The German physicist Albert Einstein (1879-1955) is a famous genius. Your image is familiar to us. His theory of relativity is celebrated. But without the ideas of three mathematicians from the 19th century, what is Einstein's main theory would simply not work.

Mathematics is the key to understanding the physical universe. As the Italian philosopher Galileo Galilei once said, without the beacon of this science, we would circle a dark labyrinth.

Pioneering mathematicians gave Einstein a map to navigate through the darkest labyrinth of everything: the fabric of the universe.

"Einstein was a good intuitive mathematician and he had a bit of a problem with these ideas, but he knew what he wanted." When he saw what he wanted to do, Riemann had done it, he knew it was, "said theoretical physicist Roger Penrose to the BBC.

** Euclid theories in Sheikh **

During 2000 years, the axiom & # 39; s contained in the great work of geometry ", from Euclid, were accepted as absolute and indisputable mathematical truths

Euclid's geometry helped us to traverse the world, build cities and countries, give man control over his environment .

But in Europe, in the middle of the 19th century, unrest arose about some of Euclid's ideas. The mathematicians began to wonder if there could be another form of geometry that he had not described, geometries in which the axioms of Euclid might be incorrect.

Science Photo Library The geometry of Euclid has helped us traverse the world, build cities and nations

It is difficult to say how radical this suggestion was. So much so that one of the first mathematicians to think about this idea, the German Carl Frederick Gauss, was reluctant to talk about the subject, although he was now considered a God in the mathematical world.

He had an impeccable reputation. He could have said something that most mathematicians would have believed, but he was silent: he did not share with anyone his suspicion that the space might be deformed.

** & # 39; Radical discoveries & # 39; **

this, in Hungary, Farkas Bolyai, another mathematical, also considered scenario in which the geometry of Euclides might be incorrect.

Bolyai studied with Gauss at the University of Göttingen, Germany, and returned to his home in Transylvania, Romania, where he struggled for years with the possibility of new geometries. * "I have traveled beyond all the reefs of this hellish Dead Sea and have always come back with the masts and damaged sails." *

] In 1823 he received a letter from his son, also a mathematician, who was in Timisoara with his army battalion.

* "My dear father, I have so many things to write about my new discoveries, that I can do nothing but write this letter without waiting for your answer to my previous letter, and maybe I should do not, but I have found beautiful things that even surprised me, and it would be a shame to lose them, he will see and know, I can no longer say, only that I have created a new and strange world out of nothing. " *

The son of Farkas Bolyai, János, had discovered what he called "imaginary worlds"; mathematical worlds that did not meet the axioms of Euclid, which seemed to be perfectly consistent and without contradictions.

Bolyai immediately wrote to his friend Gauss about the exciting discoveries that his son had made.

"* I recently received a short article from Hungary about non-Euclidean geometry: the writer is a young Austrian officer, son one of my first friends, I think the young geometrical J. Bolyai is a first-class genius. "*

But in the letter he wrote to Bolyai, the tone was very different:

*" If I should say that I can not praise this work, I would certainly be surprised for a moment, but I can not say anything else: to compliment it, it would be to myself, indeed the whole content of the work, the way your child has been taken, to compliment the results it focuses on, almost completely coincide with my reflections, who were partly occupied with my thoughts in the last 30 or 35 years. "*

The young János was completely inconsolable. His father tried to comfort him: * "Certain things have their time when they meet in different places, such as spring when violets bloom everywhere." *

Despite father's encouragement to publish, János Bolyai did not write his ideas until a few years later. It was too late.

He soon discovered that the Russian mathematician Nikolai Lobachevski published very similar ideas, two years before him.

** Beyond the three dimensions **

The radical geometries of Bolyai and

Bernhard Riemann was a timid and brilliant mathematician, who suffered from a lack of knowledge of the mathematics of mathematics and mathematics, and he was a student of Gauss, at the University of Göttingen, who brought these new geometries in an even more exotic direction. serious health problems. One of his contemporaries, Richard Dedekind, wrote about it:

* "Riemann is very unhappy." His lonely life and physical distress made him extremely hypochondical and suspicious of other people and himself. strange here only because he believes that no one can stand it. "*

In his solitude, Riemann explored the contours of the new worlds he had built

In the summer of 1854, the introvert Riemann faced a major obstacle in becoming a professor at the University of Göttingen: he had to become a public lecture at the Faculty of Philosophy The department chose the theme: "About the hypotheses that are the basis of geometry."

He was forced to present the radical ideas he had formulated on 10 June about the nature of geometry

He showed current mathematicians how to view four, five, six or more dimensions, including N-dimensions, describing forms that could only be seen with the mind of mathematicians and made them so tangible for those who listened to it as 3D objects for most people.

If you are not a mathematician, you can try a spot near the fourth dimension: the Grand Arch of La Défen se in Paris, made by the architect Johan Otto von Spreckleson.

It is a four-dimensional cube in the heart of a three-dimensional Paris, an absolutely impressive structure through which the towers could pass from the cathedral of Notre Dame

But even more surprising is the power of the idea that it represents. A super cube in the middle of the French capital, with 16 corners, 32 edges and 24 faces … extraordinary!

The architect opened a door to another world for us all. But to really understand life beyond three dimensions, Riemann's revolutionary mathematics is needed.

** Inspiration for Einstein **

Five years after the famous conference of 1854, the ideas of Riemann

Einstein tried to reflect on the structure of space when he developed the theories on the subject developed by Riemann about encountered the N-dimensional space

"He did not like it at first, he thought:" Mathematicians make life so complicated! "But he soon discovered that this was the right prism, and it was absolutely crucial, because this four-dimensional geometry fitted into the other three dimensions, and Einstein realized that he could generalize it just as Reimann had generalized Euclidean geometry by making him bend. "

With Riemann's mathematics, Einstein promoted extraordinary progress on the nature of the universe: ot he discovered that it was the fourth dimension.

The new geometry of Riemann allowed to unite space and time. geometries first thought by Gauss, described by Bolyai and Lobachevsky and generalized by Riemann, helped him to resolve relativity.

By measuring the distance between two points in space-time using the Euclid geometry, various disturbing paradoxes arise But when using the non-Euclidean geometries of Bolyai and Lobachevsky, paradoxes dissolve.

The geometries of these mathematicians of the 19th century were the key to the creation of the theory of relativity. structure of space and time to navigate.

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