View source for Geometry

{{short description|Branch of mathematics}}
{{other uses}}
{{pp-semi-indef}}
{{pp-move}}
{{Use dmy dates|date=August 2019}}
{{General geometry}}
{{Math topics TOC}}

'''Geometry''' ({{etymology|grc|''{{Wikt-lang|grc|γεωμετρία}}'' ({{grc-transl|γεωμετρία}})|land measurement}}; {{etymology||''{{Wikt-lang|grc|γῆ}}'' ({{grc-transl|γῆ}})|earth, land||''{{Wikt-lang|grc|μέτρον}}'' ({{grc-transl|μέτρον}})|a measure}})<ref>{{cite web |title=Geometry - Formulas, Examples {{!}} Plane and Solid Geometry |url=https://cuemath.com/geometry |access-date=2023-08-31 |website=Cuemath |language=en}}</ref> is a branch of [[mathematics]] concerned with properties of space such as the distance, shape, size, and relative position of figures.<!--Please, do not link these everyday words--><ref name="Risi2015">{{cite book|author=Vincenzo De Risi|title=Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age|url=https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1|year=2015|publisher=Birkhäuser|isbn=978-3-319-12102-4|pages=1–|access-date=14 September 2019|archive-date=20 February 2021|archive-url=https://web.archive.org/web/20210220094741/https://books.google.com/books?id=1m11BgAAQBAJ&pg=PA1|url-status=live}}</ref> Geometry is, along with [[arithmetic]], one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''[[List of geometers|geometer]]''. Until the 19th century, geometry was almost exclusively devoted to [[Euclidean geometry]],{{efn|Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development of [[hyperbolic geometry]] by [[Nikolai Lobachevsky|Lobachevsky]] and other [[non-Euclidean geometries]] by [[Gauss]] and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of [[Desargues]] in the 17th century, all the way back to the implicit use of [[spherical geometry]] to understand the [[geodesy|Earth geodesy]] and to navigate the oceans since antiquity.}} which includes the notions of [[point (geometry)|point]], [[line (geometry)|line]], [[plane (geometry)|plane]], [[distance]], [[angle]], [[surface (mathematics)|surface]], and [[curve]], as fundamental concepts.<ref name="Tabak 2014 xiv">{{cite book |last=Tabak |first=John |url=https://archive.org/details/geometrylanguage0000taba |title=Geometry: the language of space and form |date=2014 |publisher=Infobase Publishing |isbn=978-0-8160-4953-0 |page=xiv |url-access=registration}}</ref>

Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, [[architecture]], and other activities that are related to graphics.<ref name="Meyer2006">{{cite book|author=Walter A. Meyer|title=Geometry and Its Applications|url=https://books.google.com/books?id=ez6H5Ho6E3cC|year=2006|publisher=Elsevier|isbn=978-0-08-047803-6|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183207/https://books.google.com/books?id=ez6H5Ho6E3cC|url-status=live}}</ref> Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in [[Wiles's proof of Fermat's Last Theorem|Wiles's proof]] of [[Fermat's Last Theorem]], a problem that was stated in terms of [[elementary arithmetic]], 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 [[Carl Friedrich Gauss]]'s {{Lang|la|[[Theorema Egregium]]|italic=no}} ("remarkable theorem") that asserts roughly that the [[Gaussian curvature]] of a surface is independent from any specific [[embedding]] in a [[Euclidean space]]. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of [[manifold]]s and [[Riemannian geometry]]. Later in the 19th century, it appeared that geometries without the [[parallel postulate]] ([[non-Euclidean geometries]]) can be developed without introducing any contradiction. The geometry that underlies [[general relativity]] 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—[[differential geometry]], [[algebraic geometry]], [[computational geometry]], [[algebraic topology]], [[discrete geometry]] (also known as ''combinatorial geometry''), etc.—or on the properties of Euclidean spaces that are disregarded—[[projective geometry]] that consider only alignment of points but not distance and parallelism, [[affine geometry]] that omits the concept of angle and distance, [[finite geometry]] that omits [[continuity (mathematics)|continuity]], 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 [[space]] of the physical world and its [[model]] provided by Euclidean geometry; presently a '''geometric space''', or simply a ''space'' is a [[mathematical structure]] on which some geometry is defined.

