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Geometry

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Table of Geometry, from the 1728 Cyclopaedia.
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   Table of Geometry, from the 1728 Cyclopaedia.

   Geometry ( Greek γεωμετρία; geo = earth, metria = measure) arose as the
   field of knowledge dealing with spatial relationships. Geometry was one
   of the two fields of pre-modern mathematics, the other being the study
   of numbers.

   In modern times, geometric concepts have been extended. They sometimes
   show a high level of abstraction and complexity. Geometry now uses
   methods of calculus and abstract algebra, so that many modern branches
   of the field are not easily recognizable as the descendants of early
   geometry. (See areas of mathematics.)

History of geometry

   Woman teaching geometry. Illustration at the beginning of a medieval
   translation of Euclid's Elements, (c.1310)
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   Woman teaching geometry. Illustration at the beginning of a medieval
   translation of Euclid's Elements, (c. 1310)

   Euclid's The Elements of Geometry (c. 300 BCE), was one of the most
   important early texts on geometry, in which he presented geometry in an
   ideal axiomatic form, which came to be known as Euclidean geometry. The
   treatise is not a compendium of all that the Hellenistic mathematicians
   knew at the time about geometry; Euclid himself wrote eight more
   advanced books on geometry. We know from other references that Euclid’s
   was not the first elementary geometry textbook, but the others fell
   into disuse and were lost.

   In the early 17th century, there were two important developments in
   geometry. The first and most important was the creation of analytic
   geometry, or geometry with coordinates and equations, by Rene Descartes
   (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary
   precursor to the development of calculus and a precise quantitative
   science of physics. The second geometric development of this period was
   the systematic study of projective geometry by Girard Desargues
   (1591–1661). Projective geometry is the study of geometry without
   measurement, just the study of how points align with each other.

   Geometry is still feeling the effects of two developments from the
   nineteenth century. These were the discovery of non-Euclidean geometry,
   and the formulation of symmetry as the central consideration in the
   Erlangen Programme of Felix Klein. Two of the master geometers of the
   time were Bernhard Riemann, working primarily with tools from
   mathematical analysis, and introducing the Riemann surface, and Henri
   Poincaré, the founder of algebraic topology and the geometric theory of
   dynamical systems.

   As a consequence of these major changes in the conception of geometry,
   the concept of 'space' became something rich and varied, and the
   natural background for theories as different as complex analysis and
   classical mechanics. The traditional type of geometry was recognised as
   that of homogeneous spaces, those spaces which have a sufficient supply
   of symmetry, so that from point to point they look just the same.

Contemporary geometers

   Some of the representative leading figures in modern geometry are
   Michael Atiyah, Mikhail Gromov, and William Thurston. The common
   feature in their work is the use of smooth manifolds as the basic idea
   of space; they otherwise have rather different directions and
   interests. Geometry now is, in large part, the study of structures on
   manifolds, that have a geometric meaning in the sense of the principle
   of covariance that lies at the root of general relativity theory, in
   theoretical physics. (See Category:Structures on manifolds for a
   survey.)

   Much of this theory relates to the theory of continuous symmetry, or in
   other words Lie groups. From the foundational point of view, on
   manifolds and their geometrical structures, important is the concept of
   pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas
   introduced by Élie Cartan. A pseudogroup can play the role of a Lie
   group of infinite dimension.

Dimension

   Where the traditional geometry allowed dimensions 1 (a line), 2 (a
   plane) and 3 (our ambient world conceived of as three-dimensional
   space), mathematicians have used higher dimensions for nearly two
   centuries. Dimension has gone through stages of being any natural
   number n, possibly infinite with the introduction of Hilbert space, and
   any positive real number in fractal geometry. Dimension theory is a
   technical area, initially within general topology, that discusses
   definitions; in common with most mathematical ideas, dimension is now
   defined rather than an intuition. Connected topological manifolds have
   a well-defined dimension; this is a theorem ( invariance of domain)
   rather than anything a priori.

   The issue of dimension still matters to geometry, in the absence of
   complete answers to classic questions. Dimensions 3 of space and 4 of
   space-time are special cases in geometric topology. Dimension 10 or 11
   is a key number in string theory. Exactly why is something to which
   research may bring a satisfactory geometric answer.

Contemporary Euclidean geometry

   The study of traditional Euclidean geometry is by no means dead. It is
   now typically presented as the geometry of Euclidean spaces of any
   dimension, and of the Euclidean group of rigid motions. The fundamental
   formulae of geometry, such as the Pythagorean theorem, can be presented
   in this way for a general inner product space.

   Euclidean geometry has become closely connected with computational
   geometry, computer graphics, discrete geometry, and some areas of
   combinatorics. Momentum was given to further work on Euclidean geometry
   and the Euclidean groups by crystallography and the work of H. S. M.
   Coxeter, and can be seen in theories of Coxeter groups and polytopes.
   Geometric group theory is an expanding area of the theory of more
   general discrete groups, drawing on geometric models and algebraic
   techniques.

