Steven Strogatz The Pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world. Steven Strogatz - pleasure from x

This book is well complemented by:

Quanta

Scott Patterson

Brainiac

Ken Jennings

Moneyball

Michael Lewis

Flexible consciousness

Carol Dweck

Physics of the stock market

James Weatherall

The Joy of X

A Guided Tour of Math, from One to Infinity

Stephen Strogatz

A fascinating journey into the world of mathematics from one of the best teachers in the world

Information from the publisher

Published in Russian for the first time

Published with permission from Steven Strogatz, c/o Brockman, Inc.

Strogatz, P.

The Pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world / Steven Strogatz; lane from English - M.: Mann, Ivanov and Ferber, 2014.

ISBN 978-500057-008-1

This book can radically change your attitude towards mathematics. It consists of short chapters, in each of which you will discover something new. You will learn how useful numbers are for studying the world around you, you will understand the beauty of geometry, you will become acquainted with the grace of integral calculus, you will be convinced of the importance of statistics and you will come into contact with infinity. The author explains fundamental mathematical ideas simply and elegantly, with brilliant examples that everyone can understand.

All rights reserved.

No part of this book may be reproduced in any form without the written permission of the copyright holders.

Legal support for the publishing house is provided by the Vegas-Lex law firm.

© Steven Strogatz, 2012 All rights reserved

© Translation into Russian, publication in Russian, design. Mann, Ivanov and Ferber LLC, 2014

Preface

I have a friend who, despite his craft (he is an artist), is passionate about science. Whenever we get together, he talks enthusiastically about the latest developments in psychology or quantum mechanics. But as soon as we start talking about mathematics, he feels a trembling in his knees, which greatly upsets him. He complains that not only do these strange mathematical symbols defy his understanding, but sometimes he doesn't even know how to pronounce them.

In fact, the reason for his rejection of mathematics is much deeper. He will have no idea what mathematicians do in general and what they mean when they say that a given proof is elegant. Sometimes we joke that I just need to sit down and start teaching him from the very basics, literally 1 + 1 = 2, and go as deep into math as he can.

And although this idea seems crazy, this is exactly what I will try to implement in this book. I will guide you through all the major branches of science, from arithmetic to higher mathematics, so that those who wanted a second chance can finally take advantage of it. And this time you won't have to sit at a desk. This book will not make you a math expert. But it will help you understand what this discipline studies and why it is so fascinating for those who understand it.

We'll explore how Michael Jordan's slam dunks can help explain basic calculus. I'll show you a simple and amazing way to understand the fundamental theorem of Euclidean geometry - the Pythagorean Theorem. We'll try to get to the bottom of some of life's mysteries, big and small: did Jay Simpson kill his wife; how to reposition a mattress so that it lasts as long as possible; how many partners need to be changed before getting married - and we will see why some infinities are larger than others.

Mathematics is everywhere, you just need to learn to recognize it. You can see the sine wave on the zebra's back, hear echoes of Euclid's theorems in the Declaration of Independence; what can I say, even in the dry reports that preceded the First World War, there are negative numbers. You can also see how new areas of mathematics influence our lives today, for example, when we search for restaurants using the computer or try to at least understand, or better yet, survive the frightening fluctuations of the stock market.

A series of 15 articles under the general title “Fundamentals of Mathematics” appeared online at the end of January 2010. In response to their publication, letters and comments poured in from readers of all ages, including many students and teachers. There were also simply curious people who, for one reason or another, “lost their way” in understanding mathematical science; now they felt that they had missed something worthwhile and would like to try again. I was especially pleased by the gratitude from my parents because, with my help, they were able to explain mathematics to their children, and they themselves began to understand it better. It seemed that even my colleagues and comrades, ardent admirers of this science, enjoyed reading the articles, except for those moments when they vied with each other to offer all sorts of recommendations for improving my brainchild.

