The Euclidean geometry starts with the Éléments of Euclide , which is at the same time a sum of geometrical knowledge of the time and a mathematical attempt at formalization of this knowledge. The concepts of right, of plan, Length, surface are exposed there in an axiomatized way. The design of the Géométrie is then closely related to the vision of ambient physical space to the traditional direction of the term.

The geometrical designs undergo, starting from work of Euclide, of the evolutions along three main axes:

1. to check the current criteria of logical rigor, the axiomatic definition sudden of deep changes, the mathematical object remains nevertheless the same one.
2. more to limit themselves to dimensions two and three and to allow the development of a more powerful theory, an algebraic model of the geometry is considered. The Euclidean Espace is now defined like a vector Space or refines real Dimension finished provided with a scalar Produit.
3. Lastly, the Euclidean geometrical structure is not any more the only possible one; it is now established that there exist other coherent geometries.

More: 2000 years after its birth, Euclidean geometrical space is an always effective tool with the vast domains of applications. For example, the space of the physicists remains still mainly field of the Euclidean geometry, the Astronomie being the most notorious exception.

Euclidean approach of the science of space

The Euclidean geometry within the meaning of the antiques treats plan and space. The objects considered are the points, the segments, the right, the Demi-droite S, and their properties of incidence (the rule ), as well as the Cercle S (the compass ). The essential stakes are the study of figures and measurement.

Tools of the geometry of Euclide

The construction of Euclide is based on five postulates:

1. a segment of right can be traced by uniting two unspecified points.

2. a segment of right-hand side can be prolonged indefinitely in a straight line.
3. being given an unspecified segment of right-hand side, a Cercle can be traced by taking this segment like ray and one of its ends like center.
4. All the right angles are congruent S
5. If two lines are secant with a third in such way that the sum of the interior angles of an east coast strictly lower than two right angles, then these two lines are inevitably secant on this side.

The reasoning on the geometrical figures relates to their intersections, and their dimensions: to the incidence and the measure. From this point of view, some transformations of the figures are useful; most relevant Similitude S i.e. the transformations are the which preserve the reports/ratios of the distances. The simplest similarities are the Rotation S, the symmetries, the translations, which preserve the distances, as well as the Homothétie S. Starting from these some basic objects, all the similarities can be built by composition.

The construction of Euclide allows the development of the concept of angle, volume, surface, measure of length. There exists many surfaces of usual surfaces calculable by the techniques of the Eléments . A method, the Method of exhaustion which precedes the Intégration, makes it possible to go further. Archimedes (287 - 212 av. J.C) , for example carries out the Quadrature of the parabola. A limit of the concept of measurement comes from what the numbers considered are only the constructible numbers.

The two fundamental theorems are the Théorème of Pythagore and that of Thalès. A little analyzes makes it possible to go further with the trigonometry. It is the first example of construction of a bridge between the pure Euclidean geometry and another mathematical branch, to enrich the pallet by tools available.

An Euclidean success: the rule and the compass

See also: Triangle, Construction with the rule and the compass

An objective of the Euclidean geometry is the construction of figures to the rule and the compass. The study of the Triangle concerns this field, the richness of the results obtained is illustrated by the Liste of the remarkable elements of a triangle. A family of emblematic figures is that of certain regular polygons (see the article Partage of a tart). They are however not all constructible. The techniques of construction apply not only to the plan, but also to space as the study on the Polyèdre S. shows it.

A specificity of the Euclidean geometry lies in the fact that it uses initially only little or at all of complex and powerful theorems of algebra or analysis. It is an autonomous and independent mathematics, where the evidence comes primarily from purely geometrical reasoning. However, for the complex cases, like the construction of the figure opposite, other tools, for example the Polynomial S, will appear essential (cf Théorème of Gauss-Wantzel). The three large problems of antiquity, namely the Quadrature of the circle, the Trisection of the angle and the Duplication of the cube, with the assistance only of the rule and the compass, will be shown besides impossible only with the contribution of other branches of mathematics.

The Euclidean geometry will have many applications. The Renaissance largely uses the techniques of the Éléments . The Architecture, the Peinture through the Perspective abound in examples of this nature. The art of the interlacings of Léonard de Vinci (1452 - 1519) is another case of use. This mathematics is also used for measurement, at the same time for the Arpenteur S and in a scientific objective. The Théorème from Thalès allows Ératosthène (276 - 194 av. J. - C.) to measure the circumference of the ground. This technique, known as of triangulation also makes it possible to the sailors to know their position.

