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One names nonEuclidean geometry a geometrical theory modifying at least one of the axioms postulated by Euclide in the Éléments .

The various nonEuclidean geometries result from the will to show the fifth Postulat ( the postulate, after the four others were declared axioms ) which seemed not very satisfactory because too complex, and perhaps redundant. It what Saccheri, proceeding by the absurdity, had failed at the end of the 17th century.

In the Elements of Euclide, the Postulat resembles the Conclusion of a Théorème, but which would not comprise a Démonstration:

If a line, falling on two lines, forms the interior angles on the same side smaller than two rights, these lines, prolonged ad infinitum, will meet side where the angles are smaller than two rights

and that one can include/understand like:

By a point external on a line, it always passes a parallel on this line, and only one.

During several centuries, the Euclidean Géométrie was used without one questioning his validity. She even was regarded a long time as the prototype of the logico-deductive reasoning. She indeed had the advantage of defining the intuitive properties of the geometrical objects in a rigorous mathematical construction industry.

The development of the not-Euclidean geometries

The geometries with N dimensions and the geometries not-Euclidean are two separate branches of the geometry, which can be combined, but not obligatorily. A confusion was established in the popular literature in connection with these two geometries. Because the Euclidean Géométrie was with three dimensions, one concluded from it that the not-Euclidean geometries comprised necessarily higher dimensions.

It is Gauss which, in 1824, had formulated the possibility that there exist alternative geometries with those of Euclide.

One distinguishes the geometries with negative curve, like that of Lobatchevsky (1829) and Bolyai (1832) (nap of the angles of a triangle lower than 180°, infinite number of possible parallels on a line by a point), geometries with positive curve like that of Riemann (1867) (nap of the angles of a triangle higher than 180°, parallels meeting with the poles). The geometry commonly called “geometry of Riemann” is a spherical space with three dimensions, spaces finished and however without terminals, with regular, alternative curve with the Euclidean postulate of the parallels. Riemann in addition conceived a wide theory of the not-Euclidean geometries with N dimensions (conference of 1854).

The idea of “not-Euclidean geometry” generally implies the idea of a curved space, but the geometry of a curved space is only one representation of the not-Euclidean geometry, specifies Sommerville in the elements of the not-Euclidean geometry (London, 1914). There exist not-Euclidean spaces with three dimensions.

Various types of nonEuclidean geometry

Hyperbolic geometry

Geometry elliptic

Riemann introduced another model of nonEuclidean geometry, the elliptic Géométrie. In this case, by a point external on a line one can carry out no parallel. The model is very simple:

the points are the pairs of points antipodes of a sphere;
the lines are the large circles (i.e. to say the circles having the same center as the sphere).

Geometrical space, representative space at Poincaré

“Driving space would have as many dimensions as we have muscles” This assertion of Poincaré in science and the assumption is the mark of the clearest distinction between the two kinds of space which it considers, geometrical space and representative space.

Geometrical space

For Poincaré, geometrical space has the following properties:

it is continuous;
it is infinite;
it has three dimensions;
it is homogeneous, i.e. all its points are identical between them;
it is isotropic, i.e. all the lines which pass by the same point are identical between them.

Representative space

At Poincaré, representative space appears under triple forms: pure visual space, tactile space, driving space.

The characteristics of representative space are the following ones:

It is neither Homogène, nor Isotrope, one cannot even say that it has three dimensions.

For Poincaré, our representations are only the reproduction of our feelings (visual, tactile, driving). We thus do not represent ourselves the external bodies in the geometrical Espace (continuous, infinite, homogeneous, isotropic, with three dimensions), but we reason on these bodies, as if they were located in geometrical space.

It is as impossible for us to represent us the external bodies in geometrical space as it is impossible for a painter to paint, on a plane table, objects with their three dimensions.

the Geometric axioms are (thus) neither of the synthetic Jugements a priori, nor of the experimental facts. They are Convention S, disguised definitions. A Géométrie cannot be truer than another, it can simply be more convenient.

The fourth dimension at Poincaré

For Poincaré, the access to objects with four dimensions could be only fortuitous and our perceptive base remains space with 3 dimensions:

an experiment whatever it is, comprises an interpretation on the Euclidean assumption.

If Poincaré considers a “invariable solid with four dimensions”, the Temps like the fourth dimension, concept which already exists at D' Alembert in its Encyclopédie of 1754, at Einstein with the continuum of pseudo-Euclidean Espace-temps of Minkowski will be especially developed (rigid four-dimensional space).

Such a space time can contain to become it of a being with three dimensions in the restricted Relativité, then pseudo-riemannienne variety with its curvilinear frames of reference of space and time in General relativity. Its Intersection with a three-dimensional space gives the “Présent” of a universe.

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