# Mathematical Induction in Universe

The mathematical universe is a collection of ten documentaries of 24 minutes duration each of a mathematical nature, produced in 2000 by the program The Adventure of Knowledge, of La 2 de Televisión Española. The documentary series was awarded the Prize for scientific dissemination at the Beijing International Scientific Festival

Pythagoras: much more than a theorem

Pythagoras is undoubtedly the best known mathematician of the general public. Everyone remembers his famous theorem. But Mathematics owes Pythagoras and the Pythagoreans much more. They are the ones who laid the first scientific stones not only of Geometry but also of Arithmetic, Astronomy and Music. But before Pythagoras, two other cultures had developed very powerful practical mathematics: the Babylonians and the Egyptians.

We will explore their contributions both in the field of the numbering systems they used, as well as their astronomical and geometric skills. From the sexagesimal system of the Babylonians we have inherited both the division of the circumference into 360 degrees and the current way of measuring time in hours, minutes and seconds. Their tablets reserve a few mathematical surprises. Perhaps the most important, the Plimpton tablet, reveals the surprising fact that they knew Pythagorean triples a thousand years before Pitagoras saw the light.

We will enjoy some of the most striking graphic demonstrations of the famous theorem, which has a greater number of different demonstrations throughout history.

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If mathematics has an emblematic number that is PI:3.141592 The figure of Ramanujan, a young Indian without a university education is intimately linked to the number pi. At the beginning of the century he discovered new infinite series to obtain approximate values of pi. The same used by large computers to obtain millions of figures of this familiar and strange number.

But Pi’s true father is a Greek mathematician from 2,300 years ago, Archimedes. He discovered the famous formula of the area of the circle: A = p · r2. And also the volume and area of the sphere. In passing, I invent the first method to obtain approximate values of pi by approaching the circle by means of polygons with an increasing number of sides, but pi does not only appear in mathematics when talking about circles or spheres, its presence in numerical relationships, in the calculation of probabilities and even in statistical studies they confer an almost magical omnipresence.

Numbers and figures: a journey through time

With the arrival of the euro, the cents will return and some old acquaintances will acquire a social prominence that they had not had for a long time: the decimal numbers. Some numbers that, despite the popular belief that they exist since the beginning of mathematics, have only been between us for four centuries. And is that the history of numbers is more complex than we suspect. Throughout the program, we will take a trip through time to discover the history of the figures. We will discover the figures and how to use them from Babylonians, Egyptians, Greeks, and Romans until we reach our popular 10 figures:1,2,3,4,5… But even these figures inherited from the Arabs have not always been the habitual tool to calculate. We will know the adventures of these symbols from their birth to the present day,

Fermat: the most famous margin in history At the beginning of the 17th century a lawyer, fond of mathematics, will launch a series of challenges, based on the simplest numbers, the integers, to the entire mathematical community. It’s Pierre de Fermat. The inspiration for these challenges was found in an old math book written back in the third century, the Arithmetic of Diofanto. In one of its margins Fermat will write a phrase that will become one of the most attractive in the history of mathematics. His famous last theorem:]”There are no whole solutions to the equation xn + yn = zn when n is greater than 2″ Fermat states that he had found the proof but unfortunately he does not fit the margin. A misfortune that has brought the best mathematicians in check for over 350 years.

A **number of Fermat** is a natural number of the form:

Pierre de Fermat conjectured that all natural numbers in this way with natural *n* were prime numbers, but Leonhard Euler proved that it was not so in 1732. Indeed, taking *n*=5 gives a composite number:

Where *n* is natural.

