Geometry ( ;
geo = earth,
metria
= measure) is a part of
mathematics
concerned with questions of size, shape, 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 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.
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 19th-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.
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 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 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 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 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
sub-disciplines 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.
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 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
congruence 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
pseudo-group 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
stable in the 21st century. 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
threefold,
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
120 px
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 19th
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 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 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.
History of geometry
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;
Euclid himself wrote eight more advanced books on geometry. We know
from other references that Euclid’s was not the first elementary
geometry textbook, but the others fell into disuse and were
lost.
In the
Middle Ages,
mathematics in medieval Islam
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. The theorems
of
Ibn al-Haytham (Alhazen), Omar
Khayyam and
Nasir al-Din
al-Tusi 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 non-Euclidean geometry
among later European geometers, including
Witelo,
Levi ben
Gerson,
Alfonso,
John Wallis, and
Giovanni Girolamo Saccheri.
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 19th 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.
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 cone.
- 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.
- R. Rashed (1994), The development of Arabic mathematics:
between arithmetic and algebra, London
- Boris A. Rosenfeld and Adolf P. Youschkevitch (1996),
"Geometry", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 2, p. 447-494 [470],
Routledge, London
and New York:
External links