This is a super math relay race that spans 400 years.

Since the great god upstairs mentioned this conjecture, I will simply write down the proof process of this conjecture.

If I were scum, I wouldn't be involved in the theoretical part.

This is Kepler's conjecture: how can we pile the spheres most densely?

At the end of 1590, a British navigator named Rowley asked a seemingly simple question.

He wants to design a method of stacking shells so that he can easily count how many shells there are in each pile.

He gave this question to his assistant, Harry Riot, who is a clever young man and wants to design the most effective stacking method.

In order to store more shells in a limited space during the voyage.

Harriot has made great achievements in other natural sciences, but although this problem seems simple, it has not made progress for a long time.

So the young man wrote a letter to a mathematician, physicist and astronomer in Prague.

Of course, the recipient is not three people, he is Kepler. Mathematics, physics and astronomers.

So, the first stick of the relay race was given to the master who was born in Stuttgart.

Kepler wrote a pamphlet called Hexagonal Snowflakes in161year. This is an unofficial publication for a friend, in which he asked why snowflakes are hexagonal and why beehives are hexagonal.

After asking this question again, Kepler turned to another plant-pomegranate.

This is a study from the efficient superposition mode of two-dimensional plane to three-dimensional space.

He thinks that the accumulation mode of pomegranate seeds must be the most efficient in the limited space of pomegranate.

He reached the same conclusion as hales, a botanist after 100 years. Hales squeezed a lot of peas.

It was observed that some peas were squeezed into dodecahedron like pomegranate, but the beans were squeezed into pea paste. But it turned out that the experimental conclusion was wrong. Mendel: You don't want to give me peas. Why squeeze them?

Ok, let's have a rest here. Kepler thinks that the arrangement of nature must be the most perfect, so he thinks that one ball around twelve balls is the closest accumulation.

But he didn't prove it, nor did he say how to surround it.

For each of us, how to load the ball most efficiently seems to be a simple problem.

You put the ball on the first floor, then put the ball on the second floor in the gap on the first floor.

This is the famous face-centered cubic pair stacking. However, there is another stacking method. Although the name is cool, it turns out to be equivalent to face-centered cubic stacking. That is, the six parties are the most dense.

Let's talk about a two-dimensional plane first, and how to arrange circles is the most efficient.

This looks like 1+ 1=2.

1528, a German Renaissance artist wrote a math textbook.

According to the book, circular patterns placed on the ceiling can only be placed neatly if they are arranged in squares and hexagons. It is pointed out that hexagon is the most compact. (Kepler: there is looting in the trough.

Well, the baton was given to an Italian who had just lost all his property.

His name is Lagrange. The greatest mathematician of the eighteenth century.

So far, the research setting is based on the fact that the centers of all circles are arranged in a neat grid.

Lagrange easily proved that hexagonal packing is the tightest in this case.

Norwegian mathematician Du Xing picked up a stick and began to study the general situation, that is, how to stack the circles most tightly when they are randomly arranged.

Unfortunately, there is not much substantive progress. The baton spread to the Soviet Union, and a little boy named Minkowski immigrated to Germany with his parents.

He later became an assistant professor at the Federal Institute of Technology in Zurich, and many students in his class often skipped his classes. One of them is the greatest Patent examiners of the 20th century.

Albert Einstein.

He pointed out that the regular packing density of a circle is at least 0.8224.

But he did not point out the appearance of this arrangement. For fear that Minkowski would steal his thunder. Du was the first person to make a speech to prove it. However, the mathematical community thinks that his proof is not perfect.

Thirty years later, Hungarian mathematician Tos perfected the proof of plane filling problem.

Later, Kocheno, a math class at the University of Wisconsin, proved the problem of plane coverage. (Overlapping is allowed for overlay, but not for padding. )

It is proved that hexagonal arrangement is the best filling and the most effective covering.

Xi Tretow

The mathematical relay of the two-dimensional plane has been completed, and what needs to be solved now is the proof of the three-dimensional world.

In order to describe the three-dimensional problem, we must start with the runner on another track.

Newton and his gay friend David Gregory. They think that one ball can touch several other balls at most on the plane. We now know that the number is 6.

They expanded the problem into the air. How many balls can a ball touch in the air at most?

And had a heated debate, but their debate is only a local problem of Kepler, which is not very useful to prove the conjecture.

(Kepler guessed that there are twelve balls around a ball, and David said that a ball can touch thirteen balls at most in space. Their argument ended with 1953. )

Later, the Swiss mathematician Bender contributed to the German mathematical journal, trying to prove the above argument. His paper was perfected by Hope, the editor of the magazine, who published Bender's paper together with his own.

It seems that the stick ran smoothly, but our hope player lost the stick and his paper proved fatal.

This problem was later solved by the Dutch and the Germans.