==History==
{{Main|History of geometry}}
[[File:Westerner and Arab practicing geometry 15th century manuscript.jpg|right|thumb|A [[Ethnic groups in Europe|European]] and an [[Arab]] practicing geometry in the 15th century]]

<!--[[File:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg|right|thumb|The [[frontispiece]] of Sir Henry Billingsley's first English version of Euclid's ''[[Element (mathematics)|Elements]]'', 1570]]-->
The earliest recorded beginnings of geometry can be traced to ancient [[Mesopotamia]] and [[Ancient Egypt|Egypt]] in the 2nd millennium BC.<ref>{{Cite journal |last=Friberg |first=Jöran |date=1981 |title=Methods and traditions of Babylonian mathematics |journal=Historia Mathematica |language=en |volume=8 |issue=3 |pages=277–318 |doi=10.1016/0315-0860(81)90069-0|doi-access=free }}</ref><ref>{{Cite book |last=Neugebauer |first=Otto|author-link=Otto E. Neugebauer |chapter-url=https://books.google.com/books?id=JVhTtVA2zr8C&pg=PA71 |title=The Exact Sciences in Antiquity |publisher=[[Dover Publications]] |year=1969 |isbn=978-0-486-22332-2 |edition=2 |pages=71–96 |chapter=Chap. IV Egyptian Mathematics and Astronomy |access-date=27 February 2021 |archive-url=https://web.archive.org/web/20200814151056/https://books.google.com/books?id=JVhTtVA2zr8C |archive-date=14 August 2020 |url-status=live |orig-year=1957}}.</ref> Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in [[surveying]], [[construction]], [[astronomy]], and various crafts. The earliest known texts on geometry are the [[Egyptian mathematics|Egyptian]] [[Rhind Mathematical Papyrus|Rhind Papyrus]] (2000–1800 BC) and [[Moscow Mathematical Papyrus|Moscow Papyrus]] ({{Circa|1890 BC}}), and the [[Babylonian mathematics|Babylonian clay tablets]], such as [[Plimpton 322]] (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or [[frustum]].<ref name="Boyer 1991 loc=Egypt p. 19">{{Harv|Boyer|1991|loc="Egypt" p. 19}}</ref> Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented [[trapezoid]] procedures for computing Jupiter's position and [[Displacement (vector)|motion]] within time-velocity space. These geometric procedures anticipated the [[Oxford Calculators]], including the [[mean speed theorem]], by 14 centuries.<ref>{{cite journal |last=Ossendrijver |first=Mathieu |date=29 January 2016 |title=Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph |journal=Science |volume=351 |issue=6272 |pages=482–484 |bibcode=2016Sci...351..482O |doi=10.1126/science.aad8085 |pmid=26823423 |s2cid=206644971}}</ref> South of Egypt the [[Nubia|ancient Nubians]] established a system of geometry including early versions of sun clocks.<ref>{{cite journal |last=Depuydt |first=Leo |date=1 January 1998 |title=Gnomons at Meroë and Early Trigonometry |journal=The Journal of Egyptian Archaeology |volume=84 |pages=171–180 |doi=10.2307/3822211 |jstor=3822211}}</ref><ref>{{cite web |last=Slayman |first=Andrew |date=27 May 1998 |title=Neolithic Skywatchers |url=http://www.archaeology.org/online/news/nubia.html |url-status=live |archive-url=https://web.archive.org/web/20110605234044/http://www.archaeology.org/online/news/nubia.html |archive-date=5 June 2011 |access-date=17 April 2011 |website=Archaeology Magazine Archive}}</ref>

In the 7th century BC, the [[Greek mathematics|Greek]] mathematician [[Thales of Miletus]] used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to [[Thales's theorem]].<ref name="Boyer 1991 loc=Ionia and the Pythagoreans p. 43">{{Harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 43}}</ref> [[Pythagoras]] established the [[Pythagoreans|Pythagorean School]], which is credited with the first proof of the [[Pythagorean theorem]],<ref>Eves, Howard, ''[[iarchive:introductiontohi0000eves|An Introduction to the History of Mathematics]]'', Saunders, 1990, {{ISBN|0-03-029558-0}}.</ref> though the statement of the theorem has a long history.<ref>{{cite book |author=Kurt Von Fritz |chapter=The Discovery of Incommensurability by Hippasus of Metapontum |year=1945 |title=Classics in the History of Greek Mathematics |series=Annals of Mathematics; Boston Studies in the Philosophy of Science |volume=240 |issue=2 |pages=211–231 |publisher=Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University |doi=10.1007/978-1-4020-2640-9_11 |jstor=1969021|isbn=978-90-481-5850-8 }}</ref><ref>{{cite journal |author=James R. Choike |year=1980 |title=The Pentagram and the Discovery of an Irrational Number |url=https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468 |journal=The Two-Year College Mathematics Journal |volume=11 |issue=5 |pages=312–316 |doi=10.2307/3026893 |jstor=3026893 |access-date=9 September 2022 |archive-date=9 September 2022 |archive-url=https://web.archive.org/web/20220909203418/https://www.tandfonline.com/doi/abs/10.1080/00494925.1980.11972468 |url-status=live }}</ref> [[Eudoxus of Cnidus|Eudoxus]] (408–{{Circa|355 BC}}) developed the [[method of exhaustion]], which allowed the calculation of areas and volumes of curvilinear figures,<ref>{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 92}}</ref> as well as a theory of ratios that avoided the problem of [[incommensurable magnitudes]], which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose ''[[Euclid's Elements|Elements]]'', widely considered the most successful and influential textbook of all time,<ref>{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 119}}</ref> introduced [[mathematical rigor]] through the [[axiomatic method]] and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the ''Elements'' were already known, Euclid arranged them into a single, coherent logical framework.<ref name="Boyer 1991 loc=Euclid of Alexandria p. 104">{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 104}}</ref> The ''Elements'' was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.<ref>[[Howard Eves]], ''[[iarchive:introductiontohi0000eves|An Introduction to the History of Mathematics]]'', Saunders, 1990, {{ISBN|0-03-029558-0}} p. 141: "No work, except [[The Bible]], has been more widely used...."</ref> [[Archimedes]] ({{Circa|287–212 BC}}) of [[Syracuse, Italy]] used the method of exhaustion to calculate the [[area]] under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], and gave remarkably accurate approximations of [[pi]].<ref>{{cite web | title = A history of calculus | author1 = O'Connor, J.J. | author2 = Robertson, E.F. | publisher = [[University of St Andrews]] | url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html | date = February 1996 | access-date = 7 August 2007 | archive-url = https://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html | archive-date = 15 July 2007}}</ref> He also studied the [[Archimedes spiral|spiral]] bearing his name and obtained formulas for the [[volume]]s of [[surface of revolution|surfaces of revolution]].