Gauss: the prince of mathematicians

Early 19th century A young mathematician has just solved a problem more than 2,000 years old: the construction with a ruler and compass of the regular 17-sided polygon. This is going to be one of the first notes you will make in an old 19-page notebook. At the end of his life the notes will not reach 50, but without a doubt this notebook will be the dream of any nineteenth-century mathematician. The contributions reflected in it contain enough material to keep all the mathematicians of the century occupied. However the fame of this young man, Gauss will come from the heavens. In the late 1800s astronomers discover a new celestial object. It is not a comet, it could well be the planet sought so many years between Mars and Jupiter. Unfortunately you lose track. But with the few observations made, Gauss is put to the task of deducing its orbit and points to the place in the sky where the telescopes point a year later. And indeed there appears Ceres. Gauss’s incredible contributions are not limited to the world of Mathematics and Astronomy. Together with Weber, he will start the first operational telegraph a few years before Morse’s. In magnetism he has also left his mark: the first magnetic map of the Earth is his work. The title of Prince of Mathematicians is not undeserved, although he reigned in almost all sciences. Together with Weber, he will start the first operational telegraph a few years before Morse’s. In magnetism he has also left his mark: the first magnetic map of the Earth is his work. The title of Prince of Mathematicians is not undeserved, although he reigned in almost all sciences. Together with Weber, he will start the first operational telegraph a few years before Morse’s. In magnetism he has also left his mark: the first magnetic map of the Earth is his work. The title of Prince of Mathematicians is not undeserved, although he reigned in almost all sciences.

Euler, the most prolific genius Euler is an endearing mathematician, and not just for his works. Throughout the 18th century it widened the boundaries of mathematical knowledge in all its fields. His complete works, Opera Omnia, occupy more than 87 large volumes, and the importance of his discoveries sometimes makes us doubt that they can be the work of a single person. Although Euler was not a normal person: he was a genius. At age 19 he won the prize of the Academy of Sciences of France for a job on the best location of the ship’s masts. This is not surprising, except for the fact that Euler was born in Basel (Switzerland) and had not seen a ship in his life. He would win another eleven Academy Awards. Euler picked up the glove of all the challenges posed by Fermat and gave a satisfactory answer to all but one, The last theorem. Today his name is associated with results from almost all branches of mathematics: analysis, algebra, number theory, series, geometry, astronomy … The most surprising thing is that Euler wrote more than half of his completely blind work by mentally calculating . Nothing strange for someone who was able to recite the complete Aeneid and in Latin.

**Newton and Leibnitz: on the shoulders of giants**

No doubt Newton is the author of the first step of the space race. The Laws discovered by him are those that have allowed man to set foot on the Moon or send ships to Mars and Venus, explore the outer planets: Jupiter, Saturn, Neptune and Uranus. His telescope model has allowed us to see farther in the sky. No doubt astronomers owe Newton a lot. But mathematicians and by the way the rest of the scientists owe so much to more. He and Leibniz, although it would be better to say at the same time as Leibniz, are the discoverers of the most powerful and wonderful mathematical tool: the Calculation. Newton had a prestige and social recognition even greater than Einstein could have in our century. As the kings and very few nobles he was buried in Westminster Abbey. Leibniz died alone and abandoned by all. At his funeral in Hanover, only his servant attended. Today the two share equally the glory of being the parents of the two most powerful tools in the mathematical universe: differential calculus and integral calculus. The ideal instrument to understand and explain the functioning of the real world, from the closest things to the furthest corner of the universe.

**Mathematics in the French Revolution**

In 1791, stopping its political disputes, the French National Assembly defines what will become the measure of universal length over the years: the subway. The ten-millionth part of the quadrant of the terrestrial meridian. Thanks to French mathematicians today we buy in kilos and travel kilometers. A plethora of notable mathematicians like never before had lived in France, will live intensely the events of the French Revolution: Joseph Louis Lagrange, Gaspard Monge, Peirre Simon de Laplace, Adrien Marie Legendre, and the Marquis de Condorcet, are going to bring French mathematics to its highest peak. They are going to lay the scientific foundations of the Analysis, of the calculation of probabilities, of the descriptive Geometry and of modern Astronomy. But they will do something else: they will create the model of modern teaching of higher mathematics, a model that will survive more than two centuries. July 14, French national holiday. The French celebrate the birth of the modern state. The rest of the world should celebrate with them something perhaps more important: one of the brightest moments of Modern Science.

