Elements of a triangle Other segments are also considered in a triangle: Medians (segments connecting the vertices of the triangle with the midpoints of opposite sides.) Bisectors (segments enclosed inside the triangle that bisect its angles) Altitudes (perpendiculars dropped from the vertices of the triangle to the line containing the opposite side)
Isosceles triangle theorems In an isosceles triangle, the base angles are equal. In an isosceles triangle, the base angles are equal. In an isosceles triangle, the bisector drawn to the base is the median and altitude. In an isosceles triangle, the bisector drawn to the base is the median and altitude.
Literature used: Textbook "Geometry" grades 7-9 / L.S. Atanasyan - publishing house "Prosveshchenie", 2007 Textbook "Geometry" grades 7-9 / L.S. Atanasyan - publishing house "Prosveshchenie", 2007 Encyclopedia for children.T .11.Mathematics / Chief editor M.D. Aksenova-M.: Avanta+, 1998. Encyclopedia for children. Vol. 11. Mathematics / Chief editor. M.D. Aksenova-M.: Avanta+, 1998.
Many objects around us have a shape similar to geometric shapes. The album sheet has the shape of a rectangle. If you place a round glass on a piece of paper and trace it with a pencil, you will get a line depicting a circle. A ring or hoop resemble a circle in shape, while a circus arena, the bottom of a glass or a plate have the shape of a circle. An orange, a soccer ball, and a watermelon look like a ball. A hexagonal pencil and Egyptian pyramids are also geometric shapes.
Geometry is the science of the properties of geometric figures: triangle, square, circle, pyramid, sphere, etc.
The word “geometry” is Greek and translated into Russian means “land surveying”. It is generally accepted that geometry originated in Ancient Greece. But the Greeks adopted the basics of land surveying from the Egyptians and turned it into a scientific discipline by establishing general laws. The main work on geometry is the “Elements” of the ancient Greek scientist Euclid, compiled about 300 BC. This work was considered exemplary for a long time. Euclidean geometry studies the simplest geometric forms: points, straight lines, segments, polygons, balls, pyramids, etc. It is this section of geometry that is studied in school.
In 1877, the German mathematician Felix Klein, in his Erlanger Program, proposed a classification of various branches of geometry, which is still used today: Euclidean geometry, projective, affine, descriptive, multidimensional, Riemannian, non-Euclidean geometry, geometry of manifolds, topology.
Euclidean geometry consists of two parts: planimetry and stereometry.
Planimetry is a branch of geometry in which geometric figures on a plane are studied.
Stereometry is a branch of geometry that studies figures in space.
Projective geometry studies the properties of figures that are preserved when they are projected (replaced with similar figures of a different size).
Affine geometry studies the constant properties of figures under various changes in plane and space.
The engineering discipline - descriptive geometry uses several projections to depict an object, which allows you to make a three-dimensional image of the object.
Multidimensional geometry explores the alternative existence of the fourth dimension.
There are separate instrumental subsections: analytical geometry, which uses algebraic methods to describe geometric figures, and differential geometry, which studies the graphs of various functions.
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Geometry - a branch of mathematics that studies spatial relationships and shapes, as well as any other relationships and shapes similar to spatial ones.
In Russian (as in many others), the term “geometry” is used not only for the corresponding science, but also for the set of spatial or similar forms and properties of the object in question.
Modern geometry is divided, both according to the main objects of study and the methods used, into many disciplines, see the section Main branches of geometry, which have both fundamental and applied significance. All of them are united by a single geometric approach, consisting in the fact that attention is paid primarily to the qualitative characteristics of the objects under consideration, as well as the desire for clarity at all stages of the study, from setting the problem to formulating the result. Geometry has numerous applications, see the section The place of geometry in the modern world, which, in turn, stimulate its development.
Geometry permeates almost all areas of human activity. Our ideas about beauty and harmony, about strict proof, about an impeccable logical structure are inextricably linked with geometry. Finally, the richness of human vision greatly increases the possibilities of analysis and allows one to detect complex relationships that are not obvious without a visual image of the objects being studied. This is probably why, when solving a complex problem, we often strive to draw a picture (scheme, plan, diagram). In other words, we strive to find a successful visualization, build a geometric model, i.e. reduce the problem to a geometric one.
Development of geometry.
Geometry is one of the oldest types of human activity. Even in prehistoric times, people depicted hunting patterns on the walls of caves, as well as rather complex geometric patterns. Later, with the emergence of agriculture in Ancient Egypt and Babylon, the need arose to divide land plots. Apparently, it was then that the rudiments of science began to form in geometry: some general patterns and relationships between such geometric quantities as area and length were discovered and understood. Let us note that, in essence, these were empirical facts; evidence at that time was either absent altogether or was at a primitive level.
Finally, about two and a half thousand years ago, according to historians, geometry was brought from Egypt to Greece. Here geometry not only receives its modern name (the word “geometry” comes from the Greek language and means “to measure the earth”), but also gradually develops into a coherent system of knowledge, new facts accumulate, some requirements for evidence are developed, and the first abstract concepts of geometric figures and movements. Scientific schools appear (the most famous of them is the school Pythagoras). As a result, a qualitative leap occurs, and geometry becomes a separate mathematical science, the statements of which are provided with evidence. The brilliant result of the Greek period was the “Principles” Euclid(around 300 BC). In Euclid’s presentation, geometry (more precisely, elementary geometry) appears to us, practically, in its modern form, as the science of the simplest spatial forms and relationships, developed on the basis of clearly formulated basic principles - axioms and postulates, in a strict logical sequence. Also in Ancient Greece, the doctrine of conic sections arose ( Apollonius), the beginnings of trigonometry ( Hipparchus), etc.
During the Renaissance, interest in geometry was driven mainly by practical needs. Cartography is developing ( Mercator), astronomy ( Kepler), prospect theory ( Leonardo da Vinci, Vitruvius). However, a fundamentally new step was taken only at the beginning of the 17th century. Rene Descartes (René Descartes; Renatus Cartesius), who in his fundamental work “Discourse on the Method...” (1637) was the first to use algebraic methods in geometric research. To achieve this, Descartes introduced coordinate systems and presented curves and surfaces as sets of solutions to (algebraic) equations. With the help of his method, Descartes was able to discover a number of new facts, which made his approach very popular. In modern terms, Descartes created analytical geometry and came close to creating algebraic geometry. Also in the 17th century Desargues (Gerard Desargues) and his student Pascal (Blaise Pascal) laid the foundations of projective geometry and descriptive geometry.
