A fateful duel

The night before the duel that took his life at the age of 20, Everist Galois tried to develop the theory of bunches, which is the mathematical language of symmetry. Mario Livio, author of the book "The Language of Symmetry - The Equation for Which No Solution Has Been Found" tells in an interview with Galileo about the mathematicians who died young, about the importance of symmetry and also about breaking it

The article was published in Galileo, May

We all remember from high school the formula for finding the roots of a quadratic equation. It is vaguely known to many that there are similar formulas for finding the solutions of third and even fourth degree equations, but even the greatest mathematicians got "stuck" in the fifth degree when they ran short there.
These equations remained unsolved for over three hundred years, until two geniuses separately proved that equations of the fifth degree could not be solved by simple formulas. These geniuses - a young Norwegian named Nils Henrik Abel and a Frenchman named Oriste Galois, discovered this, and both died a tragic death. Galois, in fact, spent the last night before the fatal duel (at the age of 20) writing a summary of his proof, occasionally writing in the margin "I don't have time." Some of the mysteries surrounding his death are also solved in the book, after disappearing for 170 years.
Galawa's work marked the beginning of the mathematical field known as "bundle theory" the "language" that describes symmetry. Bunch theory describes something of the aesthetics of our world, the matching of the Hungarian cube, the music of Bach and Mozart, the physics of subatomic particles and the popularity of Anna Kournikova.
"Until now, there has not been a popular comprehensive book on the theory of bruises and there was a reason for that, because it is simply very difficult. Set theory is one of the more abstract branches of mathematics. The more abstract the things, the more difficult it is to convey them in a popular way, so it was a very serious challenge to deal with this particular material and convey it to a popular book." says Mario Livio in a special interview he gave to Galileo during a short visit he made to Israel. Livio, who in the XNUMXs filled Churchill Hall at the Technion, a hall that was usually used for performances, in his lectures in introductory undergraduate physics courses. For about a decade, he has served in various positions at the Space Telescope Scientific Institute, where the operators of the telescope and the interpreters of its photographs sit. Among other things, he has a term as the scientific director of the institute, and currently he is engaged in the field of information. This is Livio's second book. His first book - Cutting the Zeva became a bestseller, also in the Hebrew edition published by Aryeh Nir. The same publishing house is now publishing the book The Language of Symmetry - the equation for which no solution has been found.
"The unsolved equation is the fascinating part of the story." Livio says. "The theory of groups is the language of symmetry, but the path to discovering this language was difficult. They came to it out of an attempt to solve an equation that no one wanted to solve. Hence the genius of the hero of the story, Louis Galois.
Intermediate: What is symmetry
Symmetry of some kind of system is a property of the system that when you do something to the system it does not change. So you say the system is symmetric under this operation. For example, MADAM I'M ADAM is a palindrome, you read it from the end, this sentence is symmetric under the specific operation of reading from the end to the beginning. On the other hand, if we say that the laws of nature are symmetrical under rotations, what do we mean - that the laws of nature do not depend on direction. They do not depend on whether I measure the directions in relation to the adjacent building, together to the north, etc. If I take the universe and turn it around, the laws of nature will not change.
Symmetry under copying. This is the same symmetry that you see in the laws of nature - the laws here are the same as in Alabama and as at the other end of the galaxy. Even in the houses there is symmetry in the structure of the house. There is symmetry in the structure of the wallpaper found here in the cafe (Gilad and Daniel, on Dizingoff St. in Tel Aviv, and you can see a section of it in the background of the photo of Livio, AB), in many things there is symmetry under copying and also in music.

