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]] Geometry ( Greek ''γεωμετρία''; geo = earth, metria = measure) is a part of Mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning Length s, Area s, and Volume s, in the third century B.C. geometry was put into an Axiomatic Form by Euclid , whose treatment set a standard for many centuries to follow. Astronomy served as an important source of geometric problems during the next one and a half millennia. Introduction of Coordinates by Descartes and the concurrent development of Algebra marked a new stage for geometry, since geometric figures, such as Plane Curve s, could now be represented Analytically . This played a key role in the emergence of Calculus in the seventeenth century. Furthermore, the theory of Perspective showed that there is more to geometry than just the metric properties of figures. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of Topology and Differential Geometry . Since the nineteenth century discovery of Non-Euclidean Geometry , the concept of Space has undergone a spectacular transformation. Contemporary geometry considers Manifold s, spaces that are considerably more abstract than the familiar Euclidean Space , which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with Physics , exemplified by the ties between Riemannian Geometry and General Relativity . One of the youngest physical theories, String Theory , is also very geometric in flavour. The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as Algebra or Number Theory . However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, in Fractal Geometry , and especially in Algebraic Geometry .It is quite common in algebraic geometry to speak about ''geometry of Algebraic Varieties over Finite Field s'', possibly Singular . From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary Sphere s or Cone s. HISTORY OF GEOMETRY See Also: History of geometry , (c. 1310 )]] The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia , Egypt , and the Indus Valley from around 3000 BC . Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in Surveying , Construction , Astronomy , and various crafts. The earliest known texts on geometry are the Egyptian '' Rhind Papyrus '' and '' Moscow Papyrus '', the Babylonian Clay Tablets , and the Indian '' Shulba Sutras '', while the Chinese had the work of Mozi , Zhang Heng , and the '' Nine Chapters On The Mathematical Art '', edited by Liu Hui . 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 Axiom atic form, which came to be known as Euclidean Geometry . The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;1 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 Middle Ages , Muslim Mathematicians contributed to the development of geometry, especially Algebraic Geometry and Geometric Algebra . Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in Algebra . Thābit Ibn Qurra (known as Thebit in Latin ) (836-901) dealt with Arithmetic al operations applied to Ratio s of geometrical quantities, and contributed to the development of Analytic Geometry . Omar Khayyám (1048-1131) found geometric solutions to Cubic Equation s, and his extensive studies of the Parallel Postulate contributed to the development of Non-Euclidian Geometry . 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 René 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. Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of Non-Euclidean Geometries by Lobachevsky , Bolyai and Gauss and of the formulation of Symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). 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 System s. 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 recognized as that of Homogeneous Space s, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same. WHAT IS GEOMETRY? Recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages. The geometric paradigms presented below should be viewed as ' Pictures At An Exhibition ' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes. Practical geometry There is little doubt that geometry originated as a ''practical'' science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for Length s, Area s and Volume s, such as Pythagorean Theorem , Circumference and Area of a circle, area of a Triangle , volume of a Cylinder , Sphere , and a Pyramid . Development of Astronomy led to emergence of Trigonometry and Spherical Trigonometry , together with the attendant computational techniques. Axiomatic geometry A method of computing certain inaccessible distances or heights based on Similarity of geometric figures and attributed to Thales presaged more abstract approach to geometry taken by Euclid in his Elements , one of the most influential books ever written. Euclid introduced certain Axiom s or Postulate s, 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 rigour. In the twentieth century, David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry. Geometric constructions Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the Compass And Straightedge . However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found. The approach to geometric problems with geometric or mechanical means is known as Synthetic Geometry . Numbers in geometry Already Pythagoreans considered the role of numbers in geometry. However, the discovery of Incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of Coordinate s by Descartes , who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic Geometry applies methods of algebra to geometric questions, typically by relating geometric Curve s and algebraic Equation s. These ideas played a key role in the development of Calculus in the seventeenth century and led to discovery of many new properties of plane curves. Modern Algebraic Geometry considers similar questions on a vastly more abstract level. Geometry of position Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of )? What is the densest Packing Of Spheres of equal size in space ( Kepler Conjecture )? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective , Convex and Discrete geometry are three subdisciplines within present day geometry that deal with these and related questions. A new chapter in ''Geometria situs'' was opened by Leonhard Euler , who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape. Topology , which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of Hyperbolic Knot s. Geometry beyond Euclid For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of .Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) ''possibility'' of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact ''predicted'' the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai , and Lobachevsky , who demonstrated that ordinary Euclidean Space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his inaugurational lecture ''Über die Hypothesen, welche der Geometrie zu Grunde liegen'' (''On the hypotheses on which geometry is based''), published only after his death. Riemann's new idea of space proved crucial in Einstein 's General Relativity Theory and Riemannian Geometry , which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry. Symmetry of the Hyperbolic Plane ]] The theme of Symmetry in geometry is nearly as old as the science of geometry itself. The Circle , Regular Polygon s and Platonic Solid s held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher . Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized. 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 Congruence s and rigid motions, whereas in Projective Geometry an analogous role is played by Collineation s, geometric transformations that take straight lines 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 ' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in Topology and Geometric Group Theory , the latter in Lie Theory and Riemannian Geometry . Modern geometry ''Modern geometry'' is the title of a popular textbook by Dubrovin, Novikov , and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on Manifold s and their applications in contemporary Theoretical Physics . A quarter century after its publication, Differential Geometry , Algebraic Geometry , Symplectic Geometry , and Lie Theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics. CONTEMPORARY GEOMETERS Some of the representative leading figures in modern geometry are for a survey.) Much of this theory relates to the theory of ''continuous symmetry'', or in other words Lie Group s. 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 Manifold s 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 See Also: Euclidean geometry The study of traditional Euclidean Geometry is by no means dead. It is now typically presented as the geometry of Euclidean Space s of any dimension, and of the Euclidean Group of Rigid Motion s. 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 , Convex Geometry , 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 Group s and Polytope s. Geometric Group Theory is an expanding area of the theory of more general Discrete Group s, 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 Manifold s that can be described by Algebraic Equation s; or the Scheme Theory provides a technically sophisticated theory based on general Commutative Ring s. The geometric style which was traditionally called the Italian School is now known as Birational Geometry . It has made progress in the fields of Threefold s, Singularity Theory and Moduli Space s, 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 Bundle s. Fundamental to this approach is the connection between curvature and Characteristic Class es, as exemplified by the Generalized Gauss-Bonnet Theorem . TOPOLOGY AND GEOMETRY ]] The field of Topology , which saw massive development in the 20th century, is in a technical sense a type of Transformation Geometry , in which transformations are Homeomorphism s. 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 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 Group s 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 Field s, the classical groups give rise to Finite Group s, intensively studied in relation to the Finite Simple Group s; 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 s. 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 Algebra s, 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 is an approach to infinitesimals from the side of Categorical Logic , as Non-standard Analysis is by means of Model Theory . NOTES SEE ALSO Lists
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