Geometry

Etymology

The term geometry drift of the Greek of γεωμέτρης ( geometricians ) which means “geometrician, land-surveyor” and comes from γῆ ( Ge ) (“ Ground ”) and μέτρον ( métron ) “ Measurement ”). It would be thus “the science of the measurement of the ground”.

Definition of the geometry

The geometry admits many meanings according to the authors. In a strict direction, the geometry is the study of the forms and the sizes of figures . This definition is in conformity with the emergence of the geometry as a Science under the Greek Civilization during the traditional time. According to a report/ratio of Jean-Pierre Kahane, this definition coincides with the idea that are made people of the geometry like material taught: it is the place where one learns how to apprehend the Espace .

The questions put during the XIXe century resulted in reconsidering the concept of Forme S and Espace, by drawing aside the rigidity of the Euclidean distances. It summer considered the possibility of deforming a surface without preserving continuously the metric induced one, for example to deform a sphere in an ellipsoid. To study these deformations led to the emergence of the Topologie: its objects of study are units, topological spaces, of which the concept of proximity and Continuité is defined " ensemblistement" by the concept of Vicinity. According to certain mathematicians, topology forms part of the geometry fully, even in is a fundamental branch. This classification can be called into question by others.

According to the point of view of Felix Klein (1849 - 1925) , the analytical geometry synthesized in fact two later on dissociated characters: its basically metric character, and homogeneity . The first character finds in the Metrical geometry, which studies the geometrical properties of the distances. Second is with the base of the program of Erlangen, which defines the geometry as the study of the invariants of actions of group.

Current work, in fields of research bearing the name of geometry, tends to call into question the first definition given. According to Jean-Jacques Szczeciniarcz, the geometry is not built on the simple reference to space, nor even figuration or visualization but includes itself/understands through its development: the geometry is absorbed but to at the same time appears us to allot feel with the concepts by giving in addition the impression of a return to the initial direction . Jean-Jacques Sczeciniarcz raises two movements in the mathematical research which led to a widening or a parcelling out of the geometry:

procedure of idealization consisting in showing the importance of a structure by adding it to the already studied mathematical objects;
On the contrary, procedure of thematisation consisting in releasing a new structure subjacent with already studied geometrical objects.

In the prolongation, the geometry can be approached either like a unified discipline but like a vision of mathematics or an approach of the objects. According to Gerhard Heinzmann, the geometry is characterized by a use of terms and contents geometrical, like, for example, “not S”, “Distance” or “Dimension” as a linguistic framework in the most various fields , accompanied by a balance between an empirical approach and a theoretical approach.

History

Traditional geometry

For Poincaré, geometrical space has the following properties:

The traditional geometry corresponds to this definition stricto sensus of space. To build such a geometry consists in stating the rules of fitting of the four fundamental objects the not, the right , the plan and the space. This work remains the prerogative of the pure geometry which is the only one to work Ex nihilo.

Pure geometry

The pure geometry rests initially on axiomatic which defines space; then on methods of intersections, transformations and constructions of figures (Triangle, Parallelogram, Circle, Sphere, etc).

The projective geometry is more minimalist, which makes of it a joint base for the other geometries. It is founded on axioms

of incidence (or of membership) of which the most notable characteristic (and most singular) is: “ Two coplanar lines have a single point commun. ”
of order: in particular allows to order the points of a line. From this point of view, a projective line is connected with a circle because two points define two segments.
of continuity: Thus, in any geometrical space, one can join a point to another by a continuous advance. In Euclidean geometry, this axiom is the Axiome of Archimedes.

; Parallelism: To distinguish in the projective geometry from the unsuitable elements characterizes the Arguesian Géométrie. Then the geometry closely connected is born from the elimination of these unsuitable elements. This suppression of points creates the concept of parallelism since from now on certain coplanar line pairs cease intersecting. The removed unsuitable point is comparable to the direction these lines. Moreover, two points define nothing any more but one segment (that of both which does not contain the unsuitable point) and makes familiar the concept of direction or orientation (i.e., that makes it possible to distinguish $\ overline {AB}$ from $\ overline {BA}$ ).

; Congruence: August 1st

; Geometries Euclidean and not-Euclidean: The fifth axiom or “ postulate of parallels ” of the geometry of Euclide the Euclidean Géométrie founds: See the axiomatic of Hilbert or the Elements of Euclide for statements more complete of the Euclidean geometry.

Analytical geometry

The analytical geometry is most familiar. It rests on the basic principle which any line is comparable to a representation (an image) of the unit of real (or more largely, of a body. Space is then decomposable in subspaces and a point is definable by Coordonnée S. It follows that any figure is determined by a system of equations and/or inequations. For example, a curve is the representation of a function. One sees thus that this approach, resulting from the Linear algebra and based on the concept of vector Space, is with a bridge between the geometry and the analyzes.