Descartes' coordinate method made it possible to connect geometry with algebra, which was rapidly developing at that time and originated in the works Leibniz And Newton mathematical analysis. As a result, in the 18th century Euler (Leonhard Euler), Monge (Gaspard Monge) And Poncelet (Jean-Victor Poncelet) already study curves and surfaces defined by arbitrary sufficiently smooth functions (not necessarily algebraic). Thus was born differential geometry, which owes its name mainly to methods based on the use of differential calculus. In this capacity, she flourishes in her works Gauss (Johann Carl Friedrich Gauß) And Bonn (Pierre Ossian Bonnet).
The next qualitative leap occurred already in the 19th century. Apparently, the study of general surfaces and comparison of the results obtained with elementary (Euclidean) geometry led geometers to understand the possibility of the existence of other, non-Euclidean geometries. The cornerstone of the development of non-Euclidean geometries was the famous “fifth postulate” of Euclid, which states (in the formulation Prokla), that in the plane through a point not lying on a given line, one and only one line can be drawn parallel to the original one. From ancient times until the 18th century, attempts were made from time to time to derive this statement from other axioms of Euclidean geometry. Among the mathematicians who addressed this topic were Ptolemy(2nd century) and Proclus(5th century), Ibn al-Haytham And Omar Khayyam(XI century), Saccheri And Legendre(XVIII century). Finally, by the beginning of the 19th century, an understanding began to emerge that it was possible to construct a meaningful theory without the fifth postulate. The honor of discovering a new geometry belongs to N.I. Lobachevsky, who published the work “On the Principles of Geometry” in 1829, which states the impossibility of proving the fifth postulate and the existence of a consistent theory based on the opposite statement. A Hungarian mathematician independently came to the same conclusion Bolyayi (Janos Bolyai), who published his work in 1832. Later it turned out that Gauss realized the possibility of the existence of non-Euclidean geometries somewhat earlier, but did not publish works on this topic. The geometry created by Lobachevsky is now called Lobachevsky geometry.
Opening Lobachevsky And Bolyayi stimulated interest in the general theory of surfaces. It becomes clear that Lobachevsky’s “imaginary geometry” is real in curved spaces. The concept of curvature arose in the works Gauss on the theory of surfaces in the 20s of the 19th century. Gauss studies the internal geometry of the surface, i.e. a geometry that does not depend on the location of the surface in the surrounding space and does not change during bending. Proven Gauss Theorema Egregium ("Brilliant Theorem") states that the (Gaussian) curvature of a surface does not change when it is bent. In particular, it follows that no piece of a sphere can be laid on a plane without distorting the distances, which is important, for example, in cartography.
The theory of surfaces was further developed in the works Riman (GeorgFriedrichBernhard Riemann), which laid the foundations of modern multidimensional Riemannian geometry (“multidimensional theory of surfaces”). It is in the works Riman For the first time, such fundamental concepts as manifold, Riemannian metric, and curvature tensor appear. He was one of the first to realize the connection between metrics, space curvature and physical forces, which anticipated the creation of the general theory of relativity. Riemann understood that the geometries of the microcosm and macrocosm can differ significantly from Euclidean, which is in good agreement with modern physical data. Riemann He was also actively involved in complex analysis. In his works, Riemann surfaces of multivalued complex functions were constructed for the first time.
At the same time, topology is born. The first results of a topological nature were obtained back in the 18th century (for example, Euler's formula for a convex polyhedron, Euler graphs). The study of manifolds, in particular, Riemann surfaces, led to the discovery of such properties as connectedness and orientability, which are not determined by either metric or curvature. Considerations of a topological nature were already used in works Gauss , Riman, Moebius, Jordana And Cantora. However, as an independent science, topology was formed already in the 20th century, thanks to the works of Hausdorff(described an important class of topological spaces, today called Hausdorff spaces), Kuratovsky(defined a general topological space), Poincare(laid the foundations of the theory of homotopies and homology, introduced the fundamental group and Betti numbers into consideration), Alexandrova And Uryson(created the modern theory of dimensions and the theory of compact spaces).
Thus, the 19th century can be characterized as the century of the heyday of geometry. As a result, many different geometries were discovered, which, while actively developing, seemed to move further and further from each other. Felix Klein in his famous Erlangen program (1872), he proposed a unified algebraic approach that reduces geometric studies to the description of invariants of a predetermined group of transformations of a manifold. By changing the transformation group, we change the geometry under consideration. For example, from this point of view, Euclidean geometry corresponds to the group of motions of Euclidean space, projective geometry - to the group of projective transformations, topology - to the group of homeomorphisms, etc. Note that for his work on the foundations of geometry Klein was awarded the Lobachevsky Prize (1897).
Made a significant contribution to the theory of invariants Gilbert(the famous Theorem on Invariants). Gilbert He also dealt with problems of formalization of mathematics in general, in particular, he created the modern axiomatics of Euclidean geometry (fundamental work “Foundations of Geometry”, 1899). Besides, Gilbert summed up the development of geometry (and mathematics in general) by the beginning of the 20th century. Speaking at the II International Mathematical Congress (1900, Paris), Gilbert formulated 23 problems that, in his opinion, should have become the most pressing for mathematicians of the coming century. Among them are at least six geometric problems that really largely determined the direction of further development of geometry in the 20th century.
We will describe the main directions of development and sections of geometry of the 20th century in the next section. Here we will only emphasize that geometry has continued and continues to actively develop and occupies one of the leading places among the mathematical sciences. As an illustration, we present the following interesting facts. As you know, today mathematicians have two analogues of the Nobel Prize - the Fields Prize and the Abel Prize. The Fields Medal dates back to 1936. Its first two laureates (1936) were geometers: Lars Ahlfors(the theory of Riemann surfaces) and Jesse Douglas(solution to Plateau's problem on minimal surfaces). Since then, among the Fields laureates there have always been geometers. The Abel Prize is much younger, it began to be awarded in the 21st century. In total, 8 Abel Prizes were awarded in 2010, three of them in geometry ( Jean Pierre Serre 2003, Michael Atiyah And Isadore Singer 2004, Mikhail Gromov 2009) and two for geometric methods in other sciences ( Peter Lax 2005, Lenar Carleson 2006).
