# Dictionary Definition

geometrical adj

1 of or relating to or determined by geometry
[syn: geometric]

2 characterized by simple geometric forms in
design and decoration; "a buffalo hide painted with red and black
geometric designs" [syn: geometric]

# User Contributed Dictionary

## English

### Adjective

# Extensive Definition

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.

## History

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; 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,
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.

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. 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.

### Related topics

wikisource Flatland- Flatland, a book written by Edwin Abbott Abbott about two and three-dimensional space, to understand the concept of four dimensions
- Why 10 dimensions?

## Notes

## External links

- The Math Forum — Geometry
- The Mathematical Atlas — Geometric Areas of Mathematics
- "4000 Years of Geometry", lecture by Robin Wilson given at Gresham College, 3rd October 2007 (available for MP3 and MP4 download as well as a text file)
- What Is Geometry? at cut-the-knot
- Geometry at cut-the-knot
- Geometry Step by Step from the Land of the Incas by Antonio Gutierrez.
- Islamic Geometry
- Stanford Encyclopedia of Philosophy:
- Online Interactive Geometric Objects by Elmer G. Wiens
- Arabic mathematics : forgotten brilliance?
- The Geometry Junkyard
- Geometry lessons in PowerPoint All lessons introduce mathematical concepts, step by step, with animations of text, points, lines and figures in general. Solution of problems is also given step by step. Colors are used to give hints and clues to follow the concept or the solution of the problems.

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Auxiliary Language Association): Geometria

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