Despite popular belief, there is a clear interest in mathematics in society, although little attention is paid to this phenomenon. All we hear about is fear of math, and yet many would love to try to understand it better. And once this happens, it will be difficult to tear them away.

This book will introduce you to the most complex and advanced ideas from the world of mathematics. The chapters are small, easy to read and not particularly dependent on each other. Among them are those included in that first series of articles in the New York Times. So, as soon as you feel a slight mathematical hunger, don’t hesitate to pick up the next chapter. If you want to understand the issue that interests you in more detail, then at the end of the book there are notes with additional information and recommendations on what else you can read about it.

For the convenience of readers who prefer a step-by-step approach, I have divided the material into six parts in accordance with the traditional order of studying topics.

Part I, Numbers, begins our journey with arithmetic in kindergarten and primary school. It shows how useful numbers can be and how magically effective they are in describing the world around us.

Part II, “Ratios,” shifts attention from the numbers themselves to the relationships between them. These ideas lie at the heart of algebra and are the first tools for describing how one thing affects another, showing the cause-and-effect relationship of a variety of things: supply and demand, stimulus and response - in short, all the kinds of relationships that make the world so rich and varied .

Part III “Figures” tells not about numbers and symbols, but about figures and space - the domain of geometry and trigonometry. These topics, along with the description of all observable objects through shapes, logical reasoning and proof, take mathematics to a new level of precision.

In Part IV, Time for a Change, we'll look at calculus, the most exciting and diverse branch of mathematics. Calculus makes it possible to predict the trajectory of planets, the cycles of tides and make it possible to understand and describe all periodically changing processes and phenomena in the Universe and within us. An important place in this part is given to the study of infinity, the pacification of which became a breakthrough that allowed calculations to work. Computing helped solve many problems that arose in the ancient world, and this ultimately led to a revolution in science and the modern world.

Part V, “The Many Faces of Data,” deals with probability, statistics, networks, and data science—still relatively new fields, born out of the less-always orderly aspects of our lives, such as opportunity and luck, uncertainty, risk, variability, chaos, interdependence. Using the right tools of mathematics and the appropriate types of data, we will learn to detect patterns in the flow of randomness.

At the end of our journey in Part VI, “The Limits of the Possible,” we will approach the limits of mathematical knowledge, the border region between what is already known and what is as yet elusive and unknown. We will again go through the topics in the order we are already familiar with: numbers, ratios, figures, changes and infinity - but at the same time we will look at each of them in more depth, in its modern incarnation.

I hope that all the ideas described in this book will seem fascinating to you and will make you exclaim more than once: “Wow!” But you always have to start somewhere, so let's start with a simple but fascinating activity like counting.

1. Number Basics: Fish Addition

The best demonstration of number concepts I have ever seen (the clearest and funniest explanation of what numbers are and why we need them) was in an episode of the popular children's show Sesame Street called 123: Counting Together "(123 Counter with Me). X...

In 2010, Steven Strogatz wrote a series of articles about the basics of mathematics for The New York Times. The articles caused a storm of delight. Each column became the most popular story in the newspaper and attracted hundreds of comments. Readers asked for more, and Stephen did not disappoint - this book appeared, which included both already published parts and completely new chapters.

Mathematics permeates everything in this world, including ourselves, but, unfortunately, few people understand this universal language well enough to appreciate its wisdom and beauty. Steven Strogatz is the math teacher you dreamed of in high school. A teacher who is able to ignite a spark of interest and instill a lifelong love for his subject. In this incredibly easy and fun book, he gives us all a second chance to get to know mathematics. In each short chapter, you'll discover something new, from why numbers are needed in the first place to topics such as geometry, integral calculus, statistics, and infinity. The author explains great mathematical ideas simply and elegantly, with brilliant examples that everyone can understand. This book is for everyone. Those who are little familiar with mathematics will become closely acquainted with it, and those who love mathematics will enjoy reading about the “queen of sciences.”