One of the analogues of Hilbert's list in the 21st century is the so-called Millennium Problems ( MillenniumPrizeProblems), formulated by the Clay Institute, founded in 1998 by a businessman named Landon Clay (Landon T. Clay) and mathematician Arthur Jaffey (Arthur Jaffe) for the purpose of promoting mathematical knowledge. Of the 7 problems of the millennium, three are in geometry, namely, the Hodge conjecture (the structure of cohomology classes of a projective variety, realized by algebraic subvarieties), the Poincaré conjecture (on the homological sphere, solved G. Perelman), Birch and Swinnerton-Dyer conjecture (on rational points of elliptic curves). The problem concerning the study of Yang-Mills fields can also be classified as geometric.
Main sections of modern geometry.
In the modern Universal Decimal Classification (http://udk-codes.net/) there are more than 50 items that include the word “geometry” in their name. Here we will list only a few of them, corresponding to the most significant and, in our opinion, actively developing sections of geometry.
- Algebraic geometry studies solutions to systems of equations of the form P=0, where P is a polynomial in several variables. At the same time, both questions of the existence of such solutions and the properties of the set of all solutions are investigated. Such sets are called algebraic sets or algebraic varieties. The main difference between algebraic geometry and other branches of geometry is that, in addition to other geometric methods, it very much uses the ideas and methods of abstract algebra, especially its subsections such as commutative algebra and homological algebra. One of the most famous achievements of algebraic geometry is the proof of Fermat's Last Theorem.
- Analytical geometry created Descartes, was conceived by him as algebraic geometry in the modern sense. Today, analytical geometry is a subsection of algebraic geometry that studies solutions to systems of linear or quadratic equations on the plane and in space. Thus, the objects of analytical geometry are straight lines, planes, as well as curves and surfaces of the second order. The problem of classifying these objects has been completely solved, however, analytical geometry has not lost its significance. It is important both for specific calculations and for the learning process, since it contains the foundations of such important methods as the coordinate method and the method of invariants.
- Convex geometry deals with the study of the geometry of convex sets, primarily in Euclidean spaces. Starting with works Minkowski (Hermann Minkowski) And Bruna (Hermann Brunn) it became clear that the property of convexity allows us to build an independent theory, without additional assumptions about differentiability. One of the most striking results of convex geometry is the Minkowski-Aleksandrov theorem on the reconstruction of a convex polyhedron from the properties of its faces. Convex geometry has numerous applications in optimization problems, most notably in convex programming and linear programming.
- Computational geometry deals with the construction and study of combinatorial algorithms for solving geometric problems, as well as geometric modeling, i.e. the study of discrete models of continuous curves and surfaces. Classic results in computational geometry include algorithms for constructing the convex hull, Euclidean minimum spanning tree, Delaunay triangulation, Voronoi diagram, solving the nearest neighbor problem, etc. The most well-known methods of geometric modeling use splines and Bezier curves. Computational geometry has numerous applications, primarily in robotics, pattern recognition, computer graphics, etc.
- The geometry of Banach and Hilbert spaces studies infinite-dimensional analogues of normed and Euclidean spaces. Closely related to functional analysis, measure theory, probability theory, calculus of variations. Uses ideas from convex analysis, linear algebra, topology and, of course, function theory. The most striking results include the Hahn-Banach theorem on the continuation of a continuous linear functional, the theorem Banach on a fixed point, the Ries-Fréchet theorem on the isomorphism of the dual Hilbert space to the original one.
- The geometry of groups and Lie algebras studies the geometry of varieties equipped with an additional algebraic structure, namely, the structure of a group. In this case, group operations are assumed to be smooth. This algebraic operation generates an additional algebraic structure on the tangent space at the unit of the group and turns it into a Lie algebra. Named after the Norwegian mathematician Sophusa Lee (Marius Sophus Lie). The simplest examples of Lie groups are transformation groups, such as, say, the groups of motions of Euclidean space or Lobachevsky space. The richness of the internal structure of Lie groups allows, on the one hand, to obtain deep non-trivial results, such as the classification theorem for compact Lie groups, and, on the other hand, to carry out many specific calculations to the end. Lie groups also appear frequently in applications, primarily in mechanics and physics.
- The geometry of dynamic systems studies the qualitative (i.e. geometric and topological) properties of dynamic systems of various types. Examples of such properties of a dynamic system can be the number of equilibrium positions or periodic solutions, their stability or instability, chaotic or regular behavior of solutions, the topology of invariant manifolds of the system or its entire phase space. Usually, in a qualitative study of dynamical systems, they are considered up to some equivalence (trajectory, topological, smooth, etc.), and the task is to find invariants corresponding to this equivalence (in particular, to find a complete set of invariants, i.e. . classification of systems up to appropriate equivalence).
- Number geometry deals with the geometric aspects of number theory. A typical problem in number geometry is the arrangement of integer vectors with respect to convex bodies in multidimensional space. First appeared in the works Minkowski, who proved the presence of an integer point (integer basis) in a symmetric body of sufficiently large volume. Closely related to functional analysis, Diophantine and rational approximations.
- The geometry of optimization problems studies geometric objects that are critical points of certain geometric functionals, such as the length of a curve, surface area, and energy functional. Objects of this type include minimal and harmonic surfaces, geodesics, extremal networks, minimal fillings, etc. The most striking results of this theory include the solution to Plateau’s problem on minimal surfaces, the proof of the existence of three closed nested geodesics on a manifold homeomorphic to a two-dimensional sphere, and the classification of locally closed minimal networks on surfaces of constant non-negative curvature. Problems of this type have numerous applications in physics, mechanics, chemistry, biology, logistics, etc.
