Geometry

Geometry (from the <a href="https://en.wikipedia.org/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a>: γεωμετρία; <a href="https://en.wiktionary.org/wiki/%CE%B3%E1%BF%86" class="extiw" title="wikt:γῆ">geo-</a> "earth", <a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%84%CF%81%CE%BF%CE%BD" class="extiw" title="wikt:μέτρον">-metron</a> "measurement") is a branch of <a href="https://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a> concerned with questions of shape, size, relative position of figures, and the properties of space.

Geometry (from the <a href="https://en.wikipedia.org/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a>: γεωμετρία; <a href="https://en.wiktionary.org/wiki/%CE%B3%E1%BF%86" class="extiw" title="wikt:γῆ">geo-</a> "earth", <a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%84%CF%81%CE%BF%CE%BD" class="extiw" title="wikt:μέτρον">-metron</a> "measurement") is a branch of <a href="https://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a> concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a <a href="https://en.wikipedia.org/wiki/Geometer" class="mw-redirect" title="Geometer">geometer</a>.

Geometry arose independently in a number of early cultures as a body of practical knowledge concerning <a href="https://en.wikipedia.org/wiki/Length" title="Length">lengths</a>, <a href="https://en.wikipedia.org/wiki/Area" title="Area">areas</a>, and <a href="https://en.wikipedia.org/wiki/Volume" title="Volume">volumes</a>, with elements of formal <a href="https://en.wikipedia.org/wiki/Mathematical_science" class="mw-redirect" title="Mathematical science">mathematical science</a> emerging in the West as early as <a href="https://en.wikipedia.org/wiki/Thales" title="Thales">Thales</a> (6th century BC).

By the 3rd century BC, geometry was put into an <a href="https://en.wikipedia.org/wiki/Axiomatic_system" title="Axiomatic system">axiomatic form</a> by <a href="https://en.wikipedia.org/wiki/Euclid" title="Euclid">Euclid</a>, whose treatment—<a href="https://en.wikipedia.org/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>—set a standard for many centuries to follow.<a href="https://en.wikipedia.org/wiki/Geometry#cite_note-1">[1]</a>

<a href="https://en.wikipedia.org/wiki/Archimedes" title="Archimedes">Archimedes</a> developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern <a href="https://en.wikipedia.org/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>.

The field of <a href="https://en.wikipedia.org/wiki/Astronomy" title="Astronomy">astronomy</a>, especially as it relates to mapping the positions of <a href="https://en.wikipedia.org/wiki/Star" title="Star">stars</a> and <a href="https://en.wikipedia.org/wiki/Planet" title="Planet">planets</a> on the <a href="https://en.wikipedia.org/wiki/Celestial_sphere" title="Celestial sphere">celestial sphere</a> and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.

In the classical world, both geometry and astronomy were considered to be part of the <a href="https://en.wikipedia.org/wiki/Quadrivium" title="Quadrivium">Quadrivium</a>, a subset of the seven <a href="https://en.wikipedia.org/wiki/Liberal_arts#History" class="mw-redirect" title="Liberal arts">liberal arts</a> considered essential for a free citizen to master.

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 <a href="https://en.wikipedia.org/wiki/Compass_%28drafting%29" class="mw-redirect" title="Compass (drafting)">compass</a> and <a href="https://en.wikipedia.org/wiki/Ruler" title="Ruler">straightedge</a>. 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.

The introduction of <a href="https://en.wikipedia.org/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> by <a href="https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> and the concurrent developments of <a href="https://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a> marked a new stage for geometry, since geometric figures such as <a href="https://en.wikipedia.org/wiki/Plane_curve" title="Plane curve">plane curves</a> could now be represented <a href="https://en.wikipedia.org/wiki/Analytic_geometry" title="Analytic geometry">analytically</a> in the form of functions and equations. This played a key role in the emergence of <a href="https://en.wikipedia.org/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> in the 17th century. Furthermore, the theory of <a href="https://en.wikipedia.org/wiki/Perspective_%28graphical%29" title="Perspective (graphical)">perspective</a> showed that there is more to geometry than just the metric properties of figures: perspective is the origin of <a href="https://en.wikipedia.org/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>. The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with <a href="https://en.wikipedia.org/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> and <a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> and led to the creation of <a href="https://en.wikipedia.org/wiki/Topology" title="Topology">topology</a> and <a href="https://en.wikipedia.org/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>.

In Euclid's time, there was no clear distinction between physical and geometrical <a href="https://en.wikipedia.org/wiki/Space" title="Space">space</a>. Since the 19th-century discovery of <a href="https://en.wikipedia.org/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space. With the rise of formal mathematics 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. Contemporary geometry considers <a href="https://en.wikipedia.org/wiki/Manifold" title="Manifold">manifolds</a>, spaces that are considerably more abstract than the familiar <a href="https://en.wikipedia.org/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, which they only approximately resemble at small scales. These spaces may be endowed with additional structure which allow one to speak about length. Modern geometry has many ties to <a href="https://en.wikipedia.org/wiki/Physics" title="Physics">physics</a> as is exemplified by the links between <a href="https://en.wikipedia.org/wiki/Pseudo-Riemannian" class="mw-redirect" title="Pseudo-Riemannian">pseudo-Riemannian</a> geometry and <a href="https://en.wikipedia.org/wiki/General_relativity" title="General relativity">general relativity</a>. One of the youngest physical theories, <a href="https://en.wikipedia.org/wiki/String_theory" title="String theory">string theory</a>, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or <a href="https://en.wikipedia.org/wiki/Number_theory" title="Number theory">number theory</a>, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in <a href="https://en.wikipedia.org/wiki/Fractal_geometry" class="mw-redirect" title="Fractal geometry">fractal geometry</a> and <a href="https://en.wikipedia.org/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>).<a href="https://en.wikipedia.org/wiki/Geometry#cite_note-2">[2]</a>

Geometry

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