**Math women**

Do you understand the mathematics of sexes?

Are the great mysteries of Mathematics unique to men?

Why, throughout history, there are so few women who have excelled in such an ancient scientific discipline? Although it seems that there is now a balance between the number of boys and girls studying mathematics, this is a relatively recent phenomenon. Of course forty years ago this did not happen. To discover the presence of women in the Mathematics Universe we will make a historical tour that begins with the birth of mathematics, with Pythagoras and his wife Theano, and continues with Hypatia in Alexandria, with Madame de Chatelet in France and with Mary Caetana Agnesi in Bologna in the 18th century. Even in the 19th century, Sophie Germain had to adopt the identity of a former student of the Polytechnic School of Paris, Monsieur Leblanc, to get the materials and problems and to present their own results and works. His works surprised mathematicians from the height of Lagrange and Gauss. Already at the end of the century Sophia Kovaleskaya suffered the marginalization of women in the academic world despite being one of the best brains of the time. Only at the gates of the nineteenth century, a woman Marie Curie will make one of the most important discoveries in the history of mankind, a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. His works surprised mathematicians from the height of Lagrange and Gauss. Already at the end of the century Sophia Kovaleskaya suffered the marginalization of women in the academic world despite being one of the best brains of the time. Only at the gates of the nineteenth century, a woman Marie Curie will make one of the most important discoveries in the history of mankind, a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. His works surprised mathematicians from the height of Lagrange and Gauss. Already at the end of the century Sophia Kovaleskaya suffered the marginalization of women in the academic world despite being one of the best brains of the time. Only at the gates of the nineteenth century, a woman Marie Curie will make one of the most important discoveries in the history of mankind, a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. Already at the end of the century Sophia Kovaleskaya suffered the marginalization of women in the academic world despite being one of the best brains of the time. Only at the gates of the nineteenth century, a woman Marie Curie will make one of the most important discoveries in the history of mankind, a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. Already at the end of the century Sophia Kovaleskaya suffered the marginalization of women in the academic world despite being one of the best brains of the time. Only at the gates of the nineteenth century, a woman Marie Curie will make one of the most important discoveries in the history of mankind, a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries. a discovery that will change the life of a human being in the twentieth century in many aspects: radioactivity. And he achieved something perhaps so important: for the first time in history, mankind, scientific circles opened their doors wide to a woman. And with her many so unfairly ignored for centuries.

**Order and chaos.**

The search for a dream Cosmos and Chaos:

Order and disorder. That is what those two Greek words mean. The history of science comes down to this: an eternal struggle to discover the functioning of Nature, an endless attempt to bring order to chaos. And mathematics will be an essential tool. We will attend the most important mathematical battles in this eternal war. From Pythagoras seeking in the numbers the harmony of the Universe, to Plato associating the universal balance with the regular polyhedra. We will stop in a fundamental battle: the struggle of Copernicus, Galileo and Kepler to bring order to the chaotic movement of the planets. And we will witness Newton’s great triumph discovering the world system, putting the apple and the Moon at the same level. Since Newton published his Principle Mathematica in 1687, an idea will permeate every corner of all scientific disciplines: Nature has its mathematical laws and human beings can find them. But unfortunately Nature always keeps some trick. Who can predict when and where a whirlwind will occur in a stream of water, how the flames of a bonfire dance, what scrolls will describe the smoke of a cigar, when and where a storm will form, where a lightning will discharge, what strange figure will draw in the sky. They are definitely phenomena on the other side of the border of chaos. But Mathematics has already put its advance on that other shore: Chaos theory and fractal Geometry. Chaos and order, order and chaos. Are not two sides of the same and wonderful coin in the background:

Hidden dimension

Enter the incredible world of fractals, a somewhat unknown concept a priori, but closer to our environment than we think. Fractals, discovered by Polish mathematician Benoït Mandelbrot, are defined as flat or spatial figures, composed of infinite elements, which have the property that their appearance and statistical distribution do not change whatever the scale with which it is observed. In nature we observe countless examples, such as snowflakes, clouds, neurons or a simple cauliflower. This interesting documentary brings us closer to the unknown world of fractal mathematics, explaining its origin, the importance of its discovery, as well as its application today in fields so diverse that they range from artistic to medical.