- Discrete and combinatorial geometry combines geometric problems that study the combinatorial properties of discrete geometric objects, such as sets of points, lines, balls, etc. In this case, as a rule, questions about the relative position or the optimal location of these objects in the surrounding space are considered. Among the most famous problems of this type are the problem of Kepler and Newton on the maximum possible number of spheres touching a given one, the problem on the optimal packing of balls in space or in a limited volume, Tamm’s problem on the spherical code. Discrete geometry also includes questions related to one or another arrangement of graphs in ambient spaces. This also includes a number of problems of computational geometry related, for example, to Voronoi diagrams, Delaunay triangulations, etc.
- Differential geometry studies smooth manifolds with certain additional structures. It stands out, first of all, for its methods, which are closely related to mathematical analysis, in particular, to the differential properties of functions. Developed from the classical theory of curves and surfaces created by Gauss and Monge. Differential geometry is conventionally divided into local, i.e. which studies the properties of a manifold in a small neighborhood of a point, and global (the so-called geometry “as a whole”), which studies the connections between the properties of small fragments of the manifold and the characteristics of the entire manifold. In a sense, some of the individual sections of geometry that we have identified, such as Riemannian geometry and symplectic geometry, can also be considered as subsections of differential geometry.
- Integral geometry studies problems inverse to classical integration, namely, it explores the possibility of restoring a function from a set of values of its integrals over certain subsets of the domain of definition of the original function. The term “integral geometry” arose in the 30s of the XX century in the works plaque and originally meant something completely different: the calculation of integrals of functions over certain subsets of manifolds or, more generally, spaces with measure. Modern integral geometry is closely related to the theory of homogeneous spaces, the theory of fiber spaces, representation theory, and measure theory. It has numerous applications, for example in computed tomography.
- Complex geometry studies the geometry of manifolds with complex structure. Its initial branch is the theory of Riemann surfaces, created Riemann and studying the properties of one-dimensional complex manifolds. Complex geometry is characterized by close connections with complex analysis and algebra. Recently, close connections between complex geometry (in particular, geometry Teichmuller spaces) with modern theoretical physics.
- Computer geometry deals with general computer modeling associated with the visualization of geometric models. Computer geometry includes, but is not limited to, computational geometry. Within the framework of computer geometry, models of such complex objects as manifolds, non-Euclidean geometries, geodesic flow on a surface, many solutions of a differential equation, etc. are created. Computer geometry gives the modern researcher a powerful tool for conducting a variety of computer experiments, as a result of which certain hypotheses.
- Metric geometry studies the geometry of classical objects such as curves and surfaces in terms of the distance function naturally defined on them. In this case, properties defined in differential terms, such as curvature, are interpreted in terms of certain relations for the distance function. As a result, on the one hand, it is possible to transfer many results of differential geometry to the case of significantly more general objects without assumptions about smoothness, which makes it possible in many cases to achieve completeness of the spaces of the objects under consideration. As a result, unexpected connections arise between seemingly distant mathematical objects. For example, the properties of a finitely generated discrete group can be described in terms of the geometry of space with the so-called Manhattan metric ( Gromov). On the other hand, such an interpretation allows us to rethink differential geometric results and make progress in understanding such complex objects as, say, the curvature tensor.
- Descriptive geometry studies spatial figures using their multiple orthogonal projections. Originated in engineering as the main tool for constructing and reading drawings. The foundations of descriptive geometry were laid Monge, who was then teaching at an engineering school and fulfilling an order for the calculation of fortifications. Recently, in connection with the development of computer-aided design systems, the role of descriptive geometry is increasingly reduced to purely educational.
- Noncommutative geometry studies the properties of noncommutative analogues of function algebras on certain classes of spaces. The starting point that brought this idea to life is the Gelfand-Naimark theorem, proven in the early 1940s, on the equivalence of the category of compact topological spaces and commutative C * -algebras. It turned out that the algebraic structures arising here remain meaningful even after the commutativity property is abandoned. Within the framework of non-commutative geometry, methods from various departments of modern mathematics were combined: topology, differential geometry, functional analysis, measure theory, representation theory and some others. The idea of non-commutative generalization is fundamental, since thanks to it, not only many important problems were solved, but also the above-mentioned areas were mutually enriched with new methods and results. The term “non-commutative geometry” apparently arose thanks to the monograph A. Konna"Noncommutative geometry".
- Riemannian and Finslerian geometry studies manifolds on which an additional structure is given that allows one to calculate the lengths of tangent vectors. The main examples of such structures are Riemannian and pseudo-Riemannian metrics (non-degenerate symmetric bilinear forms on tangent spaces smoothly depending on a point of the manifold) and the Finsler structure (a family of norms on tangent spaces smoothly depending on the point of the manifold, which has a number of additional properties). The foundations of Riemannian geometry were laid Riemann, who generalized the theory of surfaces to the multidimensional case, transferring classical results to it Gauss, Bonn etc. Within the framework of Riemannian geometry, it is possible to obtain restrictions on the global structure of manifolds in terms of its local characteristics, similar to the curvature of a two-dimensional surface (sectional curvature, Ricci curvature, Riemann curvature).
- Symplectic geometry studies symplectic manifolds, i.e. manifolds on which a closed non-degenerate 2-form (symplectic structure) is given. In fact, symplectic geometry as a separate branch of geometry arose about 200 years ago as a convenient language for problems in classical mechanics. And now one of the main incentives for studying symplectic manifolds is that it is natural to consider them as phase spaces of dynamical systems that describe various problems of mechanics, mathematical physics, and geometry. However, starting from the 1970s-80s (after work V.I. Arnold, A. Weinstein(A. Weinstein), M.L. Gromova) symplectic geometry has turned into a separate independent field of mathematics, the development of which is stimulated by close connections with mathematical physics, low-dimensional topology, the theory of dynamical systems, algebraic geometry, and complex analysis.
