From Wikipedia, the free encyclopedia
Geometry 

Oxyrhynchus papyrus (P.Oxy. I 29) showing fragment of Euclid's Elements 
History of geometry

This box: view • talk • edit

Geometry (
Ancient Greek:
γεωμετρία;
geo "earth",
metria "measurement") "
Earth
Measuring" is a part of
mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning
lengths,
areas, and
volumes, in the 3rd century BC 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. A mathematician who works in the field of geometry is called a geometer.
The introduction of
coordinates by
René Descartes and the concurrent development of
algebra marked 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
calculus in the 17th 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 19thcentury discovery of
nonEuclidean geometry, the concept of
space has 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 geometry and
general relativity. One of the youngest physical theories,
string theory, is also very geometric in flavor.
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.
^{[1]}
Overview
Recorded development of geometry spans more than two
millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.
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
lengths,
areas and
volumes, 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
axioms, or
postulates, expressing primary or selfevident 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 20th 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 those with
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 favor 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 geometry applies methods of algebra to geometric questions, typically by relating geometric
curves and algebraic
equations. These ideas played a key role in the development of
calculus in the 17th 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
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.
Leonhard Euler, in studying problems like the
Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry
geometria situs (geometry of place), but it is now known as
topology. Topology grew out of geometry, but turned into a large independent discipline. It 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
space remained essentially the same.
Immanuel Kant argued 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.
^{[2]} This dominant view was overturned by the revolutionary discovery of nonEuclidean 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 inauguration 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
The theme of
symmetry in 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 19th 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
congruences and rigid motions, whereas in
projective geometry an 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 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
manifolds 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
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
Shiingshen Chern in pursuing ideas introduced by
Élie Cartan. A pseudogroup can play the role of a Lie group of 'infinite' dimension.
Dimension
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of
spacetime 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
Algebraic geometry
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 GaussBonnet theorem.
Topology and geometry
Axiomatic and open development
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 19th century taken to be the prospective master geometric theory, is just one aspect of the general
representation theory of
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 algebrogeometric (Cartesian) sides.
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
nonstandard analysis is by means of
model theory.
History of geometry
Woman teaching geometry. Illustration at the beginning of a medieval translation of
Euclid's Elements, (c.
1310)
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 Elements (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
Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;
^{[3]} 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.
^{[citation needed]}
In the
Middle Ages,
mathematics in medieval Islam contributed to the development of geometry, especially
algebraic geometry^{[4]}^{[5]} and
geometric algebra.
^{[6]} AlMahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in
algebra.
^{[5]} Thābit ibn Qurra (known as Thebit in
Latin) (836901) dealt with
arithmetical operations applied to
ratios of geometrical quantities, and contributed to the development of
analytic geometry.
^{[7]} Omar Khayyám (10481131) found geometric solutions to
cubic equations, and his extensive studies of the
parallel postulate contributed to the development of
nonEuclidian geometry.
^{[8]} The theorems of
Ibn alHaytham (Alhazen), Omar Khayyam and
Nasir alDin alTusi on
quadrilaterals, including the
Lambert quadrilateral and
Saccheri quadrilateral, were the first theorems on
elliptical geometry and
hyperbolic geometry, and along with their alternative postulates, such as
Playfair's axiom, these works had a considerable influence on the development of nonEuclidean geometry among later European geometers, including
Witelo,
Levi ben Gerson,
Alfonso,
John Wallis, and
Giovanni Girolamo Saccheri.
^{[9]}
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.
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 spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same. Although various laws concerning lines and angles were known to the Egyptians and the Pythagoreans, the systematic treatment of geometry by the axiomatic method began with the Elements of Euclid. From a small number of explicit axioms, postulates, and definitions Euclid deduces theorems concerning the various figures of geometrical interest. Until the 19th century this work stood as a supreme example of the exercise of reason, which all other intellectual achievements ought to take as a model. With increasing standards of formal rigor it was recognized that Euclid does contain gaps, but fully formalized versions of his geometry have been provided. For example, in the axiomatization of David Hilbert, there are six primitive terms, in that of E. V. Huntington only two: ‘sphere’ and ‘includes’.
In the work of Kant, Euclidean geometry stands as the supreme example of a synthetic a priori construction, representing the way the mind has to think about space, because of the mind's own intrinsic structure. However, only shortly after Kant was writing nonEuclidean geometries were contemplated. They were foreshadowed by the mathematician K. F. Gauss (17771855), but the first serious nonEuclidean geometry is usually attributed to the Russian mathematician N. I. Lobachevsky, writing in the 1820s. Euclid's fifth axiom, the axiom of parallels, states that through any point not falling on a straight line, one straight line can be drawn that does not intersect the first. In Lobachevsky's geometry several such lines can exist. Later G. F. B. Riemann (182266) realized that the twodimensional geometry that would be hit upon by persons confined to the surface of a sphere would be different from that of persons living on a plane: for example, π would be smaller, since the diameter of a circle, as drawn on a sphere, is relatively large compared to the circumference. In the figure, BCB, the circumference of the circle, is less than 2π AB, where AB is the radius. Generalizing, Riemann reached the idea of a geometry in which there are no straight lines that do not intersect a given straight line, just as on a sphere all great circles (the shortest distance between two points) intersect.
The way then lay open to separating the question of the mathematical nature of a purely formal geometry from the question of its physical application. In 1854 Riemann showed that space of any curvature could be described by a set of numbers known as its metric tensor. For example, ten numbers suffice to describe the point of any fourdimensional manifold. To apply a geometry means finding coordinative definitions correlating the notions of the geometry, notably those of a straight line and an equal distance, with physical phenomena such as the path of a light ray, or the size of a rod at different times and places. The status of these definitions has been controversial, with some such as Poincaré seeing them simply as conventions, and others seeing them as important empirical truths. With the general rise of holism in the philosophy of science the question of status has abated a little, it being recognized simply that the coordination plays a fundamental role in physical science. See also relativity theory, spacetime
See also
Lists
Related topics
References
 ^ 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.
 ^ 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 nonEuclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, NonEuclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of nonEuclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
 ^ Boyer (1991). "Euclid of Alexandria". pp. 104. "The Elements was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all elementary mathematics"
 ^ R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, London
 ^ ^{a} ^{b} O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/HistTopics/Arabic_mathematics.html .
 ^ Boyer (1991). "The Arabic Hegemony". pp. 241–242. "Omar Khayyam (ca. 10501123), the "tentmaker," wrote an Algebra that went beyond that of alKhwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all thirddegree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""
 ^ O'Connor, John J.; Robertson, Edmund F., "AlSabi Thabit ibn Qurra alHarrani", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Thabit.html .
 ^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Khayyam.html .
 ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447494 [470], Routledge, London and New York:
"Three scientists, Ibn alHaytham, Khayyam and alTusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European counterparts. The first European attempt to prove the postulate on parallel lines  made by Witelo, the Polish scientists of the 13th century, while revising Ibn alHaytham's
Book of Optics (
Kitab alManazir)  was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the abovementioned Alfonso from Spain directly border on Ibn alHaytham's demonstration. Above, we have demonstrated that
PseudoTusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
External links