Preface

I have a friend who, despite his craft (he is an artist), is passionate about science. Whenever we get together, he talks enthusiastically about the latest developments in psychology or quantum mechanics. But as soon as we start talking about mathematics, he feels a trembling in his knees, which greatly upsets him. He complains that not only do these strange mathematical symbols defy his understanding, but sometimes he doesn't even know how to pronounce them.

In fact, the reason for his rejection of mathematics is much deeper. He will have no idea what mathematicians do in general and what they mean when they say that a given proof is elegant. Sometimes we joke that I just need to sit down and start teaching him from the very basics, literally 1 + 1 = 2, and go as deep into math as he can.

And although this idea seems crazy, this is exactly what I will try to implement in this book. I will guide you through all the major branches of science, from arithmetic to higher mathematics, so that those who wanted a second chance can finally take advantage of it. And this time you won't have to sit at a desk. This book will not make you a math expert. But it will help you understand what this discipline studies and why it is so fascinating for those who understand it.

We'll explore how Michael Jordan's slam dunks can help explain basic calculus. I'll show you a simple and amazing way to understand the fundamental theorem of Euclidean geometry - the Pythagorean Theorem. We'll try to get to the bottom of some of life's mysteries, big and small: did Jay Simpson kill his wife; how to reposition a mattress so that it lasts as long as possible; how many partners need to be changed before getting married - and we will see why some infinities are larger than others.

Mathematics is everywhere, you just need to learn to recognize it. You can see the sine wave on the zebra's back, hear echoes of Euclid's theorems in the Declaration of Independence; what can I say, even in the dry reports that preceded the First World War, there are negative numbers. You can also see how new areas of mathematics influence our lives today, for example, when we search for restaurants using the computer or try to at least understand, or better yet, survive the frightening fluctuations of the stock market.

— Read the book “The Pleasure of X” by Stephen Strogatz online —

A series of 15 articles under the general title “Fundamentals of Mathematics” appeared online at the end of January 2010. In response to their publication, letters and comments poured in from readers of all ages, including many students and teachers. There were also simply curious people who, for one reason or another, “lost their way” in understanding mathematical science; now they felt they had missed something worthwhile and wanted to try again. I was especially pleased by the gratitude from my parents because, with my help, they were able to explain mathematics to their children, and they themselves began to understand it better. It seemed that even my colleagues and comrades, ardent admirers of this science, enjoyed reading the articles, except for those moments when they vied with each other to offer all sorts of recommendations for improving my brainchild.

Despite popular belief, there is a clear interest in mathematics in society, although little attention is paid to this phenomenon. All we hear about is fear of math, and yet many would love to try to understand it better. And once this happens, it will be difficult to tear them away.

This book will introduce you to the most complex and advanced ideas from the world of mathematics. The chapters are small, easy to read and not particularly dependent on each other. Among them are those included in that first series of articles in the New York Times. So, as soon as you feel a slight mathematical hunger, don’t hesitate to pick up the next chapter. If you want to understand the issue that interests you in more detail, then at the end of the book there are notes with additional information and recommendations on what else you can read about it.

The Pleasure of X - Steven Strogatz (download)

(introductory version)

And finally, we suggest you watch an interesting video

How useful are numbers for studying the world around us, what is the beauty of geometry, how elegant are integral numbers and how important is statistics? Steven Strogatz talks about all this in his book The Pleasure of X. The author explains fundamental mathematical ideas simply and elegantly, providing examples that everyone can understand. the site publishes one of the chapters of the book published by Mann, Ivanov and Ferber.

Statistics suddenly became a trendy field. With the advent of the Internet, e-commerce, social networks, the human genome project, and the development of digital culture in general, the world has become overwhelmed with data. Marketers study our tastes and habits. Intelligence agencies collect information about our location, emails and phone calls. Sports statisticians juggle numbers to decide which players to buy, who to draft, and who to bench. Everyone strives to connect the dots into a graph and discover a pattern in a jumbled collection of data.