Every person who deals with physics comes across the theory of bunches. You encounter it most if you deal with elementary particle physics or high-energy physics, but in fact in most branches of physics, people even in a profession that is quite far from mine - people who deal with crystallography, properties of crystals - all the description there is done using bunch theory because this whole field consists of symmetries. The fundamental theories of physics are built on symmetries and therefore the language used is the theory of bundles. I wanted to convey in the book the message of the importance of symmetry and its language.
"Suppose there was an ancient man here a million years ago. This person turned his head, there were no buildings, there were no cars. There were trees, sand. The man turned his head and saw something symmetrical, what was it, almost certainly another animal. His brain had to be developed in such a way that he would recognize such symmetries quickly and learn them quickly. If he didn't learn them it could cost him his life. This is one of the reasons why our brain has evolved in such a way that it is very sensitive to symmetries. Another reason why the brain evolved in this way is mate selection. For example - when the female peacock needs to find a partner, she looks for a partner with strong genes. The way to identify new genes is by looking at its tail. If his tail is completely symmetrical she can be pretty sure he has strong genes. The reason is simple. There are millions of poultry parasites. When these parasites attack the peacock, its tail becomes shaggy and asymmetrical. If she sees the tail symmetrical, it is a sign that this particular male has managed to control the parasites, a sign that he has strong genes. It is important to understand that symmetries play a very important role not only in the laws of physics - not all people deal with the laws of physics every day but also in other areas of life. It is not a coincidence that Mozart or other composers create this symmetry in their music. This is the kind of thing that gives us pleasure. They also give us comfort."

"One of the main features of is the closure, if you take two members of the group and unite them by the operation of the group you get another member of the group. For example all integers, with a simple arithmetic addition operation form a bunch. If you take a whole number and add another whole number to it, you also get a whole number. You took one member of the gang and performed the action on him with another member of the gang, you got a third member of the gang."
"What is there in symmetry that gives it this matter of closures? If you look at the set of all symmetries of some system, the set of all symmetries of the system is always a bunch. How can I show the closures? Let's take the action of doing something after something else. Suppose I have one action that the system is symmetric about. If you have a system and it has some kind of symmetry, it means that when you perform the operation on the system it does not change. For example if you take a snowflake and you rotate it sixty degrees around its center it stays the same. This is the symmetry of the snowflake. If I rotate the snowflake 120 degrees it also stays the same. We will perform the actions one after the other. We will rotate it by sixty degrees and after that by 120 degrees. We turned it 180 degrees and even then it remained the same flake. This was the genius of Galois, to understand that these bruises suddenly describe all the symmetries."

Intermediate: breaking the symmetry
However, despite everything that has been said, nature is not symmetrical. If it were perfectly symmetrical, there would be no room for complexity. And here Livio provides an example from the field close to his heart: "If, for example, the number of particles in the universe were exactly the same as the number of antiparticles in the universe, they would all annihilate each other, that is, from ionization there would be nothing left and we would not be left to talk about it. There is a tiny fraction - one part in three billion, there were a few more particles than anti-particles. All the particles and anti-particles destroyed each other and what remains is the tiny fraction - one in 3 billion. It creates all the baryonic matter we see in the universe.”
The symmetry of the human body and my husband's body is also not symmetrical and it is clear that if you cross the face in the picture and duplicate the other half there, you do not get the exact same face. The inside of the body is certainly not symmetrical and neither is the functioning of the brain. Livio also provides a mechanical explanation for the external symmetry. In the gravitational conditions of the earth there is a difference between the upper part of the body and the lower part. In animals there is a preference to place the sense organs in the front of the body. For a fast moving animal it is important for it to have all the senses in the front to know where to move forward, and so on. And the evidence, among creatures that don't move much like plants, there is usually no difference between the front and the back, but they obviously have a difference between top and bottom. And animals that move very slowly, like jellyfish, have cylindrical symmetries. You don't see much change between forward and backward. But Livio admits that this is not a perfect explanation either, because it turns out that even in the process of development from the cell to the complete being, there are symmetrical and not so symmetrical stages. There is another famous breaking of the symmetry known as chirality - the amino acids are compounds in proteins. All animals and plants on Earth, without exception, use amino acids called "left-handed". This entire array could be built from "right-handed" amino acids. This is probably due to the same reason why Microsoft is used all over the world. This is not to say that Microsoft's operating system is the best but once one has taken over, all the others must also know how to speak Microsoft, because otherwise they cannot function.