[[File:Woman teaching geometry.jpg|left|thumb|upright=.85|''Woman teaching geometry''. Illustration at the beginning of a medieval translation of [[Euclid's Elements]], ({{Circa|1310}}).]]

[[Indian mathematics|Indian]] mathematicians also made many important contributions in geometry. The ''[[Shatapatha Brahmana]]'' (3rd century BC) contains rules for ritual geometric constructions that are similar to the ''[[Shulba Sutras|Sulba Sutras]]''.<ref name="Staal 1999">{{cite journal  | last=Staal  | first=Frits  | author-link=Frits Staal  | title=Greek and Vedic Geometry  | journal=Journal of Indian Philosophy  | volume=27  | issue=1–2  | year=1999  | pages=105–127  | doi=10.1023/A:1004364417713  | s2cid=170894641 }}
</ref> According to {{Harv|Hayashi|2005|p=363}}, the ''Śulba Sūtras'' contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of [[Pythagorean triples]],{{efn|reference=Pythagorean triples are triples of integers <math> (a,b,c) </math> with the property: <math>a^2+b^2=c^2</math>. Thus, <math>3^2+4^2=5^2</math>, <math>8^2+15^2=17^2</math>, <math>12^2+35^2=37^2</math> etc.}} which are particular cases of [[Diophantine equations]].<ref name=cooke198>{{Harv|Cooke|2005|p=198}}: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."</ref>
In the [[Bakhshali manuscript]], there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<ref name="hayashi2005-371">{{Harv|Hayashi|2005|p=371}}</ref> [[Aryabhata]]'s ''[[Aryabhatiya]]'' (499) includes the computation of areas and volumes.
[[Brahmagupta]] wrote his astronomical work ''[[Brahmasphutasiddhanta|{{IAST|Brāhmasphuṭasiddhānta}}]]'' in 628. Chapter 12, containing 66 [[Sanskrit]] verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<ref name="hayashi2003-p121-122">{{Harv|Hayashi|2003|pp=121–122}}</ref> In the latter section, he stated his famous theorem on the diagonals of a [[cyclic quadrilateral]]. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of [[Heron's formula]]), as well as a complete description of [[rational triangle]]s (''i.e.'' triangles with rational sides and rational areas).<ref name="hayashi2003-p121-122"/>

