GEOMETRY (from the
While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, and curves, as well as the more advanced notions of manifolds and topology or metric.
CONTENTS * 1 Overview * 2 History * 3 Important concepts in geometry * 3.1 Axioms
* 3.2 Points
* 3.3 Lines
* 3.4 Planes
* 3.5 Angles
* 3.6 Curves
* 3.7 Surfaces
* 3.8 Manifolds
* 3.9 Topologies and metrics
* 3.10
* 4 Contemporary geometry * 4.1
* 5 Applications * 5.1 Art
* 5.2
* 6 See also * 6.1 Lists * 6.2 Related topics * 6.3 Other fields * 7 Notes * 8 Sources * 9 Further reading * 10 External links OVERVIEW Contemporary geometry has many subfields: *
HISTORY Main article:
The earliest recorded beginnings of geometry can be traced to ancient
In the 7th century BC, the Greek mathematician
Indian mathematicians also made many important contributions in
geometry. The _
In the
In the early 17th century, there were two important developments in
geometry. The first was the creation of analytic geometry, or geometry
with coordinates and equations , by
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 Nikolai Ivanovich Lobachevsky,
IMPORTANT CONCEPTS IN GEOMETRY The following are some of the most important concepts in geometry. AXIOMS An illustration of Euclid's parallel postulate See also:
POINTS Main article:
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part' and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points. LINES Main article:
PLANES Main article:
A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis ; and so on. ANGLES Main article:
In
In differential geometry and calculus , the angles between plane curves or space curves or surfaces can be calculated using the derivative . CURVES Main article:
A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the
real numbers to another space. In differential geometry, the same
definition is used, but the defining function is required to be
differentiable
SURFACES Main article:
A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . MANIFOLDS Main article:
A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory TOPOLOGIES AND METRICS Main article:
A topology is a mathematical structure on a set that tells how
elements of the set relate spatially to each other. The best-known
examples of topologies come from metrics , which are ways of measuring
distances between points. For instance, the
COMPASS AND STRAIGHTEDGE CONSTRUCTIONS Main article:
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. 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. DIMENSION Main article:
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.
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 .
SYMMETRY Main article:
The theme of symmetry in geometry is nearly as old as the science of
geometry itself. Symmetric shapes such as the circle , regular
polygons and platonic solids held deep significance for many ancient
philosophers and were investigated in detail before the time of
Euclid. Symmetric patterns occur in nature and were artistically
rendered in a multitude of forms, including the 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 was
recognized.
A different type of symmetry is the principle of duality in
projective geometry (see
NON-EUCLIDEAN GEOMETRY
In the nearly two thousand years since Euclid, while the range of
geometrical questions asked and answered inevitably expanded, the
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-
CONTEMPORARY GEOMETRY EUCLIDEAN GEOMETRY
DIFFERENTIAL GEOMETRY
TOPOLOGY AND GEOMETRY 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
ALGEBRAIC GEOMETRY Quintic Calabi–Yau threefold The field of algebraic geometry is the modern incarnation of the
The study of low-dimensional algebraic varieties, algebraic curves ,
algebraic surfaces and algebraic varieties of dimension 3 ("algebraic
threefolds"), has been far advanced.
APPLICATIONS
ART Main article:
ARCHITECTURE Main articles:
PHYSICS Main article:
The field of astronomy , especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history. Modern geometry has many ties to physics as is exemplified by the
links between pseudo-
OTHER FIELDS OF MATHEMATICS
An important area of application is number theory . In ancient Greece
the
While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or number theory , geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry ).
SEE ALSO *
LISTS * Category:Algebraic geometers * Category:Differential geometers * Category:Geometers * Category:Topologists *
RELATED TOPICS *
OTHER FIELDS NOTES * ^ _A_ _B_ (Boyer 1991 , "Ionia and the Pythagoreans" p. 43)
* ^ Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews
(1998). _
* ^ 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 ,
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the
most considerable contribution to this branch of geometry whose
importance came to be completely recognized only in the 19th century.
In essence, their propositions concerning the properties of
quadrangles which they considered, assuming that some of the angles of
these figures were acute of obtuse, embodied the first few theorems of
the hyperbolic and the elliptic geometries. Their other proposals
showed that various geometric statements were equivalent to the
Euclidean postulate V. It is extremely important that these scholars
established the mutual connection between this postulate and the sum
of the angles of a triangle and a quadrangle. By their works on the
theory of parallel lines
SOURCES * Boyer, C. B. (1991) . _A History of Mathematics_ (Second edition, revised by Uta C. Merzbach ed.). New York: Wiley. ISBN 0-471-54397-7 . * Nikolai I. Lobachevsky, _Pangeometry_, translator and editor: A.
Papadopoulos, Heritage of European
FURTHER READING *
EXTERNAL LINKS Find more aboutGEOMETRYat's sister projects * _Definitions from Wiktionary * |