It is not surprising that these trends are reflected in teaching. “Let's look at the statistics,” admonishes Greg Mankiw, an economist at Harvard University, in a New York Times column.

“The high school math curriculum spends too much time on traditional topics like Euclidean geometry and trigonometry. These mental exercises, useful for the average person, are, however, of little use in everyday life. Students would benefit greatly from learning more about probability and statistics.” David Brooks goes even further. In his article on disciplines that deserve attention to obtain a decent education, he writes: “Take statistics. You’ll see, it turns out that knowing what standard deviation is will be very useful to you in life.”

Quite likely, and it’s also a good idea to understand what distribution is. This is the first thing I intend to talk about. And I would like to focus on it, because this is one of the main lessons of statistics: things seem hopelessly random and unpredictable when viewed individually, but taken together they reveal a pattern and predictability.

You may have seen a demonstration of this principle in a science museum (if not, videos can be found online). A typical exhibit is a contraption called a Galton board, which is somewhat reminiscent of a pinball machine without the flippers. Inside it, there are even rows of pins at regular intervals.

Galton's board

The experiment begins with hundreds of balls being launched into the top of a Galton board. As they fall, they collide with the pins and are equally likely to bounce to the right or to the left, and then are distributed at the bottom of the board, falling into compartments of the same width. The height of a column of balls shows how likely the ball is to end up in a given location. Most of the balls are placed approximately in the middle, there are fewer on the sides, and even fewer on the edges.

In general, the picture is extremely predictable: the balls always form a bell-shaped distribution, although it is impossible to predict where each individual ball will end up.

How do individual accidents turn into general patterns? But this is how chance works. The middle column contains the most balls because, before rolling down, many of them will make approximately the same number of jumps to the right and left and, as a result, end up somewhere in the middle. Several lonely balls located at the edges form the tails of the distribution - these are those balls that, when colliding with the pins, always bounced in the same direction. Such bounces are unlikely, which is why there are so few balls at the edges.

Just as the location of each ball is determined by the sum of many random events, many phenomena in this world are the result of many small circumstances and also obey a bell-shaped curve. Insurance companies operate on this principle. They can accurately estimate the number of their clients who die each year. However, they don’t know who exactly will be unlucky this time.

Or take, for example, human height. It depends on countless accidents related to genetics, biochemistry, nutrition and the environment. Therefore, there is a good chance that, when considered together, the heights of adult men and women will form a bell-shaped curve.

In a blog post called "Misthings People Tell About Themselves Online," dating site OkCupid's statistics service recently published a graph of the growth of its clients, or rather their self-reported values. It was found that the growth rates of both sexes, as expected, form a bell-shaped curve. What is surprising, however, is that both distributions were shifted about two inches to the right of expected values.

Strogatz S. Pleasure from H. - M.: Mann, Ivanov and Ferber, 2014.

So either the customers surveyed by OkCupid are taller than average or they add a couple more inches to their height when describing themselves online.

An idealized version of such bell curves is what mathematicians call the normal distribution. This is one of the most important concepts in statistics, which has a theoretical basis. It can be proven that a normal distribution occurs when a large number of small random factors are added together, each of them acting independently of the others. And many events happen this way.

But not all. And this is the second point that I would like to draw attention to. The normal distribution is not as ubiquitous as it seems. For hundreds of years, and especially in the last few decades, scientists and statisticians have noted the existence of many phenomena that deviate from this curve and follow their own schedule. It is curious that such types of distributions are practically not mentioned in textbooks on elementary statistics, and if they are found, they are usually considered as some kind of pathology.

This is weird. I will try to explain that many phenomena of modern life become more meaningful if these "pathological" distributions are understood. This is the new normal. Take, for example, the distribution of city sizes in the United States. Rather than clustering around some average bell curve, the vast majority of cities are small in size and therefore cluster on the left side of the graph.

Strogatz S. Pleasure from H. - M.: Mann, Ivanov and Ferber, 2014.

