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Felix Klein’s fame is based on work that he carried out over a period of one decade. Klein stopped working actively in mathematics at the age of 33, but to the end of his days he remained at the center of life in scientific organizations and devoted himself completely to pedagogical and literary activities.

When the history of hyperbolic geometry is discussed today, one can get the impression that had the creators of non-Euclidean geometry proved its consistency that would have been favorably received. But it was not the lack of proof that disturbed the critics above all. People were used to having geometry deal with our real space and having this space described by Euclidean geometry. It was characteristic that Gauss distinguished geometry from the other branches of mathematics, considering it similar to mechanics in experimental science. But here Gauss, as well as Lobachevsky and Bolyai, understood that first of all, logical, orderly geometric constructions that have no physical reality are possible—“imaginary” geometries—and second, that it is not so unquestionable that on an astronomical scale Euclidean geometry governs our world. However, what was understood by only a few mathematicians was absolutely inaccessible to nonprofessionals. They measured the claims of hyperbolic geometry with the Euclidean ruler of their own geometric intuition—and came up with an inexhaustible source for their wit. Nikolai Chernyshevsky wrote to his sons from exile that the entire city of Kazan was laughing at Lobachevsky: “What is the ‘curvature of a ray’ or ‘curved space’? What is geometry without the axiom of parallel lines?” He compared this to “squaring a boot” and “extracting the roots of a boot-top,” and said that it was as ridiculous as “writing Russian without verbs.”

At the very beginning of 1913, Professor Godfrey H. Hardy of Cambridge University received a letter from far-away Madras, in India. At the age of 36, Hardy (1877–1947) was already one of the leading specialists in analysis and number theory and had written a series of excellent mathematical works. The sender of the letter, Srinivasa Ramanujan, worked as a clerk in the accounts section of the Madras Port Trust at the paltry salary of 20 pounds a year. He wrote about himself that he had no university education and had studied mathematics on his own after finishing school, not according to the accepted system but “striking out a new path for myself.” The mathematical content of the letter looked awkward enough-the author could certainly be taken to be a self-confident amateur.

The pioneering ideas of the great mathematicians underwent many changes before ending up in the pages of textbooks. In their refined form it is easier to master them and the areas in which they can be applied are clearer, but something that is hard to perceive has been lost. Perhaps this is the logic of their discovery, a feeling for the material, and simply an excitement in face of the possibilities that are opening up. How different is the enthusiasm of the creators of analytic geometry from the feeling of the student who studies it today! We will recall here only a few episodes from the history of the creation of analytic geometry without trying to recreate this history completely, and we will finish the story with an étude in the style of 19th century analytic projective geometry, but with more up-to-date material. It is very tempting to try to argue the way they could a hundred years ago! Namely, we will prove the theorem of five hyperboloids in five-dimensional space. Two two-dimensional hyperboloids of one sheet are said to be chained if they have a common generator that is the line of intersection of the three-dimensional planes they generate. Correspondingly the minimal dimension in which chained hyperboloids exist is equal to 5. Several hyperboloids are called chained if they are pairwise chained and if the generators of the chainings for each pair belong to the same family.

At the International Mathematical Congress in Helsinki in the summer of 1978, Roger Penrose (1931-) gave a plenary address entitled, “The Complex Geometry of the Real World.” Penrose’s fundamental idea was that it is natural to interpret the points of the four-dimensional space-time of Minkowski or Euclid (in Euclidean field theory) as complex lines in a three-dimensional complex space. This idea was developed by Penrose over the years into his “twistor program.” (He called the points of the auxiliary three-dimensional complex space twistors.) Not long before the congress, the first results appeared that could not be considered purely interpretive (instanton solutions of the Yang-Mills equations and complex self-dual solutions of the Einstein equations).