**Mathematical universe** is a collection of ten documentaries of 24 minutes duration each of a mathematical nature, produced in 2000 by the program The Adventure of Knowledge, of La 2 de Televisión Española. The documentary series was awarded the Prize for scientific dissemination at the Beijing International Scientific Festival.

MECHANICAL UNIVERSE

Collection of 52 videos made in 1985 by the California Institute of Technology funded by the Annenberg / CPB Project and produced by the same CALTECH and INTELECOM (a non-profit consortium that brings together California community colleges).

The series presents physics at the university level, covering topics from Copernicus to quantum mechanics. For this it uses historical dramatizations and animations that explain concepts of physics. The latter were one of the most advanced animations of the time: almost 8 hours of computer animation by NASA Jet Propulsion Laboratory expert James F. Blinn. Each episode opens and closes with a “ghost” lecture by Professor David L. Goodstein of the California Institute of Technology.

Despite its age, the series is often used even today as a supplementary aid to explain phenomena such as special relativity. During the course of the videos, contents such as electricity, classical mechanics, electromagnetism, thermodynamics, relativity and quantum mechanics are covered.

MORE FOR LESS

The golden number.

The program introduces this exotic number already known to the Greeks. We will see how it is obtained, what are the golden rectangles and their presence in countless artistic manifestations, in Painting, Architecture, Sculpture … throughout history. But the number of gold is not a mere invention of man, nature surprises us in a way that can not be casual, both in the plant world and in the animal, as in many physical phenomena, with events in which this famous Number makes an appearance.

Movements in the plane.

We enter the attractive world of Dynamic Geometry. All cultures have used symmetries, translations and turns in their artistic manifestations, they have played, almost always with surprising plastic results, with movements in the plane. Nature also gives us an exquisite sample of these movements. Dynamic Geometry art is made in the friezes and especially in the mosaics that fill the plane. In the program we investigate how to build them and the mathematical laws that allow these authentic works of art.

Geometry becomes Art.

The friezes, mosaics and geometric ornaments of Hispanic-Muslim art constitute one of the most spectacular manifestations of geometry in Art. Walking through the Alhambra we will study the techniques to build the Nasrid mosaics deforming polygons. From the hand of Prof. Rafael Pérez we will discover that the Nasrid artists knew all the possible ways to fill the plan using symmetries, turns and translations. Another great genius, the painter MC Escher, uses the technique of filling the plane with animated motifs in a surprising and disturbing way. We will make an excursion for its striking mosaics and for its magical worlds of impossible geometries.

The world of spirals.

The spirals are some of the most suggestive curves of the mathematical world. We find them among the ornamental motifs of almost all cultures, from the most remote to the present. But where the spirals shine spectacularly is in their multiple appearances in Nature. In this program we will discover the different types of spirals and the ways to build them.

Tapered basketball kites.

The curves obtained by cutting a conical surface by means of a plane have captivated mathematicians since the time of the Greeks. We investigate in this program the properties and the way of building them, their manifestations and their applications in fields as diverse as astronomy, communications and sports.

Tapered basketball kites.

The curves obtained by cutting a conical surface by means of a plane have captivated mathematicians since the time of the Greeks. We investigate in this program the properties and the way of building them, their manifestations and their applications in fields as diverse as astronomy, communications and sports.

06- Fibonacci. The magic of numbers

Leonardo de Pisa, better known as Fibonacci, is the author of the first mathematical summa of the Middle Ages, Liber Abaci. With this book he introduces nine Hindu figures and the sign of zero into Christian Europe. But it also gives the calculators of the time clear rules to perform operations with these figures with both whole numbers and fractions.