And the larger the population of a city, the less common such cities are. In other words, in the aggregate the distribution will be more of an L-shaped curve than a bell-shaped curve.

And this is not surprising. Everyone knows that there are much fewer megacities than small cities. Although it's not so obvious, city sizes follow a nice simple distribution - when you look at them on a logarithmic scale.

We will assume that the difference between two cities is the same if their population differs by the same number of times (just as any two piano keys separated by an octave always differ by half in frequency). And let's do the same on the vertical axis.

Strogatz S. Pleasure from H. - M.: Mann, Ivanov and Ferber, 2014.

The data now lies on a curve that is an almost perfect straight line. Based on the properties of logarithms, it is easy to deduce that the original L-shaped curve is a power-law dependence, which is described by a function of the form

where x is the population of the city, y is the number of cities of this size, c is a constant, and the exponent a (power-law exponent) determines the negative slope of the straight line.

Power distributions have some illogical properties from the point of view of traditional statistics. For example, unlike a normal distribution, their modes, medians, and means do not coincide due to the skewed, asymmetrical shape of L-shaped curves.

President Bush benefited greatly from this, saying in 2003 that tax cuts saved each family an average of $1,586. Although this is mathematically correct, he took advantage of the average deduction, which hid huge deductions of hundreds of thousands of dollars received by the richest 0.1% of the country's population. It is known that the tail on the right side of the income distribution follows a power law, and in such a situation, using the average is misleading because it is far from its real value. In reality, most families received less than $650 back. In this distribution, the median is significantly less than the mean.

This example demonstrates a crucial property of power law distributions: they have heavy tails compared to at least the small liquid tails of a normal distribution. Large tails like this, although rare, are more common in data distributions than regular bell-shaped curves.

On Black Monday, October 19, 1987, the Dow Jones Industrial Average fell 22%. Compared to the usual level of volatility in the stock market, this drop was more than twenty standard deviations. According to traditional statistics (which uses the normal distribution), such an event is almost impossible: its probability is less than one in 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (10 to the 50th power). However, this happened - because price fluctuations in the stock market did not follow a normal distribution.

Heavy-tailed distributions are better suited to describe them. This happens with earthquakes, fires and floods, making it difficult for insurance companies to manage risk.

The same mathematical model describes the death toll from wars and terrorist attacks, as well as other, much more peaceful things, such as the number of words in a novel or the number of sexual partners a person has.

Although the adjectives used to describe long tails do not paint them in a very favorable light, tailed distributions wear their tails proudly. Fat, heavy and long? Yes it is. But in this case, show me which one is normal?

This book is well complemented by:

Quanta

Scott Patterson

Brainiac

Ken Jennings

Moneyball

Michael Lewis

Flexible consciousness

Carol Dweck

Physics of the stock market

James Weatherall

The Joy of X

A Guided Tour of Math, from One to Infinity

Stephen Strogatz

The pleasure of X

A fascinating journey into the world of mathematics from one of the best teachers in the world

Information from the publisher

Published in Russian for the first time

Published with permission from Steven Strogatz, c/o Brockman, Inc.

Strogatz, P.

The pleasure of X. A fascinating journey into the world of mathematics from one of the best teachers in the world / Stephen Strogatz; lane from English - M.: Mann, Ivanov and Ferber, 2014.

ISBN 978-500057-008-1

This book can radically change your attitude towards mathematics. It consists of short chapters, in each of which you will discover something new. You will learn how useful numbers are for studying the world around you, you will understand the beauty of geometry, you will become acquainted with the grace of integral calculus, you will be convinced of the importance of statistics and you will come into contact with infinity. The author explains fundamental mathematical ideas simply and elegantly, with brilliant examples that everyone can understand.

All rights reserved.

No part of this book may be reproduced in any form without the written permission of the copyright holders.

Legal support for the publishing house is provided by the Vegas-Lex law firm.