The book is interspersed with many examples from many fields: from arts such as painting, poetry, music, etc. to the original historical documents related to the lives of the two heroes - Galois and Abel. "I used people to read original documents." Livio said. "I speak French but I do not have full control of French. Certainly when it comes to French from the 18th and 19th centuries, so I used people who speak French. I have a basic understanding of understanding Italian, but there are enough Italians around me anyway. And I was also helped by people who know Latin, and who speak Swedish, which is a language close to Norwegian, and I also called a Norwegian researcher who wrote a comprehensive biography of Abel, and I talked to him. The knowledge of music and art is a result of my hobby. I love art, I love music, I have a lot of books on these subjects, I regularly visit exhibitions, concerts. It came quite naturally, I didn't have to put any effort into it. I don't present myself as an expert in these things, but I know quite a bit, simply because I love it."

A theory that was not in the air
When you look at the history of science, mathematics, you find that a lot of important things that were discovered were already in the air in one way or another. Even special relativity for example, if Einstein hadn't done it, someone else would have arrived at the equation, and Poincaré almost succeeded. There are a lot of things that were up in the air. There are very few discoveries that, when they were discovered, were really not in the air. It's not that they didn't have some roots, but that if they hadn't been discovered by whoever discovered them, it would have taken many years for them to be discovered.
An example of this is the theory of general relativity. If Einstein hadn't thought of it, many years would probably have passed before someone would have come up with such an idea, and Galois's bunch theory is such a thing. Nobody talked about bruises until Galois came up with the theory of bruises nobody else talked about bruises. It was nothing in the air. Suddenly it was discovered that these are the symmetry properties of the equation that reveal which of the equations has a solution or not. Because of that, this discovery is so beautiful because it's one of those rare discoveries that grew inside someone's mind and weren't already ripe and just waiting for someone to write it down on paper."
"They say that mathematicians and poets die young. Perhaps it is not meant that they must die a biological death as actually happened to the two heroes, but that most of their important work happens at a young age, for example painters, writers, usually middle-aged workers, and philosophers at older ages. In mathematics and poetry you don't need to read entire encyclopedias to be able to do something meaningful. It is precisely the fact that you are young, you have not yet been influenced by other people and you do not feel that you have to work in areas that others work in and so on, which gives poets and mathematicians the opportunity to create breakthroughs at very young ages. This does not mean that there are no exceptions. Gauss continued to work and did fantastic work in mathematics into his old age. There are enough exceptions, but by and large this is true. In mathematics and poetry you see people who create their best work at a young age. Sometimes a little in music, you can't say that Mozart composed his best works when he was the youngest but even when he died he was 35 years old, young enough.

