Geometry

Geometry is a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a>, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a <a href="/facts/List_of_geometers/lg9o1Pjo">geometer</a>. Until the 19th century, geometry was almost exclusively devoted to <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, which includes the notions of <a href="/facts/Point_(geometry)/6YPAvX2a">point</a>, <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a>, <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a>, <a href="/facts/Distance/hbXlaEFk">distance</a>, <a href="/facts/Angle/gFUTEQYq">angle</a>, <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a>, and <a href="/facts/Curve/xBQ0pTBW">curve</a>, as fundamental concepts.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, <a href="/facts/Architecture/FccThs8G">architecture</a>, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in <a href="/facts/Wiles%27s_proof_of_Fermat%27s_Last_Theorem/Xgr6wAf2">Wiles's proof</a> of <a href="/facts/Fermat%27s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a>, a problem that was stated in terms of <a href="/facts/Elementary_arithmetic/aUn5tGtp">elementary arithmetic</a>, and remained unsolved for several centuries.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>'s <a href="/facts/Theorema_Egregium/pKPCA0QI">Theorema Egregium</a> ("remarkable theorem") that asserts roughly that the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> of a surface is independent from any specific <a href="/facts/Embedding/HKeVKc42">embedding</a> in a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of <a href="/facts/Manifold/rXPERJtu">manifolds</a> and <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>. Later in the 19th century, it appeared that geometries without the <a href="/facts/Parallel_postulate/xhSxUX4u">parallel postulate</a> (<a href="/facts/Non-Euclidean_geometries/efKa7De2">non-Euclidean geometries</a>) can be developed without introducing any contradiction. The geometry that underlies <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> is a famous application of non-Euclidean geometry.
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—<a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, <a href="/facts/Computational_geometry/eeaotQtl">computational geometry</a>, <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, <a href="/facts/Discrete_geometry/Ixnl3GBc">discrete geometry</a> (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—<a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a> that consider only alignment of points but not distance and parallelism, <a href="/facts/Affine_geometry/yUGgTx2n">affine geometry</a> that omits the concept of angle and distance, <a href="/facts/Finite_geometry/GL0nnEk8">finite geometry</a> that omits <a href="/facts/Continuity_(mathematics)/n8CGmpos">continuity</a>, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional <a href="/facts/Space/tq6LUM3C">space</a> of the physical world and its <a href="/facts/Model/Hj8KMwdD">model</a> provided by Euclidean geometry; presently a geometric space, or simply a space is a <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a> on which some geometry is defined.

Geometry open-in-new

Geometry