- Stochastic geometry is a branch of stochastic analysis. She studies random processes in infinite-dimensional Hilbert spaces and on smooth Hilbert manifolds, described by stochastic Ito equations. The smoothness properties of transition probabilities of such processes are studied, and the construction of quasi-invariant measures on infinite-dimensional Lie groups is introduced. The foundations of stochastic differential geometry are laid Yu.L. Daletsky And Y.I.Belopolskaya in the 70s of the XX century
- Fractal geometry studies so-called fractals (self-similar sets). The first examples of such sets with unusual properties appeared in the 19th century (for example, the Cantor set). The term "fractal" was introduced B. Mandelbrot in 1975 and gained widespread popularity with the publication of his book Fractal Geometry of Nature in 1977. However, “fractal” (lat. fractus- crushed, broken, broken) is not a mathematical term and does not have a generally accepted strict mathematical definition. A fractal is a complex geometric figure that has the property of self-similarity, that is, composed of several parts, each of which is similar to the entire figure. In a broader sense, fractals are understood as sets of points in Euclidean space that have a fractional metric dimension (in the sense Minkowski or Hausdorff), or a metric dimension strictly greater than the topological one. In modern fractal geometry, random fractals are also studied. Fractal geometry has deep connections to number theory and modern physics
All these very different areas of knowledge are united geometric methods.
Geometric research methods.
The most important feature of geometric objects is their invariance (independence of the coordinate system). In this regard, geometry forms a special, characteristic picture of the world, based primarily not on formulas and calculations, but on qualitative analysis; This picture is characterized by a combination of complete mathematical rigor with extensive use of intuition. Let us list some, in our opinion, fundamental methods for studying geometric objects.
- Definition and description of the properties of the space of homogeneous objects (points), equipped with one or another additional structure. For example, a description of the geometry of Euclidean spaces, regular surfaces, smooth manifolds (in particular, manifolds with various structures - Riemannian, pseudo-Riemannian, complex, algebraic, symplectic, contact, Finsler Kähler, etc.), Hilbert spaces, Lie groups, general topological spaces, cellular complexes, etc.
- One of the central methods of geometry (and mathematics in general) is the method of coordinateization. To study a geometric object, a coordinate system is introduced that makes it possible to describe its properties using analytical or algebraic apparatus. The terms “analytic geometry”, “differential geometry”, “algebraic geometry”, “symplectic geometry” themselves are associated, in particular, with those variants of the coordinate method that are used in these sections of geometry. With this approach, the presence of different geometric structures is reflected in different classes of coordinate systems and coordinate substitutions (for example, in symplectic geometry, symplectic coordinates and canonical transformations are considered, in complex geometry - analytic coordinates and holomorphic substitutions, etc.). Since geometric objects themselves are inherently invariant, an important part of the coordinate method is to describe how certain formulas change when coordinates are changed.
- The most important characteristic of a geometric object is the set of its “symmetries”, i.e. a group of transformations that preserves its properties. Thus, a group of orthogonal operators is associated with a Euclidean space, a group of diffeomorphisms is associated with a smooth manifold, a group of isometries is associated with a Riemannian manifold, etc. Studying a group of transformations allows you to obtain important information about the object itself; for example, when studying homogeneous spaces, the properties of the transformation group play a key role.
- The metric approach in geometry is associated with the introduction of an analogue of the distance between points and the study of the properties of this distance (the general theory of metric spaces, the geometry of Banach spaces and the properties of operators in them, seminorms and Fréchet spaces, etc.).
- The axiomatic method has been used in geometry since its inception. It consists in the fact that geometric structures are described using a list of axioms, from which other properties are subsequently derived. For example, Euclidean geometry is defined in a linear space (that is, a set with the operations of addition and multiplication by a number that satisfy a certain set of axioms) with a scalar product (a function of a pair of vectors that also satisfies some axioms). Another example: an affine connection on a manifold is defined as an operation of differentiation of vector fields that satisfies the linearity axioms and Leibniz's rule.
- In recent decades, computer geometric modeling has been actively developing. Many programs have been developed that allow you to visualize geometric objects that arise when modeling a variety of processes, clearly demonstrate their properties and conduct computer experiments to test mathematical, physical, biological, economic and other hypotheses. Moreover, computer modeling is also used to prove mathematical theorems (however, such proofs invariably raise doubts among many mathematicians); famous examples - proof Appel And Haken in 1976, the four-color conjecture and proof in 1989 Lam non-existence of a finite projective plane of the 10th order.
The place of geometry in the modern world.
Mathematics. The geometric view of the world permeates all modern mathematics; Most of its sections use geometric language and apply geometric methods. Often the penetration of geometric ideas leads to the creation of new theories, the formulation of new problems and unexpected results: in particular, geometric ideas in the theory of ordinary differential equations led to the creation of a qualitative theory and the theory of dynamical systems; in the theory of partial differential equations - to microlocal analysis, the theory of non-standard characteristics, the theory of solitons and Yang-Mills fields; in the calculus of variations - to geometric variational problems, the theory of geodesic flows.
Natural sciences. Modern physics is closely connected with geometry. Classical mechanics uses the language, methods and results of Riemannian and symplectic geometry, optics and thermodynamics - symplectic and contact geometry, quantum mechanics uses complex geometry, symplectic geometry and the geometry of Hilbert spaces, quantum field theory - differential, complex, algebraic and symplectic geometry. In almost all sections of theoretical physics, geometric ideas, methods or structures are encountered in one way or another. Let us note that physical ideas, in turn, appear in geometry; often the analysis of physical theories gave impetus to the development of geometric constructions (for example, symplectic and contact geometry are directly related to physics).
Geography has always used geometric language; in particular, the idea of describing a surface using maps and coordinates closely links these sciences. Spherical geometry is used in designing routes for ships and aircraft.
Geometry is used in chemistry and molecular biology; complex compounds (for example, proteins) have a rich geometric structure, which, as it turns out, significantly affects the chemical and biological properties of the substance in question; geometry is also used to describe the energetic and quantum properties of molecules.
Technique. Modern technology actively uses geometric methods and results. Computer geometry is used in the design of cars, airplanes, bridges and many other technical objects; geometric problems arise when cutting precious stones, in matters of mobile navigation, etc. Geometric methods of pattern recognition are widely used, and modern ciphers and codes are often based on the algebraic properties of elliptic curves.
Medicine. The task of reconstructing the picture of internal organs from their projections visible on photographs (medical tomography) is geometric in nature and is associated with integral geometry (describing the properties of a function on a manifold by integrals of it over given families of submanifolds). In medicine, geometric models of various parts of the skeleton are used (for example, a moving jaw for dental prosthetics, knee and elbow joints, etc.). The development of modern 3D technologies has made it possible to create individual bone prostheses based on the results of a 3D scan of a patient. Computer models of individual organs and their systems also play an important role in modern medicine. For example, when developing major operations on the heart, its geometric computer model is often used.
