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 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 curves, 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 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 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 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 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.[citation needed]
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.[citation needed]
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 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 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
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 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.
Notes
- ^ 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.
- ^ 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-”
- ^ 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.
See also
Lists
Related topics
External links
eml:Geometrîzh-classical:幾何nov:Geometria
pms:Geometrìa
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