Buffon experiment was the first example of expressing a probability problem in geometric form . It was the first time that a random experiment was used to deal with a deterministic mathematical problem. This was not only the prototype of the Monte Carlo method , but also promoted the birth of integral geometry .

But don’t forget, all of this started with our desire to find the value of π.

It seems that something is pulling us along. In the process of constantly exploring pi, we have touched upon a more vast and boundless world.

5

Supercomputing Warm-up Exercises

Our exploration of pi has spanned thousands of years and has never stopped. As we turn the clock and fast forward to this era, the story of pi has new participants:

Supercomputer.

In August 2021, Swiss scientists broke the world record by using a supercomputer to calculate pi to 62.8 trillion decimal places, which took 108 days and 9 hours.

Unexpectedly, just over half a year later, the record was broken again!

In March 2022, Google Cloud calculated all 100 trillion decimal places , which took less than 158 days, and the 100 trillionth decimal place was exactly 0.

The last 100 digits of pi to the 100 trillionth decimal place, source [3]

In fact, from the perspective of actual measurement, if the value of pi is accurate to 39 digits, the circumference of the observable universe can be calculated accurately to the size of an atom , which can meet the computing needs of most cosmology at present.

In that case, what is the point of calculating to trillions of decimal places?

Have you ever thought about how we can test a series of indicators such as the reliability, accuracy and computing speed of supercomputers when they are developing so rapidly?

Now it’s π’s turn to come on stage!

Using supercomputers to calculate multi-digit values ​​of π is a common method currently used to test computer performance and improve calculation methods.

Just as we continue to break records for climbing Mount Everest, as a supercomputer, the value of π is the peak they need to climb.

Simply put, we first need to use the π value calculation program on a working supercomputer and conduct multiple experiments to confirm that there is no problem with the program;

Then use this program on the test machine. If the test machine makes an error when calculating pi, it means that there is a problem with the hardware of this supercomputer and further inspection and adjustment are needed.

From this perspective, the infinite value of pi is probably a warm-up exercise for supercomputers .

When a supercomputer breaks the world record for the value of π, the warm-up is over, and the next step is to show its brilliance in other research fields.

6

Cosmic Code

In The Three-Body Problem, Yang Dong asks before committing suicide: "Is nature really natural?"

Carl Sagan hinted in his novel Contact that the creator of the universe had hidden a message in the number of π .

Therefore, for many π fans, nature may not be natural, and the ultimate code may be hidden in π .

For example, the mass ratio of protons to electrons is approximately 1836, which is exactly equal to the rounded value of 6π5.

Wait, is this really just a coincidence?

Could the intrinsic properties of elementary particles be closely related to some geometric features in the universe?

There is no theoretical basis for this statement so far, and it is very likely that it is just a coincidence ...

In comparison, what is more interesting is that the value of π2 is very close to the value of gravitational acceleration g .

This is no coincidence! It has to do with the definition of the unit of length m.

In 1660, the Royal Society of London proposed that a simple pendulum about one meter long on the surface of the earth would swing once in about one second.

That is to say, the original definition of length m is: the length of a simple pendulum with a swing time of 1s .

Let's look at the period formula of a simple pendulum:

Since T describes the time it takes to complete a round trip, we substitute T=2s, ignore the units, and simply transform it to get:

Since we define the length L of the simple pendulum at this time as 1m, we can get that the values ​​of π2 and g are equal!

That is to say, at the very beginning, π2=g.

Later, we continued to adjust the definition of unit length m, which led to changes in the value, but the difference was not large, so the current π2 is very close to the value of gravitational acceleration g, but not completely equal.

In addition, π also appears in various physical worlds:

Fine structure constant

Heisenberg Uncertainty Principle

Maxwell rate distribution function

In fact, π appears not only in various physical formulas, but our daily lives are also closely related to it.

There is a very intriguing clip in "Person of Interest" that is worth our careful consideration.

Now, eating a $3.14 pie.

Let’s take a closer look at the magic of π.

It is an irrational number, infinite and non-repeating;

It is also a transcendental number and is not the root of any polynomial with rational coefficients.

It contains all the infinite possibilities in the universe.

So, the meaning of raising a party to propose marriage is...

In the Name of Pi

“

Like you,

I don't know where it came from.

Beyond everything,

Infinite,

Run towards you.

”

I see!

What are you standing there for? Why are you still eating pie?

I'm talking about you!

Why don't you hurry up and go and express your feelings!

Oh, that's right!

Remember to discuss quietly:

Is nature really natural?

References:

[1] Wikipedia: Pi

[2] The Code of Pi

[3] Google Cloud Blog

Editor: Mueller's nanny