Application and new tools: Euclidean and physical space

See also: Mechanical Newtonian

From the 16th century mathematics moves away more and more from the geometry of the triangle. The Euclidean geometry keeps its utility because it always models with relevance the ambient physical world.

However, the purely ancient approach becomes too restrictive. It does not offer a framework sufficient for the development of mathematics. To further go, such as for example for the study of conical of Blaise Pascal (1623 - 1662) or the birth of the Infinitesimal calculus, the introduction of a reference mark is obligatory. With time the algebra and the analysis become prevalent: new tools, far away from the tools inherited Euclide, are created; with regard to the modeling of physical space, they nevertheless are always used within the framework of a little formalized Euclidean geometry, and this with a broad success. All the theories going back to before the 20th century are satisfied in fact with this framework. Still now, and in a very general context, the usual geometry of physics remains Euclidean. It allows spectacular results, like Newtonian mechanics.

It is only in 1915, that another geometry, that of the General relativity, explain a phenomenon better, that of the advances Mercury perihelion. The Euclidean geometry remains now valid near with three exceptions:

* astronomical distances, within the framework of general relativity

* speeds close to the light, with the geometry of the restricted Relativity,

* dimensions lower than the size of the particles, within the framework of certain contemporary theories like the supercordes.

It should however again be noted that, if the modeling of the geometry of space remains often the same one as that of Antiquity (with the exceptions already quoted near), formalization changes radically. Other mathematics that of the triangle are used.

The linear model of the geometry

The design of space by the mathematicians is historically not fixed; the evolutions are done for several reasons: the need to better found the geometrical theory, on the one hand by making up certain deficits of rigor of the text of Euclide, on the other hand by binding the theory to other branches of mathematics; but as need for being able to use the important corpus of geometrical results in other spaces, physics as well as mathematical, as ambient physical space or the usual plan.

These the last two objectives in fact are achieved thanks to a particular branch of mathematics: linear algebra.

Motivation: the mechanics of the solid

See also: Mechanical of the solid

The mechanics of the solid brings a new point of view on the Euclidean geometry. If our space describes the position of the center of gravity, the solid can still turn around this center. It lays out three more additional degrees of freedom. It is then necessary to consider a space of dimension six, to give an account of the exact position of the solid.

It is the same for speed. It is described by the movement of its center of gravity, represented classically by a vector in space and a rotation, which one can represent by a vector perpendicular to the plan of rotation and of which the length is proportional to the angular velocity. Mathematically, the field speeds is known as équiprojectif and is represented by a Torseur. Space is still of dimension six.

This step consisting in defining an abstracted space, which does not represent our universe directly any more, but a space specific to the studied problem, is fertile. It makes it possible to use the tools of the Euclidean geometry in varied contexts. This geometry is remarkably adapted for the representation of a équiprojectif field.

The mechanical statics is another example, an object is regarded as the assembly of a whole of solids subjected to constraints which bind them between them. Dimension is then equal to six times the number of solids composing the object. This step is especially developed during the 20th century. Indeed, dimension believes quickly and a computing power accessible only since the arrival from the Ordinateur S is necessary to make the approach operational.

Motivation: statistics

See also: Analysis in principal components

The examination of the Sondage S uses also the properties of the Euclidean geometry. It allows, thanks to the concept of distance, a relevant modeling, and, thanks to the tools of the Linear algebra, algorithmic powerful for calculations.

If the criteria, represented by questions of a survey can be brought back to measurable sizes, then each probed appears as a point of a space whose dimension is equal to the number of criteria. This geometry is essential in statistics:

* It reduces the dimension of space through the choice of axes ( called here component ) particularly revealing and of number reduced. The analysis of the survey becomes realizable in a space smaller, cleansed noise nonsignificant, and graphically representable for an intuitive comprehension of the examination.

* It measures the correlations between the various questions. The figure illustrates here two criteria, each one represented by an axis. For this example, when the criterion of the horizontal axis takes positive values, then the criterion of the axis vectical takes negative values. The two criteria known as are anticorrélés.

The step consisting in analyzing data through an Euclidean geometry is used in many Social sciences. It allows the analysis of the behaviors even when those do not follow rigid laws.