Art. Geometric images have long been used in fine arts and architecture. The geometric science of perspective is found in Aeschylus And Democritus(although, of course, its elements were used much earlier - for example, in the construction of Egyptian temples and pyramids). Subsequently, this section of geometry was developed by many artists and scientists (in particular, a great contribution to its development was made by Leonardo da Vinci, Durer, Desargues, Monge and others). Now perspective geometry and descriptive geometry are standard tools for artists, architects and designers. Let's say the roof of the airport terminal in Sharm el-Sheikh (Egypt) is a minimal surface model. Geometry is also important in music: the shape of a musical instrument, a concert hall, a temple is the result of subtle geometric and acoustic calculations. Finally, 3D technologies, based on projective and computational geometry, are increasingly being used in film and television, taking them to the next stage of development.
Humanities. Geometry is also used in the humanities: economics (transport problems, optimization problems, geometric models of production, application of the properties of continuous mappings to finding economic equilibrium); linguistics (geometry of word spaces), etc.
Religion. Sacred geometry - a system of religious ideas about the forms and space of the world, reflecting its proportionality and harmony - is present in most world religions. It manifests itself in sacred architecture, painting and music, and iconography. Geometric shapes are used by almost all religions as sacred symbols.
Education. In modern school education, geometry plays an exceptional role. It is in geometry lessons that children learn what a rigorous proof is, learn to think logically and draw well-founded conclusions from premises. At the same time, school geometry demonstrates visual (i.e., invariant) mathematics, based not so much on formulas, but on a detailed study of the qualitative properties of geometric objects. This combination of rigor with clarity underlies the natural scientific picture of the world; Thus, the study of geometry is the most important stage in all scientific education. , trans. from German, M.-L., 1937.
INTRODUCTION
1862-1943 ) at the end of the CIC century.
– – measure.
Geometry construction scheme
The main undefined concepts are listed.
The properties of basic concepts - axioms - are formulated.
Other geometric concepts are defined.
The properties of geometric concepts - theorems - are formulated and proven.
AXIOMS OF STEREOMETRY. CONSEQUENCES FROM AXIOMS
Basic concepts of stereometry: point, line, plane, distance.
Definition: An axiom is a proposition that does not require proof .
The basic properties of points, lines and planes regarding their relative positions are expressed in axioms. The entire system of axioms of stereometry consists of a number of axioms known to us from the planimetry course, and axioms about the relative positions of points, lines and planes in space.
AXIOMS OF STEREOMETRY
I. Axioms of belonging
I 1. There is at least one straight line and at least one plane. Every straight line and every plane is a non-empty set of points that does not coincide with space.
Designation:
A, B, C, D – dots;
a, b, c – straight;
a, b, g – planes;
A Î A– point A belongs to line a, line a passes through point A;
E Ï A –point E does not belong to line a;
S Î a– point C belongs to plane a, plane a passes through point C;
E Ï a –point E does not belong to plane a.
Conclusion: There are points that belong to a line and those that do not belong to a line; there are points that belong to a plane and that do not belong to a plane.
I 2. Through two different points there passes one and only one straight line.
Designation:
and М a –plane a passes through line a;
b Ë a–plane a does not pass through line b.
I 4. Through three points that do not belong to the same line, there passes one and only one plane.
Designation: a = ABC
Conclusion:
Planes that have three different common points coincide.
I 5. If two different planes have a common point, then their intersection is a straight line.
Designation: M Î a , MÎ b , a ¹ b , aìü b = l.
II. Axioms of distance
II 1. For any two points A And IN there is a non-negative quantity called the distance from A to IN. Distance AB equals zero if and only if the points A And IN match.
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Designation: AB³ 0.
II 2. Distance from A to IN equal to the distance from IN to A.
Designation: AB = BA.
II 3. For any three points A, IN, WITH distance from A to WITH no more than the sum of distances from A to IN and from IN to WITH.
Designation: AC £ AB + BC.
III. Axioms of order
III 1. Any point ABOUT direct r splits the set of all things distinct from a point ABOUT points of a straight line r into two non-empty sets so that for any two points A And IN, belonging to different sets, point ABOUT lies between the points A And IN; if points A And IN belong to the same set, then one of them lies between the other and the point ABOUT.
III 3. If the point WITH lies between the points A And IN, then the points A, IN, WITH belong to the same line.
III 4. Any straight line r, lying in the plane a r r.
IV. Axiom of plane mobility
If points A, IN, A 1, B 1 lie in a plane a, and AB > 0 And AB= A 1 B 1, then there are two and only two movements of this plane, each of which displays a point A per point A 1 and point IN per point B 1.
V. Axiom of parallel
Through the point A there is at most one line parallel to a given line r.
CONSEQUENCES FROM AXIOMS
Corollary 1: Through a straight line and a point not belonging to it, one and only one plane can be drawn.
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Given: M, a, M Ï A
Prove:
2. .
Proof:
1. Let us choose points A and B on line a (axiom I 1 ): AÎ a, BÎ A.
): a = MAV.
Since points A, B belong to the plane a, then straight line a belongs to the plane a (axiom I 3 ): AÌ a.
Consequently, there is a plane a passing through the line a and a point M that does not belong to it: .
2. Plane a contains line a and point M, that is, it passes through points M, A, B. Through three points that do not belong to the same line, there is a single plane (axiom I 4 ).
Corollary 2: Through two intersecting lines one and only one plane can be drawn.
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Given: a, b, a ´ b
Prove:
2. .
Proof:
1. Let us denote the point of intersection of lines a and b: .
Let us choose point A on line a and point B on line b (axiom I 1 ): AÎ a, BÎ b.
The plane a passes through the points M, A, B (axiom I 4 ): a = MAV.
Since points A, M belong to the plane a, then straight line a belongs to the plane a (axiom I 3 ): AM = aÌ a.
Since points B, M belong to plane a, then straight line b belongs to plane a (axiom I 3 ): VM = b Ì a.
Consequently, there is a plane a passing through two intersecting lines a and b: .