In the [[Middle Ages]], [[mathematics in medieval Islam]] contributed to the development of geometry, especially [[algebraic geometry]].<ref>{{Cite book |last=Rāshid |first=Rushdī |url=https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n43/mode/2up |title=The development of Arabic mathematics : between arithmetic and algebra |date=1994 |isbn=978-0-7923-2565-9 |series=Boston Studies in the Philosophy of Science |volume=156 |location= |pages=35 |doi=10.1007/978-94-017-3274-1 |oclc=29181926}}</ref><ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" pp. 241–242}} "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."".</ref> [[Al-Mahani]] (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.<ref>{{MacTutor Biography|id=Al-Mahani|title=Al-Mahani|mode=cs1}}</ref> [[Thābit ibn Qurra]] (known as Thebit in [[Latin]]) (836–901) dealt with [[arithmetic]] operations applied to [[ratio]]s of geometrical quantities, and contributed to the development of [[analytic geometry]].<ref name="ReferenceA">{{MacTutor Biography|id=Thabit|title=Al-Sabi Thabit ibn Qurra al-Harrani|mode=cs1}}</ref> [[Omar Khayyam]] (1048–1131) found geometric solutions to [[cubic equation]]s.<ref>{{MacTutor Biography|id=Khayyam|title=Omar Khayyam|mode=cs1}}</ref> The theorems of [[Ibn al-Haytham]] (Alhazen), Omar Khayyam and [[Nasir al-Din al-Tusi]] on [[quadrilateral]]s, including the [[Lambert quadrilateral]] and [[Saccheri quadrilateral]], were part of a line of research on the [[parallel postulate]] continued by later European geometers, including [[Vitello]] ({{Circa|1230|1314}}), [[Gersonides]] (1288–1344), Alfonso, [[John Wallis]], and [[Giovanni Girolamo Saccheri]], that by the 19th century led to the discovery of [[hyperbolic geometry]].<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, pp. 447–494 [470], [[Routledge]], London and New York: {{blockquote|"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 Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines—made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's ''[[Book of Optics]]'' (''Kitab al-Manazir'')—was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that ''Pseudo-Tusi's Exposition of Euclid'' had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."}}</ref>

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with [[Coordinate system|coordinates]] and [[equation]]s, by [[René Descartes]] (1596–1650) and [[Pierre de Fermat]] (1601–1665).<ref name="Boyer2012">{{cite book|author=Carl B. Boyer|author-link=Carl Benjamin Boyer|title=History of Analytic Geometry|url=https://books.google.com/books?id=2T4i5fXZbOYC|date=2012|publisher=Courier Corporation|isbn=978-0-486-15451-0|access-date=18 September 2019|archive-date=26 December 2019|archive-url=https://web.archive.org/web/20191226215605/https://books.google.com/books?id=2T4i5fXZbOYC|url-status=live}}</ref> This was a necessary precursor to the development of [[calculus]] and a precise quantitative science of [[physics]].<ref name="Edwards2012">{{cite book|author=C. H. Edwards Jr.|title=The Historical Development of the Calculus|url=https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95|date=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-6230-5|page=95|access-date=18 September 2019|archive-date=29 December 2019|archive-url=https://web.archive.org/web/20191229201529/https://books.google.com/books?id=ilrlBwAAQBAJ&pg=PA95|url-status=live}}</ref> The second geometric development of this period was the systematic study of [[projective geometry]] by [[Girard Desargues]] (1591–1661).<ref name="FieldGray2012">{{cite book|author1=Judith V. Field|author1-link=Judith V. Field|author2=Jeremy Gray|title=The Geometrical Work of Girard Desargues|url=https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8692-6|page=43|access-date=18 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227054645/https://books.google.com/books?id=zSvSBwAAQBAJ&pg=PA43|url-status=live}}</ref> Projective geometry studies properties of shapes which are unchanged under [[projection (linear algebra)|projections]] and [[section (fiber bundle)|sections]], especially as they relate to [[perspective (graphical)|artistic perspective]].<ref name="Wylie2011">{{cite book|author=C. R. Wylie|title=Introduction to Projective Geometry|url=https://books.google.com/books?id=VVvGc8kaajEC|date=2011|publisher=Courier Corporation|isbn=978-0-486-14170-1|access-date=18 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228051716/https://books.google.com/books?id=VVvGc8kaajEC|url-status=live}}</ref>

Two developments in geometry in the 19th century changed the way it had been studied previously.<ref name="Gray2011">{{cite book|author=Jeremy Gray|title=Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century|url=https://books.google.com/books?id=3UeSCvazV0QC|date=2011|publisher=Springer Science & Business Media|isbn=978-0-85729-060-1|access-date=18 September 2019|archive-date=7 December 2019|archive-url=https://web.archive.org/web/20191207041658/https://books.google.com/books?id=3UeSCvazV0QC|url-status=live}}</ref> These were the discovery of [[non-Euclidean geometry|non-Euclidean geometries]] by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of [[symmetry]] as the central consideration in the [[Erlangen programme]] of [[Felix Klein]] (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were [[Bernhard Riemann]] (1826–1866), working primarily with tools from [[mathematical analysis]], and introducing the [[Riemann surface]], and [[Henri Poincaré]], the founder of [[algebraic topology]] and the geometric theory of [[dynamical system]]s. As a consequence of these major changes in the conception of geometry, the concept of "[[space (mathematics)|space]]" became something rich and varied, and the natural background for theories as different as [[complex analysis]] and [[classical mechanics]].<ref name="Bayro-Corrochano2018">{{cite book|author=Eduardo Bayro-Corrochano|title=Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing|url=https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4|date=2018|publisher=Springer|isbn=978-3-319-74830-6|page=4|access-date=18 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228052142/https://books.google.com/books?id=SSVhDwAAQBAJ&pg=PA4|url-status=live}}</ref>

