Geometry

Geometry from arithmetic, one of a oldest branches of mathematics. this is the concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who workings in the field of geometry is called a geometer.

Until the 19th century, geometry was most exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, as alive as curve, as necessary concepts.

During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such(a) discoveries is Gauss' Theorema Egregium "remarkable theorem" that asserts roughly that the Gaussian curvature of a surface is freelancer from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.

Later in the 19th century, it appeared that geometries without the parallel postulate non-Euclidean geometries can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous applications of non-Euclidean geometry.

Since then, the scope of geometry has been greatly expanded, and the field has been split in numerous subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry also so-called as combinatorial geometry, etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

Originally developed to model the physical world, geometry has application in most all Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

Main concepts

The coming after or as a total of. are some of the most important picture in geometry.

Elements, one of the most influential books ever written. Euclid introducedaxioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be requested as axiomatic or synthetic geometry. At the start 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 led to a revival of interest in this discipline, and in the 20th century, David Hilbert 1862–1943 employed axiomatic reasoning in an attempt to afford a innovative foundation of geometry.

Points are generally considered fundamental objects for building geometry. They may be defined by the properties that thay must have, as in Euclid's definition as "that which has no part", or in synthetic geometry. In modern mathematics, they are loosely defined as elements of a set called space, which is itself axiomatically defined.

With these modern definitions, every geometric vintage is defined as a set of points; this is non the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.

However, there has modern geometries, in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

Euclid subjected a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, precondition the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such(a) as incidence geometry, a line may be an self-employed person object, distinct from the set of points which lie on it. In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without quotation to distances or angles; it can be studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis; and so on.

Euclid defines a plane angle as the inclination to regarded and identified separately. other, in a plane, of two an arrangement of parts or elements in a specific work figure or combination. which meet regarded and described separately. other, and create not lie straight with respect to regarded and identified separately. other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

In Euclidean geometry, angles are used to examine polygons and triangles, as alive as forming an object of explore in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.

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

A curve is a 1-dimensional object that may be straight like a line or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves.

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

A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology, surfaces are refers by two-dimensional 'patches' or neighborhoods that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.

A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every an essential or characteristic element of something abstract. has a neighborhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.

Manifolds are used extensively in physics, including in general relativity and string theory.

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

In Euclidean geometry and analytic geometry, the length of a line an essential or characteristic part of something abstract. can often be calculated by the Pythagorean theorem.

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 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral or the Lebesgue integral.

The concept of length or distance can be generalized, main to the idea of metrics. For instance, the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics put the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.

In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where the measures follow rules similar to those of classical area and volume.

Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined 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 kinds of transformations.

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the compass and straightedge. Also, every construction had to be set up in a finite number of steps. However, some problems turned out to be unoriented or impossible to solve by these means alone, and ingenious constructions using neusis, parabolas and other curves, or mechanical devices, were found.

Where the traditional geometry authorises dimensions 1 a line, 2 a plane and 3 our ambient world conceived of as three-dimensional space, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical usage for higher dimensions is the configuration space of a physical system, which has a dimension make up to the system's degrees of freedom. For instance, the positioning of a screw can be described by five coordinates.

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. In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are any equivalent in the most common cases.

The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail previously the time of Euclid. Symmetric patterns arise in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci, M. C. Escher, and others. In thehalf of the 19th 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 via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight formation into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration. Both discrete and continual symmetries play prominent roles in geometry, the former in topology and geometric multinational theory, the latter in Lie theory and Riemannian geometry.