Linear model of the geometry: Euclidean spaces

Drawing to be made? --> The concept of vector Space provides a first purely algebraic structure in which the geometrical language can be expressed. Concept of coordinate introduced with? century becomes central, and the plan, for example, is modelled by a vector space of dimension two, which is primarily identified with the whole of all the couples of coordinates $(x\_1,\; x\_2)$, where $x\_1,\; x\_2$ is real numbers; a point is then simply such a couple, and a line the whole of the couples of the form $(\backslash \; lambda\; x\_1,\; \backslash \; lambda\; x\_2)$, where $(x\_1,\; x\_2)$ is a fixed couple, and $\backslash \; lambda$ varies among all the numbers; generalization is done easily with the space of dimension 3 by considering triplets of coordinates $(x\_1,\; x\_2,\; x\_3)$, but also with spaces of dimension N by considering the whole of all the tuples of numbers $(x\_1,\; \backslash \; dowries,\; x\_n)$. The properties of incidence are not however those awaited; in particular, all the lines meet in a point origin: the tuple $(0,\; \backslash \; dowries,\; 0)$. In fact, the lines have the behavior only awaited in the vicinity of the origin. It is then necessary to be able to transfer this local situation to all the other points from space. That is done by Translation, more precisely by making act the vector space on itself by translation. The local situation is then the same one in the vicinity of all the points, one obtains thus the concept of Espace refines, which makes it possible to fully give an account of the properties of incidence: for example, in a space refines real dimension 2, the lines check the fifth postulate of Euclide.

However, with simply the concept of space refines, only the properties of incidence are modelled, most of the traditional Euclidean geometry is not reached: it misses primarily a concept of measurement. A linear tool makes it possible to fill this gap; it is the scalar Produit. A space refines real provided with a scalar product is called Euclidean space , all the traditional geometrical concepts are defined in such a space, and their properties resulting from the algebra check all the Euclidean axioms: the geometrical theorems resulting from the traditional corpora, bearing on any objects checking these axioms, thus become in particular theorems for the points, right-hand sides, circles, such as defined in an Euclidean space.

Lastly, as one saw, spaces closely connected Euclidean are not limited to dimensions 2 or 3; they thus make it possible to return account of the various physical problems and statistics mentioned above, and which brought into play a greater number of variables, with a use of a geometrical language, and much of theorems of incidence and measurement spread almost automatically, in particular the theorem of Pythagore. It deals with triangle, it is now modelled by two vectors X and there . The properties of the scalar product ensure the following equality:

$(X\; there|X\; there)\; =\; (X|X)\; +\; (there|there)\; -\; 2\; (X|there)\; \backslash ;$

It is enough to notice that (X|there) is null if and only if the vectors are orthogonal to affirm the theorem of Phytagore and its reciprocal. Equivalent demonstrations in ancient formalization are given in the associated article. They are longer, particularly because of the reciprocal one, which calls upon the Théorème of Al-Kashi.

The passage to a higher degree of abstraction offers a more powerful formalism, giving access to new theorems and simplifying the demonstrations; the usual geometrical intuition of dimensions 2 or 3 is sometimes defied by these higher dimensions, is sometimes always effective. The profits are sufficient so that the sophisticated analyzes are generally expressed using the scalar product.

History of the linear approach

The concept of vector space appears gradually. Rene Descartes (1596 - 1650) and Pierre de Fermat (1601 - 1665) use the principle of Coordonnée S like a tool to solve with an algebraic approach geometrical problems. The concept of orthonormal reference mark is used in 1636. Bernard Bolzano (1781 - 1848) develops a first geometrical design where the not S, the right S and the plans are only defined by algebraic operations, i.e. the Addition and the multiplication by a number. This approach makes it possible to generalize the geometry with other dimensions that those of the plans and volumes. Arthur Cayley (1821 - 1895) is a major actor in the formalization of the vector spaces.

A contemporary William Rowan Hamilton (1805 - 1865) uses another body numbers that of the real : imaginary . He shows that this step is essential in geometry for the resolution of many problems.

Following work of Gaspard Monge (1746 - 1818) , its pupil Jean Poncelet (1788 - 1867) reform the projective Geometry. The projective geometry, geometry of the prospect, become also modélisable by the linear algebra: a projective Espace is built starting from a vector space starting from a process of identification of the points according to a rule of prospect. Projective spaces are thus generalized with unspecified dimensions. It is as interesting to note as the projective geometry is a nonEuclidean geometry, in the direction where the fifth postulate of Euclide falls there at fault; the linear algebra thus provides not only one model for the Euclidean geometry, but also, by simple adaptations, for a nonEuclidean geometry.