==Main concepts==

The following are some of the most important concepts in geometry.<ref name="Tabak 2014 xiv"/><ref name="Kline1990">{{cite book|author=Morris Kline|title=Mathematical Thought From Ancient to Modern Times: Volume 3|url=https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010|year=1990|publisher=Oxford University Press|location=US|isbn=978-0-19-506137-6|pages=1010–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183204/https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010|url-status=live}}</ref>

===Axioms===
[[File:Parallel postulate en.svg|thumb|right|An illustration of Euclid's [[parallel postulate]]]]
{{See also|Euclidean geometry|Axiom}}
[[Euclid]] took an abstract approach to geometry in his [[Euclid's Elements|Elements]],<ref name="Katz2000">{{cite book|author=Victor J. Katz|title=Using History to Teach Mathematics: An International Perspective|url=https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45|year=2000|publisher=Cambridge University Press|isbn=978-0-88385-163-0|pages=45–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183205/https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45|url-status=live}}</ref> one of the most influential books ever written.<ref name="Berlinski2014">{{cite book|author=David Berlinski|author-link=David Berlinski|title=The King of Infinite Space: Euclid and His Elements|url=https://archive.org/details/kingofinfinitesp00davi|url-access=registration|year=2014|publisher=Basic Books|isbn=978-0-465-03863-3}}</ref> Euclid introduced certain [[axiom]]s, or [[postulate]]s, expressing primary or self-evident properties of points, lines, and planes.<ref name="Hartshorne2013">{{cite book|author=Robin Hartshorne|author-link=Robin Hartshorne|title=Geometry: Euclid and Beyond|url=https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29|year=2013|publisher=Springer Science & Business Media|isbn=978-0-387-22676-7|pages=29–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183205/https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29|url-status=live}}</ref> He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as ''axiomatic'' or ''[[synthetic geometry|synthetic]]'' geometry.<ref name="HerbstFujita2017">{{cite book|author1=Pat Herbst|author2=Taro Fujita|author3=Stefan Halverscheid|author4=Michael Weiss|title=The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective|url=https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20|year=2017|publisher=Taylor & Francis|isbn=978-1-351-97353-3|pages=20–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183206/https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20|url-status=live}}</ref> At the start of the 19th century, the discovery of [[non-Euclidean geometries]] by [[Nikolai Ivanovich Lobachevsky]] (1792–1856), [[János Bolyai]] (1802–1860), [[Carl Friedrich Gauss]] (1777–1855) and others<ref name="Yaglom2012">{{cite book|author=I. M. Yaglom|author-link=Isaak Yaglom|title=A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity|url=https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-6135-3|pages=6–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183221/https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6|url-status=live}}</ref> led to a revival of interest in this discipline, and in the 20th century, [[David Hilbert]] (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.<ref name="Holme2010">{{cite book|author=Audun Holme|title=Geometry: Our Cultural Heritage|url=https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254|date=2010|publisher=Springer Science & Business Media|isbn=978-3-642-14441-7|pages=254–|access-date=14 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183209/https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254|url-status=live}}</ref>

===Spaces and subspaces===

====Points====
{{Main|Point (geometry)}}

Points are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part",<ref name=EuclidAll>''Euclid's Elements – All thirteen books in one volume'', Based on Heath's translation, Green Lion Press {{ISBN|1-888009-18-7}}.</ref> or in [[synthetic geometry]]. In modern mathematics, they are generally defined as [[element (set theory)|elements]] of a [[set (mathematics)|set]] called [[space (mathematics)|space]], which is itself [[axiomatically]] defined.

With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes.

However, there are modern geometries in which points are not primitive objects, or even without points.<ref>{{cite book|author=Gerla, G.|year=1995|chapter-url= http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf|url-status=dead|archive-url=https://web.archive.org/web/20110717210751/http://www.dmi.unisa.it/people/gerla/www/Down/point-free.pdf|archive-date=17 July 2011|chapter=Pointless Geometries|editor=Buekenhout, F.|editor2=Kantor, W.|title=Handbook of incidence geometry: buildings and foundations|publisher=North-Holland|pages=1015–1031}}</ref><ref>{{cite journal |last= Clark|first=Bowman L. |date= Jan 1985|title= Individuals and Points|journal= Notre Dame Journal of Formal Logic|volume= 26|issue=1 |pages= 61–75|doi= 10.1305/ndjfl/1093870761|doi-access= free}}</ref> One of the oldest such geometries is [[Whitehead's point-free geometry]], formulated by [[Alfred North Whitehead]] in 1919–1920.

