Геометрия 7-9 класс
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Geometry (from other Greek - γῆ - Earth and μετρέω - "measure") is a branch of mathematics that studies spatial structures, relationships, and their generalizations.
Geometry as a systematic science appeared in ancient Greece, its axiomatic constructions are described in Euclid's Elements. Euclidean geometry dealt with the study of the simplest figures on the plane and in space, the calculation of their area and volume. The coordinate method proposed by Descartes in 1637 formed the basis of analytic and differential geometry, and problems connected with drawing led to the creation of a descriptive and projective geometry. In this case, all the constructions remained within the framework of Euclid's axiomatic approach. The radical changes are connected with the work of Lobachevsky in 1826, which abandoned the axiom of parallelism and created a new non-Euclidean geometry, thus defining the path of further development of science and the creation of new theories.
The classification of geometry, proposed by Klein in the Erlangen program in 1872 and containing in its base the invariance of geometric objects with respect to various transformation groups, is still preserved.
Geometry deals with the mutual arrangement of bodies, which is expressed in the touch or fit to each other, the location "between", "inside", etc.; the size of bodies, that is, the concepts of equality of bodies, "more" or "less"; as well as transformations of bodies. The geometric body represents an abstraction from the time of Euclid, who believed that "a line is a length without a width", "a surface is that which has a length and a width." A point is an abstraction associated with an unlimited reduction of all body dimensions, or the limit of infinite division. The location, dimensions and transformations of geometric shapes are determined by spatial relationships.
Exploring real objects, geometry considers only their shape and relative position, being distracted from other properties of objects, such as density, weight, color. This allows us to move from spatial relations between real objects to any relations and forms that arise when considering homogeneous objects, and similar to spatial ones. In particular, geometry allows us to consider the distances between functions
Common in our days [the source is not indicated 370 days], the classification of various sections of geometry was proposed by Felix Klein in his Erlangen program (1872). According to Klein, each section studies those properties of geometric objects that are preserved (invariant) under the action of a certain group of transformations specific for each section. In accordance with this classification, the following main sections can be distinguished in classical geometry.
Euclidean geometry, in which it is assumed that the dimensions of segments and angles do not change when moving figures on the plane. In other words, this is the theory of those properties of figures that are preserved when they are transferred, rotated and repelled.
Planimetry is a section of Euclidean geometry that examines figures on a plane.
Stereometry is a section of Euclidean geometry in which figures in space are studied.
Projective geometry that studies the projective properties of figures, that is, properties that persist under their projective transformations.
Affine geometry that studies the properties of figures that are preserved under affine transformations.
Descriptive geometry is an engineering discipline based on the projection method. This method uses two or more projections (orthogonal or oblique), which makes it possible to represent a three-dimensional object on a plane.
Spherical triangle
Modern geometry includes the following additional sections.
Multidimensional geometry.
Non-Euclidean Geometry.
Spherical geometry.
Geometry of Lobachevsky.
Riemannian geometry.
Geometry of manifolds.
Geometry as a systematic science appeared in ancient Greece, its axiomatic constructions are described in Euclid's Elements. Euclidean geometry dealt with the study of the simplest figures on the plane and in space, the calculation of their area and volume. The coordinate method proposed by Descartes in 1637 formed the basis of analytic and differential geometry, and problems connected with drawing led to the creation of a descriptive and projective geometry. In this case, all the constructions remained within the framework of Euclid's axiomatic approach. The radical changes are connected with the work of Lobachevsky in 1826, which abandoned the axiom of parallelism and created a new non-Euclidean geometry, thus defining the path of further development of science and the creation of new theories.
The classification of geometry, proposed by Klein in the Erlangen program in 1872 and containing in its base the invariance of geometric objects with respect to various transformation groups, is still preserved.
Geometry deals with the mutual arrangement of bodies, which is expressed in the touch or fit to each other, the location "between", "inside", etc.; the size of bodies, that is, the concepts of equality of bodies, "more" or "less"; as well as transformations of bodies. The geometric body represents an abstraction from the time of Euclid, who believed that "a line is a length without a width", "a surface is that which has a length and a width." A point is an abstraction associated with an unlimited reduction of all body dimensions, or the limit of infinite division. The location, dimensions and transformations of geometric shapes are determined by spatial relationships.
Exploring real objects, geometry considers only their shape and relative position, being distracted from other properties of objects, such as density, weight, color. This allows us to move from spatial relations between real objects to any relations and forms that arise when considering homogeneous objects, and similar to spatial ones. In particular, geometry allows us to consider the distances between functions
Common in our days [the source is not indicated 370 days], the classification of various sections of geometry was proposed by Felix Klein in his Erlangen program (1872). According to Klein, each section studies those properties of geometric objects that are preserved (invariant) under the action of a certain group of transformations specific for each section. In accordance with this classification, the following main sections can be distinguished in classical geometry.
Euclidean geometry, in which it is assumed that the dimensions of segments and angles do not change when moving figures on the plane. In other words, this is the theory of those properties of figures that are preserved when they are transferred, rotated and repelled.
Planimetry is a section of Euclidean geometry that examines figures on a plane.
Stereometry is a section of Euclidean geometry in which figures in space are studied.
Projective geometry that studies the projective properties of figures, that is, properties that persist under their projective transformations.
Affine geometry that studies the properties of figures that are preserved under affine transformations.
Descriptive geometry is an engineering discipline based on the projection method. This method uses two or more projections (orthogonal or oblique), which makes it possible to represent a three-dimensional object on a plane.
Spherical triangle
Modern geometry includes the following additional sections.
Multidimensional geometry.
Non-Euclidean Geometry.
Spherical geometry.
Geometry of Lobachevsky.
Riemannian geometry.
Geometry of manifolds.
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Александр Колесников
mail4correspondence@gmail.com
Russia
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