Axioms

In Euclidean geometry, axioms and postulates are propositions that relate concepts, defined in terms of the point, the line and the plane. Euclid proposed five postulates and it was the fifth (the postulate of parallelism) that centuries later – when many geometers questioned it when analyzing it – gave rise to new geometries: the elliptic (Riemann geometry) or the hyperbolic geometry of Nikolai Lobachevsky.

In analytical geometry, axioms are defined in terms of equations of points, based on mathematical analysis and algebra. Talking about points, lines or planes takes on another new meaning. You can define any function, whether it is a line, a circle, a plane, etc.

Euclid adopted an abstract approach to geometry in his Elements,[7]​ one of the most influential books ever written.[8]​ Euclid introduced certain axioms or postulates that express primary or self-evident properties of points, lines, and planes.[9]​ He proceeded to rigorously deduce other properties through mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.[10] At the beginning of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792-1856), János Bolyai (1802-1860), Carl Friedrich Gauss (1777-1855) and others[11]​ led to There was a resurgence of interest in this discipline, and in the 19th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern basis for geometry.[12]

Points

Points "Point (geometry)") are considered fundamental objects in Euclidean geometry. They have been defined in various ways, including Euclid's definition as "that which has no part" [13] and by the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be constructed from points. However, some geometry study has been carried out without reference to points.[15]​.

Lines

Euclid described a line as "length without width" that "lies equally with respect to points on itself."[13]​ In modern mathematics, given the multitude of geometries, the concept of line is closely related to the way geometry is described. For example, in analytical geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,[16]​ but in a more abstract setting, such as incidence geometry, a line can be an independent object, distinct from the set of points lying on it.[17]​ In differential geometry, a geodesic is a generalization of the notion of line to curved spaces.[18]​.

Plans

A plane is a two-dimensional flat surface that extends infinitely.[13] Planes are used in all areas of geometry. For example, planes can be studied as a topological surface without reference to distances or angles;[19]​ it can be studied as an affine space, where collinearity and proportions can be studied but not distances;[20]​ it can be studied as the complex plane using complex analysis techniques;[21]​ and so on.

Angles

Euclid defines a plane angle as the inclination towards each other, in a plane, of two lines that meet and are not straight to each other.[13]​ In modern terms, an angle is the figure formed by two rays of light, called sides of the angle, that share a common end point, called the vertex of the angle.[22]​.

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.[13]​ The study of the angles of a triangle or the angles in a unit circle forms the basis of trigonometry.[23]​.

In differential geometry and calculus, angles between plane curves or space curves or surfaces can be calculated using the derivative.[24][25]​.

Curves

A curve is a one-dimensional object that may or may not be straight (like a line); Curves in two-dimensional space are called plane curves and those in three-dimensional space are called space curves.[26]​.

In topology, a curve is defined by a function from an interval of the real numbers to another space.[19]​ In differential geometry, the same definition is used, but the defining function is required to be differentiable.[27]​ Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.[28]​.

Surfaces

A surface is a two-dimensional object, such as a sphere or a paraboloid.[29]​ In differential geometry[27]​ and topology,[19]​ surfaces are described by two-dimensional "patches" (or neighborhoods) that are assembled using diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.[28]​.

Varieties

A variety "Variety (mathematics)") is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where each point has a neighborhood that is homeomorphic to Euclidean space.[19]​ In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.[27]​.

Manifolds are widely used in physics, including general relativity and string theory.[30]​.

Length, area and volume

Length, area and volume describe the size or extent of an object in one dimension, two dimensions and three dimensions, respectively.[31]​.

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated using the Pythagorean theorem.[32]​.

Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or three-dimensional space.[31] Mathematicians have found many explicit formulas for the area and formulas for the volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral[33]​ or the Lebesgue integral.[34]​.

Metrics and measurements

The concept of length or distance can be generalized, giving rise to the idea of ​​metrics.[35] For example, the Euclidean metric measures the distance between points on the Euclidean plane, while the hyperbolic metric measures the distance on the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metric of general relativity.[36]​.

In another direction, the concepts of length, area and volume are expanded with measurement theory, which studies methods of assigning a size or measurement to sets, where measurements follow rules similar to those of classical area and volume.[37]​.

Congruence and similarity

Congruence and similarity are concepts that describe when two shapes have similar characteristics.[38]​ In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are equal in both size and shape.[39]​ Hilbert, in his work on creating a more rigorous basis for geometry, treated congruence as an indefinite term whose properties are defined by axioms.

