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| '''Geometry''' (from the Ancient Greek: ''γεωμετρία''; geo- "earth", -metron "[[measurement]]") is a branch of [[mathematics (nonfiction)]] concerned with questions of [[shape (nonfiction)]], [[size (nonfiction)]], [[volume (nonfiction)]], [[relative position (nonfiction)]] of [[Figure (nonfiction)|figures (nonfiction)]], and the properties of [[space (nonfiction)]]. | | [[File:Woman_teaching_geometry_circa_1309.jpg|thumb|]]'''Geometry''' (from the Ancient Greek: ''γεωμετρία''; ''geo-'' "earth", ''-metron'' "[[measurement]]") is a branch of [[mathematics (nonfiction)|mathematics]] concerned with questions of shape, size, volume, relative position of figures, and the properties of space. |
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| == History == | | == History == |
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| Geometry arose independently in a number of early cultures as a body of practical knowledge concerning [[Length (nonfiction)|lengths (nonfiction)]], [[Area (nonfiction)|areas (nonfiction)]], and [[Volume (nonfiction)|volumes (nonfiction)]], with elements of formal mathematical science emerging in the West as early as [[Thales (nonfiction)]] (6th century BC). | | Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as [[Thales]] (6th century BC). |
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| == Euclid == | | == Euclid == |
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| By the 3rd century BC, geometry was put into an axiomatic form by [[Euclid (nonfiction)]], whose treatment -- [[Euclidean geometry (nonfiction)]] -- set a standard for many centuries to follow. | | By the 3rd century BC, geometry was put into an axiomatic form by [[Euclid (nonfiction)]], whose treatment -- Euclidean geometry-- set a standard for many centuries to follow. |
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| == Archimedes == | | == Archimedes == |
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| [[Archimedes (nonfiction)]] developed [[ingenious techniques for calculating areas and volumes]], in many ways anticipating modern [[integral calculus (nonfiction)]]. | | [[Archimedes (nonfiction)]] developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. |
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| == Astronomy ==
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| The field of [[astronomy (nonfiction)]], especially as it relates to mapping the positions of stars and planets on the [[celestial sphere (nonfiction)]] and describing the relationship between [[movements of celestial bodies (nonfiction)]], served as an important source of geometric problems during the next one and a half millennia.
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| == Quadrivium == | | == Quadrivium == |
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| In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master. | | In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master. |
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| == Coordinates, algebra == | | == Nonfiction cross-reference == |
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| The introduction of coordinates by [[René Descartes (nonfiction)]] and the concurrent developments of [[algebra (nonfiction)]] marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century.
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| == Projective geometry ==
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| The theory of [[perspective (nonfiction)]] showed that there is more to geometry than just the metric properties of figures: perspective is the origin of [[projective geometry (nonfiction)]].
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| == Topology, Differential geometry ==
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| The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with [[Euler (nonfiction)]] and [[Gauss (nonfiction)]] and led to the creation of [[topology (nonfiction)]] and [[differential geometry (nonfiction)]].
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| == Distinction between physical and geometrical space ==
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| In Euclid's time, there was no clear distinction between physical and geometrical space.
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| Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space.
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| == Formal mathematics ==
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| With the rise of [[formal mathematics (nonfiction)]] in the 20th century, 'space' (whether 'point', 'line', or 'plane') lost its intuitive contents, so today one has to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meanings) and abstract spaces.
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| == Contemporary geometry ==
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| [[Contemporary geometry (nonfiction)]] considers [[manifolds (nonfiction)]], spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales.
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| These spaces may be endowed with additional structure which allow one to speak about length.
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| == Modern geometry ==
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| [[Modern geometry (nonfiction)]] has many ties to [[physics (nonfiction)]] as is exemplified by the links between [[pseudo-Riemannian geometry (nonfiction)]] and [[general relativity (nonfiction)]].
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| == Exotic applications ==
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| While the visual nature of geometry makes it initially more accessible than other mathematical areas such as [[algebra (nonfiction)]] or [[number theory (nonfiction)]], [[geometric language (nonfiction)]] is also used in contexts far removed from its traditional, Euclidean provenance.
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| Examples include:
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| * [[Fractal geometry (nonfiction)]]
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| * [[Algebraic geometry (nonfiction)]] | |
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| == Fiction cross-reference == | | == Fiction cross-reference == |
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| * [[Circle (nonfiction)]] | | * [[Geometry solvent]] |
| * [[Coordinate system (nonfiction)]]
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| * [[Degrees of freedom (nonfiction)]]
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| * [[Descriptive geometry (nonfiction)]]
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| * [[Differential geometry (nonfiction)]]
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| * [[Link (geometry) (nonfiction)]]
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| * [[Mathematical notation (nonfiction)]]
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| * [[Mathematics (nonfiction)]]
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| * [[Measurement (nonfiction)]]
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| * [[Pattern (nonfiction)]]
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| * [[Scaling (geometry) (nonfiction)]]
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| * [[Simplicial complex (nonfiction)]]
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| * [[Topology (nonfiction)]]
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| * [[Two-dimensional (nonfiction)]]
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| * [[Vertex (geometry) (nonfiction)]]
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| == Nonfiction cross-reference ==
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| * [[Mathematics (nonfiction)]]
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| == External links == | | == External links == |
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, volume, relative position of figures, and the properties of space.
History
Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC).
Euclid
By the 3rd century BC, geometry was put into an axiomatic form by Euclid (nonfiction), whose treatment -- Euclidean geometry-- set a standard for many centuries to follow.
Archimedes
Archimedes (nonfiction) developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus.
Quadrivium
In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
Nonfiction cross-reference
Fiction cross-reference
External links
