Geometry (Greek "γεωμετρία"; geo = earth, metria = measure) is a part of
mathematicsconcerned 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 lengths, areas, and volumes, in the third century B.C., geometry was put into an axiomatic form by Euclid, whose treatment - Euclidean geometry- set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia.
coordinatesby René Descartesand the concurrent development of algebramarked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of calculusin 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 Eulerand Gauss and led to the creation of topologyand differential geometry.
Since the nineteenth century discovery of
non-Euclidean geometry, the concept of spacehas undergone a radical transformation. Contemporary geometry considers manifolds, 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 geometryand 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
algebraor 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 fields", 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 spheres or cones.]
The earliest recorded beginnings of geometry can be traced to ancient
Mesopotamia, Egypt, and the Indus Valley from around 3000 BCE. 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 axiomatic form, which came to be known as Euclidean geometry. The treatise is not, as is sometimes thought, a compendium of all that Hellenisticmathematicians knew about geometry at that time; rather, it is an elementary introduction to it; [cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|pages=104|quote=The "Elements" was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all "elementary" mathematics-] 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.Fact|date=July 2007
Middle Ages, Muslim mathematicians contributed to the development of geometry, especially algebraic geometryand 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 arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám(1048-1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulatecontributed to the development of Non-Euclidian geometry.Fact|date=July 2007
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 calculusand a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometryby 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
symmetryas the central consideration in the Erlangen Programmeof 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 topologyand 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 analysisand classical mechanics. The traditional type of geometry was recognized 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.
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.
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
lengths, areas and volumes, such as Pythagorean theorem, circumferenceand area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. Development of astronomyled to emergence of trigonometryand spherical trigonometry, together with the attendant computational techniques.
A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to
Thalespresaged more abstract approach to geometry taken by Euclidin his Elements, one of the most influential books ever written. Euclid introduced certain axioms, 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. In the twentieth century, David Hilbertemployed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry.
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
Pythagoreansconsidered 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 coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic geometryapplies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculusin the seventeenth century and led to discovery of many new properties of plane curves. Modern algebraic geometryconsiders 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
polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius ( kissing number problem)? 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 knots.
Geometry beyond Euclid
For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of
spaceremained essentially the same. Immanuel Kantargued that there is only one, "absolute", geometry, which is known to be true "a priori" by an inner faculty of mind: Euclidean geometry was synthetic a priori. [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 spaceis only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemannin 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 theoryand Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
The theme of
symmetryin geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids 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 programproclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry "is". Symmetry in classical Euclidean geometryis represented by congruences and rigid motions, whereas in projective geometryan analogous role is played by collineations, 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 Liethat 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 topologyand geometric group theory, the latter in Lie theoryand Riemannian 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
manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry, and Lie theorypresented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.
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 covariancethat lies at the root of general relativitytheory in theoretical physics. (See 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 Chernin pursuing ideas introduced by Élie Cartan. A pseudogroup can play the role of a Lie group of "infinite" 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 dimensionsfor 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 theoryis 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-timeare 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 geometryis by no means dead. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean groupof 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, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallographyand the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theoryis an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
The field of
algebraic geometryis the modern incarnation of the Cartesian geometryof 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 theoryprovides 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 theoryand 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 theoryand other parts of homological algebra. The Hodge conjectureis an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications, Gröbner basistheory and real algebraic geometryare major subfields.
Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physicssince 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 20th 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 topologyand differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topologyand general topologyhave 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 Steinerbeing a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axiomsand regarded as of important pedagogic value, most contemporary geometry is a matter of style. Computational synthetic geometryis 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 subgroupconcept) arranges theories according to generalization and specialization. For example affine geometryis 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 theoryof algebraic groups and 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 theoryof Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theoryon complex manifolds. In contrast, the non-commutative geometryof Alain Connesis 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 theorywhere the commutative lawof 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 Grassmannwas for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physicssuch 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 algebrabecame 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 geometryis an approach to infinitesimals from the side of categorical logic, as non-standard analysisis by means of model theory.
List of basic geometry topics
List of geometry topics
List of geometers
* Important publications in geometry
List of mathematics articles
Interactive geometry software
Flatland", a book written by Edwin Abbott Abbottabout two and three-dimensional space, to understand the concept of four dimensions
Why 10 dimensions?
* [http://www.mathforum.org/library/topics/geometry/ "The Math Forum" — Geometry]
** [http://www.mathforum.org/geometry/k12.geometry.html "The Math Forum" — K–12 Geometry]
** [http://www.mathforum.org/geometry/coll.geometry.html "The Math Forum" — College Geometry]
** [http://www.mathforum.org/advanced/geom.html "The Math Forum" — Advanced Geometry]
* [http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html "The Mathematical Atlas" — Geometric Areas of Mathematics]
* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "4000 Years of Geometry"] , lecture by Robin Wilson given at
Gresham College, 3rd October 2007 (available for MP3 and MP4 download as well as a text file)
* [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at
* [http://www.cut-the-knot.org/geometry.shtml Geometry] at
* [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
* [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
* Stanford Encyclopedia of Philosophy:
** [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
** [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
* [http://www.egwald.ca/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens
* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics : forgotten brilliance?]
* [http://www.ics.uci.edu/~eppstein/junkyard/topic.html The Geometry Junkyard]
* [http://mrperezonlinemathtutor.com/A_Geometry.html Geometry lessons in PowerPoint]
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