Limits of Euclide

The linear approach is not a handing-over in question of the Euclidean designs. It allows contrary to generalizing those, to extend their range, and to enrich them in return. Only, another great historical movement on the contrary called into question these designs.

The fifth postulate

See also: nonEuclidean Geometry

The 19th century sees the appearance of many news geometries. The origin of their births results from interrogations on the fifth postultat, that Proclus expresses in the following way: In a plan, by a point distinct from a line D, there exists a single straight line parallel with D ; this postulate, admitted by Euclide, and that the intuition supports, shouldn't it be a theorem? Or, on the contrary, can one imagine geometries where it would fall at fault?

A stake during the 19th century for the mathematicians, will be to manage to be detached from an accidentally barren physical intuition, as well as inappropriate respect of the lessons of old, to dare to invent new geometrical designs; those will not be essential without difficulty.

At the beginning of the century Carl Friedrich Gauss (1777 - 1855) wonders about this postulate. In 1813 it writes: For the theory of the parallels, we are not more advanced than Euclide, it is a shame for mathematics . In 1817 it seems that Gauss acquired the conviction of the existence of nonEuclidean geometries. In 1832, the mathematician János Bolyai (1802 - 1860) writes a report on the subject. The existence of a nonEuclidean geometry is not formally shown, but a strong presumption is acquired. The comment of Gauss is eloquent: to congratulate to You would amount congratulating me myself . Gauss forever published these results, probably to avoid a polemic. Independently, Nicolaï Lobatchevsky (1792 - 1856) precedes Bolyai on the description of a similar geometry in the Russian newspaper the messenger of Kazan in 1829. Two other ^et publications on the subject do not have nevertheless more impact on the mathematicians of the time.

Bernhard Riemann (1826 - 1866) establishes the existence of another family of nonEuclidean geometries for its work of thesis under the direction of Gauss. The impact remains weak, the thesis is published only two years after its death.

The geometries of Lobatchevsky and Bolyai correspond to hyperbolic structure where there exists an infinity of parallels passing by the same point. This situation is illustrated in the figure opposite, the lines d₁ , d₂ and d₃ is three examples of parallels to D passing by the point M . The case riemannien corresponds to the elliptic case where no parallel exists.

Unification of Klein

See also: Program of Erlangen

The situation became confused, the Eléments are not able to give an account of such a diversity. One counts many geometrical spaces: Euclidean vector spaces, the projective spaces closely connected Euclidean, spaces, elliptic and hyperbolic geometries, plus some exotic cases like the Ribbon of Möbius. Each geometry has definitions different, more or less equivalent and leading on series of more or less different theorems according to the authors and the geometries. The end of Euclidean supremacy generates an important disorder, which returns the comprehension of the difficult geometry. Young a 24 year old professor Felix Klein (1848 - 1925) lately appointed professor at the university of Erlangen orders all these geometries in his inaugural speech. This work has this time a vast repercussion on the scientific community, Euclidean supremacy disappears and the polemic born of the questioning of the fifth postulate dies out. Its work implies a reform of the formalization of Euclidean spaces. It uses work of James Joseph Sylvester (1814 - 1897) on what one calls now the scalar produced . The Euclidean geometry remains of topicality at the price of a major recasting of its construction.

In its program of Erlangen, Felix Klein finds the criterion making it possible to define all the geometries. The awaited profits are with go. The geometries are classifiéees, those which are presented as particular cases appear and the generic theorems can be expressed on the integrality of their scope of application; in particular, the axiomatic space checking Euclidean one is the limit which separates the hyperbolic families of geometries from Bolyai and Lobatchevsky from the elliptic geometries of Riemann.

Klein defines a geometry by the whole of its Isométrie S i.e. the transformations leaving the invariant distances. This approach characterizes a geometry perfectly. This unit profits from a new structure for the time, that of a particular group since it has to him even a geometry. This approach in the Euclidean case is equivalent to that of the scalar product, and if it is of a more abstract handling, it is also more general because it applies to all the geometries. The presentation of Euclidean spaces is thus limited to the study of the scalar product, perfectly sufficient to describe the geometry in this particular case.