The runner on this fork has finished the whole race. Let's review our initial trajectory.

Baseball players are a little strange to us now. His name is Augustus hippo. He tried to prove that the square of cube volume divided by the square of twisted box volume is always less than three.

For this seemingly insignificant little number, he wrote a 248-page thick book.

Then I handed the baton to the captain of this marathon relay race, the math prince Gauss.

However, Gaussian is Gaussian.

He spent a page and a half behind the 248-page proof of Hippo, pushing the limit of this ratio to 2.

What a miracle! I seem to hear Gauss draw his sword and shout, "We have penetrated the enemy's armor! Prepare to charge! "

Through this page, Gauss semi-indirectly explained that the highest density limit of the densest stacking mode of circles under regular arrangement is 74.05%. (When the ball is in a three-dimensional grid)

Then the question is, what kind of superposition can achieve this density? Kepler's Is this the only one?

For nearly a century, the baton silently stopped on a page and a half of Gauss' proof.

Until1August 8, 900, the second international congress of mathematicians was held in Paris.

German mathematician Hilbert put forward 23 famous mathematical problems.

Kepler's conjecture, question 18.

At this time, the relay race entered a hot stage, and mathematicians wanted to find a closer arrangement than Kepler's conjecture. (For example, a chaotic arrangement)

So they took the density of 74.05% as the lower limit and 100% as the initial upper limit.

What we need to do now is to narrow their distance.

Danish cloth litchfield took the stick, reduced the upper limit to 83.5%, and then passed it on to Scottish mathematician Rankin. With the help of Cambridge Mathematics Laboratory, he reduced the upper limit to 82.7%.

At this time, the research methods they said before will come to an end, and the upper bound cannot continue to decline.

Toth, who had received the baton before, came up with another method.

This method was put forward by another Russian mathematician, Voronoi, but his untimely death was not well proved.

He suggested that all we have to do is find a cube called V-unit.

This V cell needs to have two characteristics. First, it can fill three-dimensional space without gaps, just like a cube. Second, there is a ball in it.

In this way, the volume of the ball is unchanged, and as long as a smaller V unit is found, the packing density of the ball will increase.

Rogers of Birmingham University used this method to reduce the upper limit to 78%, which was a wonderful run.

After another 30 years, Lindsay of California Institute of Technology took the baton and ran 77.84% of the good results. Then the mathematician Mulder drained the potential of the V-unit method and took him to the extreme.

The upper limit has been lowered again, although it is only one in ten thousand, but it is not easy.

All of a sudden.

Xiang Wuyi, a native of Taiwan Province Province and University of California, Berkeley, took the baton and rode directly across the finish line!

It's a pity that his proof is considered incomplete and has many loopholes. Our attack failed to break through the core! Observed signs of enemy life!

The baton returned to rookie hales.

As long as the upper limit drops to 74.05, Kepler's conjecture will be proved immediately.

Hales adopted Delaunay's method. Assuming that the space is full of spheres, we connect adjacent centers with straight lines to get many tetrahedrons, and then analyze and calculate them.

However, hales has not made much substantial progress. This method does not lower the upper limit, but directly proves Kepler's conjecture. If it doesn't succeed, it will get nothing.

At the suggestion of Princeton colleagues, hales began to use computers to fight this problem that has not been solved for hundreds of years.

He made a detailed analysis of many possible arrangements.

However, the result of running the program is unexpected.

The results show that no arrangement can exceed 74.08%.

Hmm? 74.08%? This is different from the agreed 75.05. I fell! Director, did you give the wrong script?

After inspection, hales found an odd arrangement, which seemed a little closer than Kepler's stack. Call it a BUG.

Next, his works are divided into five parts. In a simple summary, he proposed a method of grading each arrangement. He only needs to prove that all four categories except Kepler's arrangement are below 8, and then prove that the arrangement of bugs is also below 8. Kepler's ranking score is 8.

The first four categories are easy to complete.

Only the BUG was left, and this powerful foreign aid appeared. One of the patients of Dr. hales's father happened to be a professor of mathematics, and his son became a student in hales.

Coincidentally, there are no books.

Hales had expected to complete the analysis of this wrong arrangement within a few months.

In fact, it took them three years.

Finally, on the morning of August 9th, 1998. An ordinary Sunday.

Hales sat down and wrote an e-mail telling colleagues all over the world that an old and complicated conjecture in discrete geometry had been proved.

The research process and computer program code are attached.

But there are still many people who have doubts about this exhaustive proof method.

At this point, Kepler's conjecture was proved to be over.

This seemingly intuitive conjecture took 400 years to be basically proved.

This group of the most outstanding geniuses in human history went forward bravely and followed the relay.

Most of them can't see the day when this conjecture is proved.

If the truth and laws of this world are hidden in the darkness,

Then thank them for lighting the bright torch for us.

May the fire never go out.