This geometrical is in conformity with the pure geometry in the direction where the vector space makes it possible to build model of geometries (as mathematical objects). August 1st

traditional Geometry: extension of the geometry of the elementary figures of the plan and space, it concerns the linear algebra now especially and it includes/understands the vectorial Géométrie, the Géométrie refines, the projective Géométrie, the Euclidean Géométrie, the nonEuclidean Géométrie, of which the spherical Géométrie, the elliptic Géométrie ~~, or the geometry of Möbius~~ and the hyperbolic Géométrie;

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Program of Erlangen

In the design of Felix Klein (author of Program of Erlangen), the geometry is the study, of spaces of points on which operate Groupe S of transformations (also called symmetries), and quantities and properties which are invariant for these groups.

Among the most known transformations, one finds the Isométrie S, the Similitude S, the Rotation S, the Réflexion S, the Translation S and the Homothétie s.

It thus is not about a discipline; but of an important work of synthesis, which allowed a clear vision of the characteristics of each geometry. This program thus characterizes more the geometry that it does not found it. It have a mediator role in the debate on the nature of the not-Euclidean geometries and controversy between geometries analytical and synthetic.

Fields of research concerned with the geometry

Geometry riemannienne

Complex geometry

Geometries symplectic and of contact

The symplectic geometry can be introduced like a generalization in higher dimension of the concept of surface met in dimension 2. Just like the complex geometry, its studied objects, the symplectic varieties, are sufficiently rigid

Geometries discrete and convex

See also: discrete Geometry, convex Geometry

Geometries algebraic and arithemetic

Noncommutative geometry

Applications of the geometry

A long time, geometry and Astronomie were dependant. On an elementary level, the calculation of the sizes of the moon, Sun and their respective distances to the Earth calls upon the Théorème from Thalès. In the first models of the solar system, with each planet a Platonic Solide was associated. Since the astronomical observations of Kepler, confirmed by work of Newton, it is proven that the planets follow a elliptic orbit whose Sun constitutes one of the hearths. Such considerations of geometrical nature can usually intervene in traditional Mécanique to describe qualitatively the Trajectoire S.

In this direction, the geometry intervenes in Ingénierie in the study of the stability of a mechanical system. But it intervenes even more naturally in the Draftsmanship. The draftsmanship shows the Coupe S or the projection S of a three-dimensional object, and is annotated lengths and angles. It is the first stage of the installation of a project of industrial Conception. Recently, the marriage of the geometry with the Informatique allowed the arrival of the Conception computer-assisted (CAD), of calculations by finite elements and of the Infographie.

Euclidean trigonometry intervenes in optics to treat for example light diffraction. It is also at the origin of the development of the Navigation: sea transport with stars (with the Sextant S), cartography, aerial navigation (piloting with the instruments starting from the signals of the beacons).

The new projections in geometry at the 19th century find echoes in physics. It is often known as that the Géométrie riemannienne was initially justified by the interrogations of Gauss on the cartography of the Earth. It gives an account in particular of the geometry of the Surface S in space. One of its extensions, the Lorentzian Geometry, provided the ideal formalism to formulate the laws of the General relativity. The differential Géométrie finds new applications in post-Newtonian physics with the Théorie of the cords or of the membranes.

The noncommutative Geometry, invented by Alain Connes, tends to be essential to have the good mathematical structures with which to work to set up new physical theories.

Teaching of the geometry

The geometry occupies a place privileged in the Enseignement of mathematics. Many teaching studies prove its interest: it makes it possible to the pupils to develop a reflection on problems, to visualize figures of the plan and space, to write Démonstration S, to deduce from the results of stated assumptions. But more still, the geometrical reasoning is much richer than the simple formal deduction , because it is based on the intuition born of observation of the figures .

In the years 1960, the teaching of mathematics in France insisted on the practical application of the problems concerned with the geometry in the everyday life. In particular, the Théorème of Pythagore was illustrated by the Règle of the 3,4,5 and its use in carpentry. The involutions, harmonic divisions, and the birapports were with the program of the secondary. But the Reform of modern mathematics, born in the United States, and adapted in Europe, resulted in reducing considerably the knowledge taught in geometry to introduce Linear algebra into the second degree. In many countries, this reform was strongly criticized and designated as person in charge of school failures. A report/ratio of Jean-Pierre Kahane denounces the lack of a true preliminary didactic reflection on the contribution of the geometry: in particular, one practical of the vectorial geometry prepare the pupil with a better assimilation of the formal concepts of vector space, bilinear form,…

The use of the figures in the teaching of other matters makes it possible to better render comprehensible with the pupils the exposed reasoning.

References

Works

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