Euclide and the rigor

See also: History of logic

The last reform is that of logic. The vectorial point of view of the geometry made it possible to find a model of the geometry of Euclide in another mathematical field which one hopes for better founded. But that does not make null and void certain questions about the intrinsic logical construction of Euclide. The postulates suggested by Euclide are not enough in fact not to entirely describe the results which it obtains; it uses in an implicit way of the not formulated postulates. This logical gap is mainly due to a deficiency of formalization of the text of Euclide: The Elements do not satisfy the current criteria of rigor. Let us note some gaps.

The invariant transformations leaving the distances cannot result from the axiomatic base. The heart even of the ancient geometrical reasoning is thus not demonstrable starting from the postulates. The Théorème of Pythagore is an example illustrated in the figure of right-hand side. The demonstration uses the fact that the triangles IBC and AEC have the same surface because one corresponds to the rotation of a quarter of turn of the other. This assertion is not in fact not demonstrable within the axiomatic framework chosen by Euclide.

The nature of the subjacent body of numbers is not explicitable. It is thus not possible to know the nature lengths. The article on the real numbers watch annoying consequences of this problem. Thus, as illustrated on the figure of left, the rotation of an angle of 45° of the diagonal of a circle from with dimensions 1 does not have, a priori its end A' .

Finally the limits covered by Euclidean construction are not clarified. The fifth postulate, whose statute is between that of axiom and of Conjecture is an example. Orientation of the space (see article Déterminant) which makes it possible to speak about right-hand side and of left is still supposed to exist. On the other hand, no element in the axiomatic base allows such a construction.

The answer of Hilbert

See also: Axioms of Hilbert

The mathematician David Hilbert then seeks to bring an answer by the definition of an axiomatic news describing Euclidean space exactly. It must fill the three following criteria:

• To be minimal, no axiom must be able to be cut off without other geometries being then possible that of Euclide.

• To be complete, all the nondemonstrable elements starting from the construction of Euclide must now be able the being. The objective is more ambitious, the axiomatic base must be complete, i.e. a demonstration must be able to exist to show the veracity or not any proposal.
• To be intrinsic, like that of Euclide, the axiomatic base should not call upon other mathematical concepts, such as for example the real numbers.

The fruit of its work is the article of 1899 qu ' it calls Fondements of the geometry and containing a list of 21 axioms. If this list describes the geometry of Euclide well, two of the three criteria are not filled. In 1902, the mathematician Eliakim Hastings Moore (1862 1932) will show the redundancy of the 21st axiom of the list of Hilbert. It is since minimal. The complétude is not reached either. In fact, the mathematician Kurt Gödel (1906 - 1978) will show in 1931 that, in such a context, the complétude is never atteignable. Thus, there exist other classes of problems for which the construction of Hilbert suffers from the same gaps that the ancient axiomatic base. The Hypothèse of continuous the is a mathematical example again field where this axiomatic base proves too incomplete to slice. And Gödel showed that the addition of a new axiom to allow a demonstration will do nothing but open one new fields containing of other indécidables proposals.

In spite of its limits, the article of 1899 is of a great influence on mathematics of the 20th century. To the detail close to the 21st axiom which it is easy to withdraw, the construction of Hilbert corresponds so that one can better do as regards axiomatic base. These bases allow a construction as rigorous as possible of the Euclidean geometry. This approach is pionnière in the world of the logical . On the other hand, the construction of Felix Klein is more operational and more easily generalizable. In geometry, the axiomatic base of Hilbert thus only little is used.

Generalizations

See also: nonEuclidean Geometry

The concept of geometry is now applied to a vast unit of spaces. If the handing-over in question of the fifth postulate is the historical example which gives contents to the concept of nonEuclidean geometry, a more precise analysis shows the existence of quantity of other cases not under consideration by Euclide respecting nevertheless the fifth postulate.

In the case of the vector spaces, the body of number can be modified, the distance is sometimes selected so as to have a new group of isométries, the number of dimensions can become infinite.

There exists in addition to many cases where space is not vectorial, Klein formalizes nondirectional geometries, Georg Cantor (1845 - 1918) discovers a together triadic whose dimension is not whole and who now is classified in the category of the geometries fractales. Topology opens the door with the construction of many other cases.

This is why the term of nonEuclidean geometry falls gradually in disuse during the 20th century. It now entered the use to describe a geometry by the properties which they have and not by one, become very specific and that it would not have, namely its Euclidean character.