2. Plane a contains lines a and b, that is, it passes through points M, A, B. Through three points that do not belong to the same line, there is a single plane (axiom I 4 ).
Definition:
Lines are called parallel if they lie in the same plane and do not have common points or coincide.
Corollary 3:
Through two parallel lines one and only one plane can be drawn.
Given: a, b,
Prove:
2. .
Proof:
1. The existence of a plane a passing through two parallel lines a and b follows from the definition of parallel lines.
2. Suppose that there is another plane containing lines a and b. Let us choose point A on line a, and points B and M on line b (axiom I 1 ): AÎ a, BÎ b, MÎ b. We found that two planes pass through points A, B, M, which contradicts the axiom I 4. Therefore, the assumption is not true, plane a–the only one.
Exercises:
c) 
;
2. Name from the picture:
a) the planes in which the straight lines PE, MK, DB, AB, EC lie;
b) the points of intersection of straight line DK with plane ABC, straight line CE with plane ADV;
c) points lying in the planes ADB and DBC;
d) 
straight lines along which planes ABC and DCB, ABD and CDA, PDC and ABC intersect.
3. Name from the picture:
a) points lying in the DCC 1 and BQC planes;
b) planes in which straight line AA 1 lies;
c) the points of intersection of the straight line MK with the plane ABD, straight lines DK and BP with the plane A 1 B 1 C 1;
d) straight lines along which planes AA 1 B 1 and ACD, PB 1 C 1 and ABC intersect;
e) points of intersection of straight lines MK and DC, B 1 C 1 and BP, C 1 M and DC.
3. RELATIVE POSITION OF TWO STRAIGHT LINES IN SPACE
SIGN OF CROSSING STRAIGHT LINES
Rice. 1. Fig. 2. Fig. 3.
Definition: Planes are parallel if they do not have common points or coincide.
TETRAHEDRON. PARALLELEPIPED
In the topic “Geometric bodies, their surfaces and volumes” we will study polyhedra - geometric bodies whose surfaces are made up of polygons. To illustrate the concepts associated with the relative position of lines and planes in space, let’s get acquainted with two polyhedra - a tetrahedron and a parallelepiped.
Consider an arbitrary triangle ABC and period D , not lying in the plane of this triangle. Connecting the dot D segments with triangle vertices ABC , we get triangles DAB , DBC ,DCA .
Surface composed of four triangles ABC , DAB , DBC ,DCA , called tetrahedron and is designated DABC .
The triangles that make up a tetrahedron are called edges, their sides – ribs, and the vertices are vertices of a tetrahedron. A tetrahedron has four faces, six edges and four vertices.
Two edges of a tetrahedron that do not have common vertices are called opposite. At the tetrahedron DABC the ribs are opposite AD And Sun , ВD And AC , CD And AB . Often one of the faces of a tetrahedron is called basis, and three others - side faces.
Consider two equal parallelograms ABCD And A 1 B 1 C 1 D 1 , located in parallel planes so that the segments AA 1 , BB 1 , SS 1 And DD 1 parallel. Quadrilaterals АВВ 1 А 1 , VSS 1 IN 1 , СDD 1 С 1 ,DAA 1 D 1 are also parallelograms, since each of them has pairs of parallel opposite sides.
Surface composed of two equal parallelograms ABCD And A 1 B 1 C 1 D 1 and four parallelograms АВВ 1 А 1 , VSS 1 IN 1 , СDD 1 С 1 ,DA A 1 D 1 , is called a parallelepiped and is denoted ABCDA 1 B 1 C 1 D 1 .
The parallelograms that make up a parallelepiped are called edges, their sides – ribs, and the vertices of parallelograms are vertices of a parallelepiped. The parallelepiped has six faces, twelve edges and eight vertices. Two faces of a parallelepiped that have a common edge are called adjacent, and not having common edges – opposite. Two vertices that do not belong to the same face are called opposite. A line segment connecting opposite vertices is called parallelepiped diagonal. Each parallelepiped has four diagonals.
Diagonals of a parallelepiped ABCDA 1 B 1 C 1 D 1 are segments AC 1 , ВD 1 , CA 1 , DВ 1 .
Often some two opposite faces are identified and called reasons, and the remaining faces are lateral faces of the parallelepiped. The edges of a parallelepiped that do not belong to the bases are called side ribs.
If as the bases of a parallelepiped ABCDA 1 B 1 C 1 D 1 select faces ABCD And A 1 B 1 C 1 D 1 , then the lateral faces will be parallelograms АВВ 1 А 1 , VSS 1 IN 1 , СDD 1 С 1 ,DA A 1 D 1 , and the lateral edges are segments AA 1 , BB 1 , SS 1 And DD 1 .
Exercises:
1. In the tetrahedron DABC the points M, N, Q, P are the midpoints of segments ВD, DC, AC, AB. Find the perimeter of the quadrilateral MNQP if AD = 12 cm, BC = 14 cm.
ANGLE BETWEEN STRAIGHTS
Definition: Angle between non-parallel lines T And n is the smallest of the adjacent angles formed by intersecting lines T" And p", Where T"|| T,p"|| n.
, , .
Comment: The angle between parallel lines is considered equal to zero.
Definition: Two lines in space are called perpendicular if the angle between them is equal .
Designation:
Perpendicular lines can intersect and can be skew.
Task: Given a cube ABCDA 1 B 1 C 1 D 1.
Find: 
; 
; .
Solution:
Based on the parallelism of two lines:
and therefore . 
.
. 
, since CDD 1 C 1 is a square.
On the basis of intersecting lines:
, hence, · .
, hence, .
Conclusion: 
From the center O of a circle of radius 3 dm, a perpendicular OB to its plane is restored. A tangent is drawn to the circle at point A and on this tangent a segment AC equal to 2 dm is laid off from the point of tangency. Find the length of the inclined BC if the length of the perpendicular OB is 6 dm.
5. From vertex D of rectangle ABCD, whose sides AB = 9 cm and BC = 8 cm, perpendicular DF = 12 cm is restored to the plane of the rectangle. Find the distances from point F to the vertices of the rectangle.