© Steven Strogatz, 2012 All rights reserved

© Translation into Russian, publication in Russian, design. Mann, Ivanov and Ferber LLC, 2014

Preface

I have a friend who, despite his craft (he is an artist), is passionate about science. Whenever we get together, he talks enthusiastically about the latest developments in psychology or quantum mechanics. But as soon as we start talking about mathematics, he feels a trembling in his knees, which greatly upsets him. He complains that not only do these strange mathematical symbols defy his understanding, but sometimes he doesn't even know how to pronounce them.

In fact, the reason for his rejection of mathematics is much deeper. He will have no idea what mathematicians do in general and what they mean when they say that a given proof is elegant. Sometimes we joke that I just need to sit down and start teaching him from the very basics, literally 1 + 1 = 2, and go as deep into math as he can.

And although this idea seems crazy, this is exactly what I will try to implement in this book. I will guide you through all the major branches of science, from arithmetic to higher mathematics, so that those who wanted a second chance can finally take advantage of it. And this time you won't have to sit at a desk. This book will not make you a math expert. But it will help you understand what this discipline studies and why it is so fascinating for those who understand it.

We'll explore how Michael Jordan's slam dunks can help explain basic calculus. I'll show you a simple and amazing way to understand the fundamental theorem of Euclidean geometry - the Pythagorean Theorem. We'll try to get to the bottom of some of life's mysteries, big and small: did Jay Simpson kill his wife; how to reposition a mattress so that it lasts as long as possible; how many partners need to be changed before getting married - and we will see why some infinities are larger than others.

Mathematics is everywhere, you just need to learn to recognize it. You can see the sine wave on the zebra's back, hear echoes of Euclid's theorems in the Declaration of Independence; what can I say, even in the dry reports that preceded the First World War, there are negative numbers. You can also see how new areas of mathematics influence our lives today, for example, when we search for restaurants using the computer or try to at least understand, or better yet, survive the frightening fluctuations of the stock market.

Mathematics is the most accurate and universal language of science, but is it possible to explain human feelings with the help of numbers? Formulas of love, seeds of chaos and romantic differential equations - T&P publishes a chapter from the book The Pleasure of X by one of the best mathematics teachers in the world, Stephen Strogatz, published by Mann, Ivanov and Ferber.

In the spring, Tennyson wrote, a young man’s imagination easily turns to thoughts of love. Unfortunately, a young man's potential partner may have his own ideas about love, and then their relationship will be full of the stormy ups and downs that make love so exciting and so painful. Some sufferers from unrequited love seek an explanation for these love swings in wine, others in poetry. And we will consult the calculus.

The analysis below will be tongue-in-cheek, but it touches on serious topics. Moreover, while understanding the laws of love may elude us, the laws of the inanimate world are now well studied. They take the form of differential equations that describe how interrelated variables change from moment to moment depending on their current values. Such equations may have little to do with romance, but they can at least shed light on why, in the words of another poet, “the path of true love never runs smooth.” To illustrate the method of differential equations, suppose that Romeo loves Juliet, but in our version of the story Juliet is a flighty lover. The more Romeo loves her, the more she wants to hide from him. But when Romeo grows cold towards her, he begins to seem unusually attractive to her. However, the young lover tends to reflect her feelings: he glows when she loves him, and cools down when she hates him.

What happens to our star-crossed lovers? How does love consume them and fade away over time? This is where differential calculus comes to the rescue. By creating equations that summarize the waxing and waning feelings of Romeo and Juliet, and then solving them, we can predict the course of the couple's relationship. The ultimate prognosis for her will be a tragically endless cycle of love and hate. At least a quarter of this time they will have mutual love.

To reach this conclusion, I assumed that Romeo's behavior could be modeled using a differential equation,

which describes how his love ® changes in the next moment (dt). According to this equation, the amount of change (dR) is directly proportional (with proportionality coefficient a) to Juliet's love (J). This relationship reflects what we already know: Romeo's love increases when Juliet loves him, but it also suggests that Romeo's love increases in direct proportion to how much Juliet loves him. This assumption of a linear relationship is emotionally implausible, but it makes solving the equation much easier.