The following examples among are most frequently used.

Infinite dimension

See also: Space of Hilbert

Spaces of functions to actual value have a structure of vector space. It is fertile to study them with geometrical tools. It is possible to associate with it a distance resulting from a scalar product if the functions are of integrable square . This scalar product is defined in the following way:

$(F|G)\; =\; \backslash \; int\; fg\; \backslash ;$ In such a space, the theorem of Pythagore spreads and allowed Joseph Fourier (1768 - 1830) to solve the equation of heat.

This approach, consisting in using the tools of the geometry to solve problems of analyzes is called now the analyzes functional. Multiple different distances are then defined on these spaces, generating different geometries then. They take, for example, for name Espace of Hilbert, Espace of Banach, Espace préhilbertien or vector Space normalized. The space of Hilbert is the most natural generalization of the Euclidean geometries.

Remark : Space given in example has a particular property. A null function everywhere safe in zero or it is worth is at a null distance from the null function. Two different points thus have a null distance. To solve this difficulty, it is possible of quotienter the vector space by the space of the functions of null measurement.

Square space

See also: Square

The real numbers suffer from a weakness, the body which they form is not algebraically closed. That means that there exist nonconstant Polynôme S which do not have there a root. This weakness complexes the analysis of the linear applications of a vector space in itself. The article on the eigenvalues clarifies this difficulty. A solution often used consists in generalizing the body of number and passing to the complex . This method is used in physics, for example for the study of the oscillating systems. The generalization of an Euclidean space to the imaginary numbers is called a space Hermitien.

Negative distance

See also: Cone of light, restricted Relativity

The Euclidean geometry supposes that the distance between two points is always positive. There exist however bilinear forms not checking this assumption. The geometry is then different, the whole of the points at a null distance from the origin forms a cone. Physics uses this geometry.

In restricted Relativity, the space part of the Espace-temps is Euclidean, but if time is integrated, it is not it any more. Indeed, the quadratic Form $x^2\; +\; y^2\; +\; z^2\; -\; c^2t^2$ defining the “square of the space-time distance” is not positive.

The famous figure this phenomenon. Point has represents observer, plan horizontal represents dimensions space ( three dimensions are represented here by a plan so that the figure is representable in dimension three ), the vertical axis represents temporal dimension. So that the observer can interact with the point C , one would need a displacement at an high speed with the light. Within the framework of this theory, it is impossible. Consequently, the point C is not in the part accessible from the universe for the observer. The point B on the other hand, is in the cone of light, consequently, an interaction between the observer and the point B is possible.

Variety

See also: Variety (geometry), Variety (algebra)

All the geometries do not satisfy the fifth postulate of Euclide. The surface of the ground is a perfect example, the shortest way between two points is always located on a large circle on the surface of the sphere and whose center is that of the ground, this type of object thus corresponds to the right for this geometry. Here is a coherent geometry, corresponding to a real case. However the fifth postulate is not checked any more. In this example, two right not confused always has an intersection containing two points.

The abandonment of the fifth postulate is fundamental. It is indeed desirable to consider the sphere, not like a subset of an Euclidean space of dimension 3 but like a geometry with whole share, having of a distance and a criterion from orthogonality. Without tool of this nature, the study of object of this nature becomes more delicate.

The mathematical solution is derived from the example illustrated in the figure. If the study is summarized at a local zone, such as for example Paris, then it is possible to use a plane chart. I.e. an Euclidean representation, which, with the proviso not of moving away too much from the point of study, will be of a precision as large as desired. A plan of Paris is not completely exact because on a sphere the sum of the angles of a triangle is always higher than 180 degrees. On the other hand, the dimension of Paris is sufficiently small so that the error is negligible. The solution consisting in describing a geometrical object with a chart associated with each point is that retained for the mathematical description of such an object.

Geometry of this nature, which does not correspond to a space as Euclide imagined it is called a nonEuclidean Géométrie, one now calls them Variétés. The mathematician Felix Klein showed that others phénomèmes that the handing-over in question of the fifth postulate can bring a nonEuclidean geometry.

Astrophysics with large scales cannot be satisfied with the Euclidean geometry. In theory of the General relativity, the models used are not Euclidean any more, the gravitation appearing by the curved trajectory followed by a mass along geodetic nonEuclidean. It is thus appropriate of susbstituer for the Euclidean distance a distance defined by a quadratic Forme more general.

See too

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