8. DIHEDRAL ANGLE. LINEAR ANGLE DIHEDRAL ANGLE
III 4. Any straight line r
, lying in the plane a
, splits the set of points of this plane that do not belong to it into two non-empty sets so that any two points belonging to different sets are separated by a straight line r
; any two points belonging to the same set are not separated by a line r
.
Sets to which there is a direct line r splits a set of points on the plane that do not belong to it a, are called open half-planes with boundary r.
Side BC of rectangle ABCD serves as side of triangle BCF, with vertex F projected onto DC. Name the linear angle of the dihedral angle formed by planes ABC and BCF (Fig. 1.).
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Rice. 1. Fig. 2.
An image of an isosceles trapezoid ABCD and a triangle ABM is given. The segment MC is perpendicular to the plane ABC. Construct a linear angle of the dihedral angle formed by planes ABC and ВСМ so that one of its sides passes through point M (Fig. 2.).
3. On the face of a dihedral angle equal to 45°, a point is given that is 4 cm away from the edge. Find the distance from this point to the other face.
The polygon is divided by diagonals drawn from one vertex into a finite number of triangles, for each of which the theorem is true. Therefore, the theorem will also be true for the sum of the areas of all triangles whose planes form the same angle with the projection plane.
Comment: The theorem proved is valid for any plane figure bounded by a closed curve.
Exercises:
1. Find the area of a triangle whose plane is inclined to the projection plane at an angle , if its projection is a regular triangle with side a.
2. Find the area of a triangle whose plane is inclined to the projection plane at an angle , if its projection is an isosceles triangle with a side of 10 cm and a base of 12 cm.
3. Find the area of a triangle whose plane is inclined to the projection plane at an angle , if its projection is a triangle with sides 9, 10 and 17 cm.
4. Calculate the area of a trapezoid, the plane of which is inclined to the projection plane at an angle , if its projection is an isosceles trapezoid, the larger base of which is 44 cm, the side is 17 cm and the diagonal is 39 cm.
5. Calculate the projection area of a regular hexagon with a side of 8 cm, the plane of which is inclined to the projection plane at an angle.
6. A rhombus with a side of 12 cm and an acute angle forms an angle with a given plane. Calculate the area of the projection of the rhombus onto this plane.
7. A rhombus with a side of 20 cm and a diagonal of 32 cm forms an angle with a given plane. Calculate the area of the projection of the rhombus onto this plane.
8. The projection of a canopy onto a horizontal plane is a rectangle with sides and . Find the area of the canopy if the side faces are equal rectangles inclined to the horizontal plane at an angle , and the middle part of the canopy is a square parallel to the projection plane.
11. Exercises on the topic “Lines and planes in space”:
The sides of the triangle are equal to 20 cm, 65 cm, 75 cm. From the vertex of the larger angle of the triangle, a perpendicular equal to 60 cm is drawn to its plane. Find the distance from the ends of the perpendicular to the larger side of the triangle.
2. From a point located at a distance of cm from the plane, two inclined ones are drawn, forming angles with the plane equal to , and a right angle between them. Find the distance between the points of intersection of the inclined planes.
3. The side of a regular triangle is 12 cm. Point M is chosen so that the segments connecting point M with all the vertices of the triangle form angles with its plane. Find the distance from point M to the vertices and sides of the triangle.
4. A plane is drawn through the side of the square at an angle to the diagonal of the square. Find the angles at which two sides of the square are inclined to the plane.
5. The leg of an isosceles right triangle is inclined to the plane a passing through the hypotenuse at an angle . Prove that the angle between plane a and the plane of the triangle is equal to .
6. The dihedral angle between the planes of triangles ABC and DBC is equal to . Find AD if AB = AC = 5 cm, BC = 6 cm, BD = DC = cm.
Test questions on the topic “Lines and planes in space”
1. List the basic concepts of stereometry. Formulate the axioms of stereometry.
2. Prove consequences from the axioms.
3. What is the relative position of two lines in space? Give definitions of intersecting, parallel, and skew lines.
4. Prove the sign of skew lines.
5. What is the relative position of the line and the plane? Give definitions of intersecting, parallel lines and planes.
6. Prove the sign of parallelism between a line and a plane.
7. What is the relative position of the two planes?
8. Define parallel planes. Prove a sign that two planes are parallel. State theorems about parallel planes.
9. Define the angle between straight lines.
10. Prove the sign of perpendicularity of a line and a plane.
11. Define the base of a perpendicular, the base of an inclined, the projection of an inclined onto a plane. Formulate the properties of a perpendicular and inclined lines dropped onto a plane from one point.
12. Define the angle between a straight line and a plane.
13. Prove the theorem about three perpendiculars.
14. Give definitions of dihedral angle, linear angle of dihedral angle.
15. Prove the sign of perpendicularity of two planes.
16. Define the distance between two different points.
17. Define the distance from a point to a line.
18. Define the distance from a point to a plane.
19. Define the distance between a straight line and a plane parallel to it.
20. Define the distance between parallel planes.
21. Define the distance between intersecting lines.
22. Define the orthogonal projection of a point onto a plane.
23. Define the orthogonal projection of a figure onto a plane.
24. Formulate the properties of projections onto a plane.
25. Formulate and prove a theorem on the projection area of a plane polygon.
INTRODUCTION
The first scientific presentation of geometry that has come down to us is contained in the work “Elements,” compiled by the ancient Greek scientist Euclid, who lived in the 3rd century BC in the city of Alexandria. It was Euclid who made the first attempt to give an axiomatic presentation of geometry. For the first time, the scientific system of Euclid's axioms was formulated by D. Hilbert ( 1862-1943 ) at the end of the CIC century.
The school geometry course consists of two parts: planimetry and stereometry. In planimetry, the properties of geometric figures on a plane are studied.
Stereometry is a branch of geometry that studies the properties of figures in space.
The word "stereometry" comes from the Greek words "stereos" – volumetric, spatial and “metreo” – measure.
The idea of geometric bodies studied in stereometry is given by the objects around us. Unlike real objects, geometric bodies are imaginary objects. By studying the properties of geometric bodies, we gain an understanding of the geometric properties of real objects and can use these properties in practical activities. Geometry, in particular stereometry, is widely used in construction, architecture, mechanical engineering, geodesy, and in many other fields of science and technology.
Geometry construction scheme