Doesn't the academic establishment wear out geniuses?
There are geniuses even today, people manage to function even in the existing establishments. On the other hand there is much more communication. Everyone is on emails. However you still see some young mathematicians doing outstanding things in physics as well. People like Ed Witten who create amazing breakthroughs (string theory). Even in the more institutionalized structure there is still room for geniuses and breakthroughs.
I think the real geniuses will find the way to create their own breakthroughs. Today it is also a little more difficult to isolate yourself, in physics today you need a deep background to even reach the possibility of creating something significant. It is not a coincidence that at the beginning of the twentieth century there was a huge period of discoveries in physics and then things stopped a bit. Whole branches were created then - relativity, quantum theory, there was a place to discover new things. Then when things become more complex it becomes more difficult to discover the local discoveries.
But beyond that, it should be remembered that today enormous instruments are required to provide these discoveries. In 2007, the big accelerator in Geneva will start operating. The expectation is that after a year of experiments we will reach significant achievements. I can guess that they will see the Higgs because if they don't see the Higgs boson - a particularly elusive subatomic particle, it will be a huge surprise. Super Symmetry may or may not find out. If they find out there will be a huge boost because it will say that this direction is correct. If they don't find out it will be more problematic because then they will have to explain the reason for the non-disclosure.
We hope that the Hubble Space Telescope will undergo another service mission and then there will be two new devices on this telescope, one of which is the more sophisticated camera that also sees in the infrared and will allow wider images with greater sensitivity in the infrared and a new spectrograph will be installed in it mainly in the ultraviolet field which will also allow Again to discover the structures, especially in the interstellar and intergalactic medium. In 2013 we hope that the James Webb telescope will be launched, which we (at the Space Telescope Science Institute) will also manage and which will show us the very first galaxies in the universe. In almost every field of physics, a device is designed to perform a large experiment. The Planck space telescope will be launched in a year or two and will examine the background radiation with much greater sensitivity than WMAP. For some of the experiments you need the treasures of a country, sometimes not even a country but an entire continent, in the accelerator it is the resources of the entire continent of Europe, in spacecrafts - the USA. I am optimistic, the next fifteen years will bring fantastic discoveries. I have no doubt about this.

Livio says that the most important message in his book is that people will understand the importance of symmetry, less in matters of wallpaper than in the laws of nature and will see the beauty of the mathematical language of symmetry and also that they will see the creativity, especially the pair of these people, but it's obvious that they did. It is also important to promote the public relations of mathematics. "I'm sure there are many people who don't see exactly where mathematics is important to them at all. But in fact, today mathematics enters literally all areas of life, starting from analyzing the behavior of the stock exchange and ending with research in the field of social sciences. There is a Journal of Mathematical Sociology - which deals with social processes and shows that it is possible to analyze them using mathematical models. People manage to make models of social relationships that look almost the same as models of molecules in chemistry. The group theory is also used in these studies, among other things, when they studied the structure of marriage and the affiliation of children in the aboriginal tribes, they discovered structures and patterns that you can only see through mathematics."

In one of the rare cases, a writer can read the translation of his book into Hebrew. Livio compliments Emanuel Lotem, the translator. "I went over the translation not to change anything in the linguistic style, but to maybe check when I wrote the text in English something that wasn't so clear. Because I happen to read Hebrew there is an added bonus, that I can also read the Hebrew edition, which I can't do in almost any other language and I want to say that the translation is wonderful."

A ratio equal to gold - on Prof. Mario Livio's previous book - The Golden Ratio

19.12.2003

Ever since middle school, we all know "Phi" (9) - the Greek letter whose value was 3.14. Less well known is the letter "pi" (a loose p, marked r), also known as the "golden ratio" or "golden ratio". It is a geometric proportion that expresses a harmonious relationship. For example, if a certain line AB is cut at some point C, the ratio between the length of the segment AC and the length of the segment CB will be the same as the ratio between AB and AC. The golden ratio is expressed in a constant whose value is about 1.618.

Throughout history, many myths have been associated with the golden ratio, which are now presented in a new book, "The Golden Ratio" (published by Aryeh Nir), written by the Israeli astrophysicist, Prof. Mario Livio. Livio, a former lecturer and researcher at the Technion, currently serves as the head of the scientific division at the US Space Telescope Institute. Livio says, for example, that the seeds of the apple are arranged in the pattern of a five-pointed star, with the ratio between each of its sides equal to the "golden ratio". So is the relationship between the position of the leaves along the stem, the scales of the pine cone and the pineapple, as well as crystals, sunflowers, and many galaxy shapes. Since the days of ancient Greece, the golden ratio has occupied mathematicians, philosophers, scientists and artists.
Painters, designers and musicians used the golden ratio to achieve a visual or sound impression in their works. The followers of Pythagoras even believed that he revealed the finger of God.