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different types of transformations.[40]​.

Constructions with compass and ruler

Classical geometers paid special attention to the construction of geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass "Compass (instrument)") and the ruler. Furthermore, each build had to be completed in a finite number of steps. However, some problems proved difficult or impossible to solve by these means alone, and ingenious constructions were found using parabolas and other curves, as well as mechanical devices.

Dimension

Where traditional geometry allowed the dimensions of a line of a plane and our environmental world conceived as a three-dimensional space, mathematicians and physicists have used higher dimensions for almost two centuries.[41] An example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the degrees of freedom of the system. For example, the configuration of a screw can be described by five coordinates.[42]​.

In general topology, the concept of dimension has been extended from natural numbers to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).[43]​ In algebraic geometry, the dimension of an algebraic variety has received several apparently different definitions, which are all equivalent in the most common cases.[44]​.

Symmetry

The topic of symmetry in geometry is almost as old as the science of geometry itself.[45] Symmetrical shapes such as the circle, regular polygons, and Platonic solids had deep meaning for many ancient philosophers[46] and were investigated in detail before the time of Euclid.[9] Symmetrical patterns occur in nature and were artistically represented in a multitude of forms, including the graphics of Leonardo da Vinci, MC Escher, and others.[47]​ In the second half of the century, the relationship between symmetry and geometry came under intense scrutiny.

Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed through the notion of a transformation group"), determines what geometry is.[48] Symmetry in classical Euclidean geometry is represented by congruencies and rigid motions, while in projective geometry an analogous role is played by hillions, geometric transformations that convert straight lines into straight lines.[49] However, it was in the new geometries by Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lieque Klein's idea of "defining a geometry through its symmetry group" found its inspiration.[50] Both discrete and continuous symmetries play a prominent role in geometry, the former in topology and the theory of geometric groups,[51][52]​ the latter in Lie theory") and the geometry of Riemann.[53][54]​.

A different type of symmetry is the duality principle "Duality (projective geometry)") in projective geometry, among other fields. This meta-phenomenon can be described roughly as follows: in any theorem, swap "point" with "plane", "join" with "meet", "meets" with "contains", and the result is an equally true theorem.[55]​ A similar and closely related form of duality exists between a vector space and its dual space.[56]​.

Axioms

In Euclidean geometry, axioms and postulates are propositions that relate concepts, defined in terms of the point, the line and the plane. Euclid proposed five postulates and it was the fifth (the postulate of parallelism) that centuries later – when many geometers questioned it when analyzing it – gave rise to new geometries: the elliptic (Riemann geometry) or the hyperbolic geometry of Nikolai Lobachevsky.

In analytical geometry, axioms are defined in terms of equations of points, based on mathematical analysis and algebra. Talking about points, lines or planes takes on another new meaning. You can define any function, whether it is a line, a circle, a plane, etc.

Euclid adopted an abstract approach to geometry in his Elements,[7]​ one of the most influential books ever written.[8]​ Euclid introduced certain axioms or postulates that express primary or self-evident properties of points, lines, and planes.[9]​ He proceeded to rigorously deduce other properties through mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.[10] At the beginning of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792-1856), János Bolyai (1802-1860), Carl Friedrich Gauss (1777-1855) and others[11]​ led to There was a resurgence of interest in this discipline, and in the 19th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern basis for geometry.[12]

Points

Points "Point (geometry)") are considered fundamental objects in Euclidean geometry. They have been defined in various ways, including Euclid's definition as "that which has no part" [13] and by the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be constructed from points. However, some geometry study has been carried out without reference to points.[15]​.

Lines

Euclid described a line as "length without width" that "lies equally with respect to points on itself."[13]​ In modern mathematics, given the multitude of geometries, the concept of line is closely related to the way geometry is described. For example, in analytical geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,[16]​ but in a more abstract setting, such as incidence geometry, a line can be an independent object, distinct from the set of points lying on it.[17]​ In differential geometry, a geodesic is a generalization of the notion of line to curved spaces.[18]​.

Plans

A plane is a two-dimensional flat surface that extends infinitely.[13] Planes are used in all areas of geometry. For example, planes can be studied as a topological surface without reference to distances or angles;[19]​ it can be studied as an affine space, where collinearity and proportions can be studied but not distances;[20]​ it can be studied as the complex plane using complex analysis techniques;[21]​ and so on.

Angles

Euclid defines a plane angle as the inclination towards each other, in a plane, of two lines that meet and are not straight to each other.[13]​ In modern terms, an angle is the figure formed by two rays of light, called sides of the angle, that share a common end point, called the vertex of the angle.[22]​.