====Lines====
{{main|Line (geometry)}}

[[Euclid]] described a line as "breadthless length" which "lies equally with respect to the points on itself".<ref name=EuclidAll /> In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in [[analytic geometry]], a line in the plane is often defined as the set of points whose coordinates satisfy a given [[linear equation]],<ref>{{cite book|authorlink=John Casey (mathematician)|author=John Casey|year=1885|url= https://archive.org/details/cu31924001520455|title=Analytic Geometry of the Point, Line, Circle, and Conic Sections}}</ref> but in a more abstract setting, such as [[incidence geometry]], a line may be an independent object, distinct from the set of points which lie on it.<ref>{{Cite book |url=https://www.worldcat.org/oclc/162589397 |title=Handbook of incidence geometry : buildings and foundations |date=1995 |publisher=Elsevier |editor=Francis Buekenhout |isbn=978-0-444-88355-1 |location=Amsterdam |oclc=162589397 |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145150/https://www.worldcat.org/title/162589397 |url-status=live }}</ref> In differential geometry, a [[geodesic]] is a generalization of the notion of a line to [[manifold|curved spaces]].<ref>{{cite web|url=https://www.oxforddictionaries.com/definition/english/geodesic|title=geodesic – definition of geodesic in English from the Oxford dictionary|publisher=[[OxfordDictionaries.com]]|access-date=2016-01-20|archive-url=https://web.archive.org/web/20160715034047/http://www.oxforddictionaries.com/definition/english/geodesic|archive-date=15 July 2016|url-status=dead}}</ref>

====Planes====
{{main|Euclidean plane}}

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely;<ref name= EuclidAll /> the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a [[surface (topology)|topological surface]] without reference to distances or angles;<ref name="Munkres">{{Cite book |last=Munkres |first=James R.|author-link=James Munkres |url=https://www.worldcat.org/oclc/42683260 |title=Topology |date=2000 |publisher=Prentice Hall, Inc |isbn=0-13-181629-2 |edition=2nd |volume=2 |location=Upper Saddle River, NJ |oclc=42683260}}</ref> it can be studied as an [[affine space]], where collinearity and ratios can be studied but not distances;<ref>{{Cite book |last=Szmielew |first=Wanda |author-link=Wanda Szmielew|url=https://books.google.com/books?id=xDJPAQAAIAAJ |title=From Affine to Euclidean Geometry |year=1983 |publisher=Springer |isbn=978-90-277-1243-1 |language=en |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145204/https://books.google.com/books?id=xDJPAQAAIAAJ |url-status=live }}</ref> it can be studied as the [[complex plane]] using techniques of [[complex analysis]];<ref>{{Cite book |last=Ahlfors |first=Lars V.|author-link=Lars Ahlfors |url=https://books.google.com/books?id=2MRuus-5GGoC |title=Complex analysis : an introduction to the theory of analytic functions of one complex variable |date=1979 |publisher=McGraw-Hill |isbn=9780070006577 |edition=3rd |location=New York |oclc=4036464 |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145208/https://books.google.com/books?id=2MRuus-5GGoC |url-status=live }}</ref> and so on.

====Curves====
{{main|Curve (geometry)}}

A [[curve (geometry)|curve]] is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called [[plane curve]]s and those in 3-dimensional space are called [[space curve]]s.<ref>Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.</ref>

In topology, a curve is defined by a function from an interval of the real numbers to another space.<ref name = Munkres /> In differential geometry, the same definition is used, but the defining function is required to be differentiable.<ref name = Carmo /> Algebraic geometry studies [[algebraic curve]]s, which are defined as [[algebraic varieties]] of [[dimension of an algebraic variety|dimension]] one.<ref name = mumford />

====Surfaces====
{{main| Surface (mathematics)}}
[[File:Sphere wireframe.svg|thumb|upright=0.85|A sphere is a surface that can be defined parametrically (by {{nowrap|''x'' {{=}} ''r'' sin ''θ'' cos ''φ'',}} {{nowrap|''y'' {{=}} ''r'' sin ''θ'' sin ''φ'',}} {{nowrap|''z'' {{=}} ''r'' cos ''θ'')}} or implicitly (by {{nowrap|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> − ''r''<sup>2</sup> {{=}} 0}}).]]

A [[surface (mathematics)|surface]] is a two-dimensional object, such as a sphere or paraboloid.<ref>Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." {{ISBN|978-0-321-57056-7}}.</ref> In [[differential geometry]]<ref name="Carmo">{{Cite book |last=Carmo |first=Manfredo Perdigão do |url=https://books.google.com/books?id=1v0YAQAAIAAJ |title=Differential geometry of curves and surfaces |date=1976 |publisher=Prentice-Hall |isbn=0-13-212589-7 |volume=2 |location=Englewood Cliffs, N.J. |oclc=1529515 |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145145/https://books.google.com/books?id=1v0YAQAAIAAJ |url-status=live }}</ref> and [[topology]],<ref name=Munkres /> surfaces are described by two-dimensional 'patches' (or [[neighborhood (topology)|neighborhoods]]) that are assembled by [[diffeomorphism]]s or [[homeomorphism]]s, respectively. In algebraic geometry, surfaces are described by [[polynomial equation]]s.<ref name=mumford>{{cite book |last=Mumford |first=David |author-link=David Mumford |title=The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians |edition=2nd |year=1999 |publisher=[[Springer Science+Business Media|Springer-Verlag]] |isbn=978-3-540-63293-1 |zbl=0945.14001}}</ref>

