Mathematics Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! === Geometry === {{Main|Geometry}} [[File:Triangles (spherical geometry).jpg|thumb|On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.]] Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as [[line (geometry)|lines]], [[angle]]s and [[circle]]s, which were developed mainly for the needs of [[surveying]] and [[architecture]], but has since blossomed out into many other subfields.<ref name="Straume_2014">{{Cite arXiv|last=Straume |first=Eldar |date=September 4, 2014 |title=A Survey of the Development of Geometry up to 1870 |class=math.HO |eprint=1409.1140 }}</ref> A fundamental innovation was the ancient Greeks' introduction of the concept of [[mathematical proof|proofs]], which require that every assertion must be ''proved''. For example, it is not sufficient to verify by [[measurement]] that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ([[theorem]]s) and a few basic statements. The basic statements are not subject to proof because they are self-evident ([[postulate]]s), or are part of the definition of the subject of study ([[axiom]]s). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by [[Euclid]] around 300 BC in his book ''[[Euclid's Elements|Elements]]''.<ref>{{cite book |last=Hilbert |first=David |author-link=David Hilbert |year=1902 |title=The Foundations of Geometry |page=1 |publisher=[[Open Court Publishing Company]] |doi=10.1126/science.16.399.307 |lccn=02019303 |oclc=996838 |s2cid=238499430 |url={{GBurl|id=8ZBsAAAAMAAJ}} |access-date=February 6, 2024}} {{free access}}</ref><ref>{{cite book |last=Hartshorne |first=Robin |author-link=Robin Hartshorne |year=2000 |chapter=Euclid's Geometry |pages=9–13 |title=Geometry: Euclid and Beyond |publisher=[[Springer New York]] |isbn=0-387-98650-2 |lccn=99044789 |oclc=42290188 |url={{GBurl|id=EJCSL9S6la0C|p=9}} |access-date=February 7, 2024}}</ref> The resulting [[Euclidean geometry]] is the study of shapes and their arrangements [[straightedge and compass construction|constructed]] from lines, [[plane (geometry)|planes]] and circles in the [[Euclidean plane]] ([[plane geometry]]) and the three-dimensional [[Euclidean space]].{{efn|This includes [[conic section]]s, which are intersections of [[circular cylinder]]s and planes.}}<ref name=Straume_2014/> Euclidean geometry was developed without change of methods or scope until the 17th century, when [[René Descartes]] introduced what is now called [[Cartesian coordinates]]. This constituted a major [[Paradigm shift|change of paradigm]]: Instead of defining [[real number]]s as lengths of [[line segments]] (see [[number line]]), it allowed the representation of points using their ''coordinates'', which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: [[synthetic geometry]], which uses purely geometrical methods, and [[analytic geometry]], which uses coordinates systemically.<ref>{{cite book |last=Boyer |first=Carl B. |author-link=Carl B. Boyer |year=2004 |orig-date=1956 |chapter=Fermat and Descartes |pages=74–102 |title=History of Analytic Geometry |publisher=[[Dover Publications]] |isbn=0-486-43832-5 |lccn=2004056235 |oclc=56317813}}</ref> Analytic geometry allows the study of [[curve]]s unrelated to circles and lines. Such curves can be defined as the [[graph of a function|graph of functions]], the study of which led to [[differential geometry]]. They can also be defined as [[implicit equation]]s, often [[polynomial equation]]s (which spawned [[algebraic geometry]]). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.<ref name=Straume_2014/> In the 19th century, mathematicians discovered [[non-Euclidean geometries]], which do not follow the [[parallel postulate]]. By questioning that postulate's truth, this discovery has been viewed as joining [[Russell's paradox]] in revealing the [[foundational crisis of mathematics]]. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.<ref>{{cite journal |last=Stump |year=1997 |first=David J. |title=Reconstructing the Unity of Mathematics circa 1900 |journal=[[Perspectives on Science]] |volume=5 |issue=3 |page=383–417 |doi=10.1162/posc_a_00532 |eissn=1530-9274 |issn=1063-6145 |lccn=94657506 |oclc=26085129 |s2cid=117709681 |url=https://philpapers.org/archive/STURTU.pdf |access-date=February 8, 2024}}</ref><ref name=Kleiner_1991/> In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that [[Invariant (mathematics)|do not change]] under specific transformations of the [[space (mathematics)|space]].<ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=February 1996 |title=Non-Euclidean geometry |website=MacTuror |publisher=[[University of St. Andrews]] |publication-place=Scotland, UK |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/ |url-status=live |archive-url=https://web.archive.org/web/20221106142807/https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/ |archive-date=November 6, 2022 |access-date=February 8, 2024}}</ref> Today's subareas of geometry include:<ref name=MSC/> * [[Projective geometry]], introduced in the 16th century by [[Girard Desargues]], extends Euclidean geometry by adding [[points at infinity]] at which [[parallel lines]] intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines. * [[Affine geometry]], the study of properties relative to [[parallel (geometry)|parallelism]] and independent from the concept of length. * [[Differential geometry]], the study of curves, surfaces, and their generalizations, which are defined using [[differentiable function]]s. * [[Manifold theory]], the study of shapes that are not necessarily embedded in a larger space. * [[Riemannian geometry]], the study of distance properties in curved spaces. * [[Algebraic geometry]], the study of curves, surfaces, and their generalizations, which are defined using [[polynomial]]s. * [[Topology]], the study of properties that are kept under [[continuous deformation]]s. ** [[Algebraic topology]], the use in topology of algebraic methods, mainly [[homological algebra]]. * [[Discrete geometry]], the study of finite configurations in geometry. * [[Convex geometry]], the study of [[convex set]]s, which takes its importance from its applications in [[convex optimization|optimization]]. * [[Complex geometry]], the geometry obtained by replacing real numbers with [[complex number]]s. Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page