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.[13]​ The study of the angles of a triangle or the angles in a unit circle forms the basis of trigonometry.[23]​.

In differential geometry and calculus, angles between plane curves or space curves or surfaces can be calculated using the derivative.[24][25]​.

Curves

A curve is a one-dimensional object that may or may not be straight (like a line); Curves in two-dimensional space are called plane curves and those in three-dimensional space are called space curves.[26]​.

In topology, a curve is defined by a function from an interval of the real numbers to another space.[19]​ In differential geometry, the same definition is used, but the defining function is required to be differentiable.[27]​ Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.[28]​.

Surfaces

A surface is a two-dimensional object, such as a sphere or a paraboloid.[29]​ In differential geometry[27]​ and topology,[19]​ surfaces are described by two-dimensional "patches" (or neighborhoods) that are assembled using diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.[28]​.

Varieties

A variety "Variety (mathematics)") is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where each point has a neighborhood that is homeomorphic to Euclidean space.[19]​ In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.[27]​.

Manifolds are widely used in physics, including general relativity and string theory.[30]​.

Length, area and volume

Length, area and volume describe the size or extent of an object in one dimension, two dimensions and three dimensions, respectively.[31]​.

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated using the Pythagorean theorem.[32]​.

Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or three-dimensional space.[31] Mathematicians have found many explicit formulas for the area and formulas for the volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral[33]​ or the Lebesgue integral.[34]​.

Metrics and measurements

The concept of length or distance can be generalized, giving rise to the idea of ​​metrics.[35] For example, the Euclidean metric measures the distance between points on the Euclidean plane, while the hyperbolic metric measures the distance on the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metric of general relativity.[36]​.

In another direction, the concepts of length, area and volume are expanded with measurement theory, which studies methods of assigning a size or measurement to sets, where measurements follow rules similar to those of classical area and volume.[37]​.

Congruence and similarity

Congruence and similarity are concepts that describe when two shapes have similar characteristics.[38]​ In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are equal in both size and shape.[39]​ Hilbert, in his work on creating a more rigorous basis for geometry, treated congruence as an indefinite term whose properties are defined by axioms.

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different types of transformations.[40]​.

Constructions with compass and ruler

Classical geometers paid special attention to the construction of geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass "Compass (instrument)") and the ruler. Furthermore, each build had to be completed in a finite number of steps. However, some problems proved difficult or impossible to solve by these means alone, and ingenious constructions were found using parabolas and other curves, as well as mechanical devices.

Dimension

Where traditional geometry allowed the dimensions of a line of a plane and our environmental world conceived as a three-dimensional space, mathematicians and physicists have used higher dimensions for almost two centuries.[41] An example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the degrees of freedom of the system. For example, the configuration of a screw can be described by five coordinates.[42]​.

In general topology, the concept of dimension has been extended from natural numbers to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).[43]​ In algebraic geometry, the dimension of an algebraic variety has received several apparently different definitions, which are all equivalent in the most common cases.[44]​.

Symmetry

The topic of symmetry in geometry is almost as old as the science of geometry itself.[45] Symmetrical shapes such as the circle, regular polygons, and Platonic solids had deep meaning for many ancient philosophers[46] and were investigated in detail before the time of Euclid.[9] Symmetrical patterns occur in nature and were artistically represented in a multitude of forms, including the graphics of Leonardo da Vinci, MC Escher, and others.[47]​ In the second half of the century, the relationship between symmetry and geometry came under intense scrutiny.

Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed through the notion of a transformation group"), determines what geometry is.[48] Symmetry in classical Euclidean geometry is represented by congruencies and rigid motions, while in projective geometry an analogous role is played by hillions, geometric transformations that convert straight lines into straight lines.[49] However, it was in the new geometries by Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lieque Klein's idea of "defining a geometry through its symmetry group" found its inspiration.[50] Both discrete and continuous symmetries play a prominent role in geometry, the former in topology and the theory of geometric groups,[51][52]​ the latter in Lie theory") and the geometry of Riemann.[53][54]​.

A different type of symmetry is the duality principle "Duality (projective geometry)") in projective geometry, among other fields. This meta-phenomenon can be described roughly as follows: in any theorem, swap "point" with "plane", "join" with "meet", "meets" with "contains", and the result is an equally true theorem.[55]​ A similar and closely related form of duality exists between a vector space and its dual space.[56]​.