But Fibonacci is best known among mathematicians for the curious succession of numbers that bears his name and in which each term is the sum of the previous two.

This succession is a true source of pleasant surprises. We will analyze the suggestive relationships that exist between their terms and discover their presence in natural phenomena such as the branching of some plants, the distribution of pine nuts in pineapples and pipes in sunflowers. And, although in principle it is hard to believe it, we will see that she is directly related to an old friend of ours: the golden number

The Laws of Chance.

The human being has always been worried about what the future will hold. Mathematics has tried to illuminate, at least in part, the guidelines that govern the immediate future subject to chance. In our country we spend every week billions of pesetas in lotteries, bonolotos, primitive, raffles … We put our luck and our money in the hands of chance. But chance has its laws and in some of those laws we will deepen this program. We will discover, among other things, the probability of hitting a full in the primitive. What started as a game, a dice problem posed to Pascal, has become the Probability Theory, one of the most widely used mathematical tools today. From fans to gambling,

Natural numbers. Prime numbers.

The numbers that serve to count, natural numbers, one of the oldest inventions of Humanity. What would our lives be like without the existence of these numbers … From the Pythagoreans, who considered them as the beginning and explanation of the entire Universe, to this day these numbers have exerted a powerful influence on mathematicians of all ages. One of the fields that the great mathematicians have in check is that of prime numbers; A real box of surprises. Even today, using powerful computers, some of the conjectures made on these numbers have not been proven more than two hundred years ago. We will see some of them and discover one of the strangest applications of prime numbers today, its use in cryptography.

Fractals … the geometry of chaos.

The computer has made them fashionable. And yet they were already known at the beginning of the century. We refer to fractals. They are the most attractive, spectacular and enigmatic mathematical objects. Halfway between the line and the plane, between the plane and the space, they break even with the classical concept of dimension. Its dimensions are not integers, hence its strange name. And yet they can be obtained by simple iterations, that is, by repeating very simple geometric or functional procedures indefinitely. They have given rise to a new geometry fractal geometry. A new mathematical tool capable of shedding some light on chaotic phenomena and showing us that even in chaos it is possible to find a certain order.

Electoral Mathematics

When elections are announced, a powerful mathematical machine starts up. It is Statistics through surveys and opinion polls. We will analyze in this program the most important mathematical aspects of this type of surveys and their margins of reliability. But after casting the vote, mathematics continues to act. The Spanish electoral system is based on the D´Hont law, a sophisticated mechanism in which arithmetic intervenes decisively. We will study the mathematical characteristics of this system and its influence on the parliamentary map in our country.

A number called e.

There are numbers that surprise us by their tendency to appear in the most unexpected situations. What can power lines have in common, bank accounts, the development of a colony of bacteria, the carbon 14 test to date organic debris, population surveys, the probability of taking out an even number 70 when launching a dice 100 times … Apparently nothing. However, in all these situations there is a strange number between 2 and 3, which has infinite decimal figures and a somewhat exotic origin. Like the most famous pi number, mathematicians know him by letter. It is a number called e.

The language of the graphics.

Mathematical content graphics have become the most universal language of the late twentieth century. In any means of communication every time you want to give quantitative information about a process, a mathematical graph appears. Its advantages are unquestionable, they are able to offer a large amount of information at a glance. They are an essential instrument in fields as diverse as medicine, economics, physics, biology and even in sports. In this program we will investigate its relatively recent origin, have a little more than 200 years of existence, and its different applications and give some advice to critically interpret the information presented in the form of graphs.

Mathematics and reality.

The beauty of geometric shapes in the Alhambra in Granada is unquestionable; but a group of students from the School of Architecture will surprise us by giving some of the Nasrid geometric figures a practical and functional application, such as the design of a school or a housing development. We will also see how mathematics helps to measure and quantify natural phenomena as different as the intensity of an earthquake, the brightness of the stars or the noise of our streets.