Geometry
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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
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.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 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 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 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.
(1)History
missing image!
- Woman teaching geometry.jpg -
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 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
Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;
(2) 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}}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
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 postulate contributed 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
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 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 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.
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 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 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
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 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
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
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.
(3) 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
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 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
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
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,
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 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
missing image!
- Trefoil knot arb.png -
120 px|A thickening of the trefoil knot
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 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
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 theory of
Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of
sheaf theory on
complex manifolds. 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.
See also
{{sisterlinks|Geometry}}
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References
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[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.]
-
[BOOK, Boyer, Carl Benjamin Boyer, 1991, Euclid of Alexandria, 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-, ]
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[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.]
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