Algebraic geometry

   The field of algebraic geometry is the modern incarnation of the
   Cartesian geometry of co-ordinates. After a turbulent period of
   axiomatization, its foundations are in the twenty-first century on a
   stable basis. Either one studies the 'classical' case where the spaces
   are complex manifolds that can be described by algebraic equations; or
   the scheme theory provides a technically sophisticated theory based on
   general commutative rings.

   The geometric style which was traditionally called the Italian school
   is now known as birational geometry. It has made progress in the fields
   of threefolds, singularity theory and moduli spaces, as well as
   recovering and correcting the bulk of the older results. Objects from
   algebraic geometry are now commonly applied in string theory, as well
   as diophantine geometry.

   Methods of algebraic geometry rely heavily on sheaf theory and other
   parts of homological algebra. The Hodge conjecture is an open problem
   that has gradually taken its place as one of the major questions for
   mathematicians. For practical applications, Gröbner basis theory and
   real algebraic geometry are major subfields.

Differential geometry

   Differential geometry, which in simple terms is the geometry of
   curvature, has been of increasing importance to mathematical physics
   since the suggestion that space is not flat space. Contemporary
   differential geometry is intrinsic, meaning that space is a manifold
   and structure is given by a Riemannian metric, or analogue, locally
   determining a geometry that is variable from point to point.

   This approach contrasts with the extrinsic point of view, where
   curvature means the way a space bends within a larger space. The idea
   of 'larger' spaces is discarded, and instead manifolds carry vector
   bundles. Fundamental to this approach is the connection between
   curvature and characteristic classes, as exemplified by the generalized
   Gauss-Bonnet theorem.

Topology and geometry

   The field of topology, which saw massive development in the twentieth
   century, is in a technical sense a type of transformation geometry, in
   which transformations are homeomorphisms. This has often been expressed
   in the form of the dictum 'topology is rubber-sheet geometry'.
   Contemporary geometric topology and differential topology, and
   particular subfields such as Morse theory, would be counted by most
   mathematicians as part of geometry. Algebraic topology and general
   topology have gone their own ways.

Axiomatic and open development

   The model of Euclid's Elements, a connected development of geometry as
   an axiomatic system, is in a tension with René Descartes's reduction of
   geometry to algebra by means of a coordinate system. There were many
   champions of synthetic geometry, Euclid-style development of projective
   geometry, in the nineteenth century, Jakob Steiner being a particularly
   brilliant figure. In contrast to such approaches to geometry as a
   closed system, culminating in Hilbert's axioms and regarded as of
   important pedagogic value, most contemporary geometry is a matter of
   style. Computational synthetic geometry is now a branch of computer
   algebra.

   The Cartesian approach currently predominates, with geometric questions
   being tackled by tools from other parts of mathematics, and geometric
   theories being quite open and integrated. This is to be seen in the
   context of the axiomatization of the whole of pure mathematics, which
   went on in the period c.1900–c.1950: in principle all methods are on a
   common axiomatic footing. This reductive approach has had several
   effects. There is a taxonomic trend, which following Klein and his
   Erlangen program (a taxonomy based on the subgroup concept) arranges
   theories according to generalization and specialization. For example
   affine geometry is more general than Euclidean geometry, and more
   special than projective geometry. The whole theory of classical groups
   thereby becomes an aspect of geometry. Their invariant theory, at one
   point in the nineteenth century taken to be the prospective master
   geometric theory, is just one aspect of the general representation
   theory of Lie groups. Using finite fields, the classical groups give
   rise to finite groups, intensively studied in relation to the finite
   simple groups; and associated finite geometry, which has both
   combinatorial (synthetic) and algebro-geometric (Cartesian) sides.

   An example from recent decades is the twistor theory of Roger Penrose,
   initially an intuitive and synthetic theory, then subsequently shown to
   be an aspect of sheaf theory on complex manifold. In contrast, the
   non-commutative geometry of Alain Connes is a conscious use of
   geometric language to express phenomena of the theory of von Neumann
   algebras, and to extend geometry into the domain of ring theory where
   the commutative law of multiplication is not assumed.

   Another consequence of the contemporary approach, attributable in large
   measure to the Procrustean bed represented by Bourbakiste
   axiomatization trying to complete the work of David Hilbert, is to
   create winners and losers. The Ausdehnungslehre (calculus of extension)
   of Hermann Grassmann was for many years a mathematical backwater,
   competing in three dimensions against other popular theories in the
   area of mathematical physics such as those derived from quaternions. In
   the shape of general exterior algebra, it became a beneficiary of the
   Bourbaki presentation of multilinear algebra, and from 1950 onwards has
   been ubiquitous. In much the same way, Clifford algebra became popular,
   helped by a 1957 book Geometric Algebra by Emil Artin. The history of
   'lost' geometric methods, for example infinitely near points, which
   were dropped since they did not well fit into the pure mathematical
   world post- Principia Mathematica, is yet unwritten. The situation is
   analogous to the expulsion of infinitesimals from differential
   calculus. As in that case, the concepts may be recovered by fresh
   approaches and definitions. Those may not be unique: synthetic
   differential geometry is an approach to infinitesimals from the side of
   categorical logic, as non-standard analysis is by means of model
   theory.

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