In contrast, Juliet's behavior can be modeled using the equation

The negative sign in front of the constant b reflects that her love is cooling as Romeo's love intensifies.

The only thing left to determine is their initial feelings (that is, the values ​​of R and J at time t = 0). After this, all the necessary parameters will be set. We can use the computer to move forward slowly, step by step, changing the values ​​of R and J according to the differential equations described above. In fact, using the fundamental theorem of integral calculus, we can find the solution analytically. Because the model is simple, integral calculus produces a pair of comprehensive formulas that tell us how much Romeo and Juliet will love (or hate) each other at any point in time in the future.

The differential equations presented above should be familiar to physics students: Romeo and Juliet behave as simple harmonic oscillators. Thus, the model predicts that the functions R (t) and J (t), which describe the change in their ratios over time, will be sinusoids, each of them increasing and decreasing, but their maximum values ​​do not coincide.

“The stupid idea to describe love relationships using differential equations came to me when I was in love for the first time and was trying to understand the incomprehensible behavior of my girlfriend.”

The model can be made more realistic in different ways. For example, Romeo may react not only to Juliet's feelings, but also to his own. What if he is one of those guys who is so afraid of being abandoned that he begins to cool his feelings. Or he belongs to another type of guy who loves to suffer - that’s why he loves her.

Add to these scenarios two more behaviors of Romeo: he responds to Juliet's affection by either increasing or weakening his own affection - and you will see that there are four different styles of behavior in a love relationship. My students and the students of Peter Christopher's group at Worcester Polytechnic Institute suggested calling representatives of these types like this: the Hermit or Evil Misanthrope for the Romeo who cools his feelings and distances himself from Juliet, and the Narcissistic Blockhead and Flirting Fink for the one who warms up his ardor, but rejected by Juliet. (You can come up with your own names for all of these types.)

Although the examples given are fantastic, the types of equations that describe them are quite insightful. They represent the most powerful tools humanity has ever created for making sense of the material world. Sir Isaac Newton used differential equations to discover the secret of planetary motion. Using these equations, he unified the terrestrial and celestial spheres, showing that the same laws of motion apply to both.

Almost 350 years after Newton, humanity has come to understand that the laws of physics are always expressed in the language of differential equations. This is true for the equations that describe the flow of heat, air and water, for the laws of electricity and magnetism, even for the atom, where quantum mechanics reigns.

In all cases, theoretical physics must find the correct differential equations and solve them. When Newton discovered this key to the secrets of the Universe and realized its great significance, he published it in the form of a Latin anagram. Loosely translated, it sounds like this: “It is useful to solve differential equations.”

The stupid idea to describe love relationships using differential equations came to me when I was in love for the first time and was trying to understand the incomprehensible behavior of my girlfriend. It was a summer romance at the end of my sophomore year of college. I then very much resembled the first Romeo, and she - the first Juliet. The cyclical nature of our relationship drove me crazy until I realized that we were both acting out of inertia, in accordance with a simple push-pull rule. But by the end of the summer, my equation began to fall apart, and I became even more confused. It turned out that an important event happened that I did not take into account: her ex-lover wanted her back.

In mathematics we call this problem the three-body problem. It is obviously unsolvable, especially in the context of astronomy, where it first arose. After Newton solved the differential equations for the two-body problem (which explains why the planets move in elliptical orbits around the Sun), he turned his attention to the three-body problem for the Sun, Earth, and Moon. Neither he nor other scientists were able to solve it. It was later discovered that the three-body problem contained the seeds of chaos, meaning that in the long term their behavior was unpredictable.

Newton knew nothing about chaos dynamics, but according to his friend Edmund Halley, he complained that the three-body problem gave him a headache and kept him up so often that he would no longer think about it.

Here I am with you, Sir Isaac.