====Solids====
{{main|Solid geometry}}
[[File:Blue ball.png|thumb|left|In [[Euclidean space]], a ball is the volume bounded by a sphere.]]

A [[solid (mathematics)|solid]] is a three-dimensional object bounded by a closed surface; for example, a [[ball (mathematics)|ball]] is the volume bounded by a sphere.

====Manifolds====
{{main | Manifold}}

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 (topology)|neighborhood]] that is [[homeomorphism|homeomorphic]] to Euclidean space.<ref name = Munkres /> In [[differential geometry]], a [[differentiable manifold]] is a space where each neighborhood is [[diffeomorphism|diffeomorphic]] to Euclidean space.<ref name=Carmo/>

Manifolds are used extensively in physics, including in [[general relativity]] and [[string theory]].<ref>[[Shing-Tung Yau|Yau, Shing-Tung]]; Nadis, Steve (2010). ''The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions''. Basic Books. {{ISBN|978-0-465-02023-2}}.</ref>

===Angles===
{{main|Angle}}
[[File:Angle obtuse acute straight.svg|thumb|Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.]]

[[Euclid]] defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other.<ref name= EuclidAll /> In modern terms, an angle is the figure formed by two [[Ray (geometry)|rays]], called the ''sides'' of the angle, sharing a common endpoint, called the ''[[vertex (geometry)|vertex]]'' of the angle.<ref>{{SpringerEOM|id=Angle&oldid=13323|title=Angle|year=2001|last=Sidorov|first=L.A.|mode=cs1}}</ref>
The size of an angle is formalized as an [[angular measure]].

In [[Euclidean geometry]], angles are used to study [[polygon]]s and [[triangle]]s, as well as forming an object of study in their own right.<ref name= EuclidAll /> The study of the angles of a triangle or of angles in a [[unit circle]] forms the basis of [[trigonometry]].<ref>{{Cite book |last=Gelʹfand |first=I. M. |author-link=Israel Gelfand|url=https://books.google.com/books?id=ZCYtwHFVZHgC |title=Trigonometry |date=2001 |publisher=Birkhäuser |others=Mark E. Saul |isbn=0-8176-3914-4 |location=Boston |pages=1–20 |oclc=41355833 |access-date=10 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145230/https://books.google.com/books?id=ZCYtwHFVZHgC |url-status=live }}</ref>

In [[differential geometry]] and [[calculus]], the angles between [[plane curve]]s or [[space curve]]s or [[surface (geometry)|surfaces]] can be calculated using the [[derivative (calculus)|derivative]].<ref name=Stewart>[[James Stewart (mathematician)|Stewart, James]] (2012). ''Calculus: Early Transcendentals'', 7th ed., Brooks Cole Cengage Learning. {{ISBN|978-0-538-49790-9}}</ref><ref>{{cite book |last=Jost |first=Jürgen |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-540-42627-1}}.</ref>

===Measures: length, area, and volume{{anchor|Measures}}===
{{main|Length|Area|Volume}}
{{See also|Area#List of formulas|Volume#Volume formulas}}

[[Length]], [[area]], and [[volume]] describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.<ref name="Treese2018">{{cite book|author=Steven A. Treese|title=History and Measurement of the Base and Derived Units|url=https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101|year=2018|publisher=Springer International Publishing|isbn=978-3-319-77577-7|pages=101–|access-date=25 September 2019|archive-date=30 December 2019|archive-url=https://web.archive.org/web/20191230065433/https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101|url-status=live}}</ref>

In [[Euclidean geometry]] and [[analytic geometry]], the length of a line segment can often be calculated by the [[Pythagorean theorem]].<ref name="Cannon2017">{{cite book|author=James W. Cannon|author-link=James W. Cannon|title=Geometry of Lengths, Areas, and Volumes|url=https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11|year=2017|publisher=American Mathematical Soc.|isbn=978-1-4704-3714-5|page=11|access-date=25 September 2019|archive-date=31 December 2019|archive-url=https://web.archive.org/web/20191231135911/https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11|url-status=live}}</ref>

Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space.<ref name="Treese2018"/> Mathematicians have found many explicit [[Area#List of formulas|formulas for area]] and [[Volume#Formulas|formulas for volume]] of various geometric objects. In [[calculus]], area and volume can be defined in terms of [[integral]]s, such as the [[Riemann integral]]<ref name="Strang1991">{{cite book|author=Gilbert Strang|author-link=Gilbert Strang|title=Calculus|url=https://books.google.com/books?id=OisInC1zvEMC|date=1991|publisher=SIAM|isbn=978-0-9614088-2-4|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224134500/https://books.google.com/books?id=OisInC1zvEMC|url-status=live}}</ref> or the [[Lebesgue integral]].<ref name="Bear2002">{{cite book|author=H. S. Bear|title=A Primer of Lebesgue Integration|url=https://books.google.com/books?id=__AmiGnEEewC|year=2002|publisher=Academic Press|isbn=978-0-12-083971-1|access-date=25 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225032645/https://books.google.com/books?id=__AmiGnEEewC|url-status=live}}</ref>

Other geometrical measures include the [[curvature]] and [[compactness measure|compactness]].

====Metrics and measures====
{{main|Metric (mathematics)|Measure (mathematics)}}
[[File:Chinese pythagoras.jpg|thumb|right|Visual checking of the [[Pythagorean theorem]] for the (3, 4, 5) [[triangle]] as in the [[Zhoubi Suanjing]] 500–200&nbsp;BC. The Pythagorean theorem is a consequence of the [[Euclidean metric]].]]

The concept of length or distance can be generalized, leading to the idea of [[metric space|metrics]].<ref>Dmitri Burago, [[Yuri Dmitrievich Burago|Yu D Burago]], Sergei Ivanov, ''A Course in Metric Geometry'', American Mathematical Society, 2001, {{ISBN|0-8218-2129-6}}.</ref> For instance, the [[Euclidean metric]] measures the distance between points in the [[Euclidean plane]], while the [[hyperbolic metric]] measures the distance in the [[hyperbolic plane]]. Other important examples of metrics include the [[Lorentz metric]] of [[special relativity]] and the semi-[[Riemannian metric]]s of [[general relativity]].<ref>{{cite book|last=Wald|first=Robert M.|author-link=Robert Wald|title=General Relativity|publisher=University of Chicago Press|date=1984|isbn=978-0-226-87033-5|title-link=General Relativity (book)}}</ref>

In a different direction, the concepts of length, area and volume are extended by [[measure theory]], which studies methods of assigning a size or ''measure'' to [[Set (mathematics)|sets]], where the measures follow rules similar to those of classical area and volume.<ref name="Tao2011">{{cite book|author=Terence Tao|author-link=Terence Tao|title=An Introduction to Measure Theory|url=https://books.google.com/books?id=HoGDAwAAQBAJ|year=2011|publisher=American Mathematical Soc.|isbn=978-0-8218-6919-2|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ|url-status=live}}</ref>

===Congruence and similarity===
{{main|Congruence (geometry)|Similarity (geometry)}}

[[Congruence (geometry)|Congruence]] and [[Similarity (geometry)|similarity]] are concepts that describe when two shapes have similar characteristics.<ref name="Libeskind2008">{{cite book|author=Shlomo Libeskind|title=Euclidean and Transformational Geometry: A Deductive Inquiry|url=https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255|year=2008|publisher=Jones & Bartlett Learning|isbn=978-0-7637-4366-6|page=255|access-date=25 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225090248/https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255|url-status=live}}</ref> In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.<ref name="Freitag2013">{{cite book|author=Mark A. Freitag|title=Mathematics for Elementary School Teachers: A Process Approach|url=https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614|year=2013|publisher=Cengage Learning|isbn=978-0-618-61008-2|page=614|access-date=25 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228123854/https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614|url-status=live}}</ref> [[Hilbert]], in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by [[axiom]]s.

Congruence and similarity are generalized in [[transformation geometry]], which studies the properties of geometric objects that are preserved by different kinds of transformations.<ref name="Martin2012">{{cite book|author=George E. Martin|title=Transformation Geometry: An Introduction to Symmetry|url=https://books.google.com/books?id=gevlBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-5680-9|access-date=25 September 2019|archive-date=7 December 2019|archive-url=https://web.archive.org/web/20191207041210/https://books.google.com/books?id=gevlBwAAQBAJ|url-status=live}}</ref>

===Compass and straightedge constructions===
{{Main|Compass and straightedge constructions}}

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the [[Compass (drafting)|compass]] and [[ruler|straightedge]].{{efn|The ancient Greeks had some constructions using other instruments.}} 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 [[Neusis construction|neusis]], parabolas and other curves, or mechanical devices, were found.

===Rotation and orientation===
{{main|Rotation (geometry)|Orientation (geometry)}}
The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.

===Dimension===
{{broader|Dimension (mathematics)}}

[[File:Von Koch curve.gif|thumb|The [[Koch snowflake]], with [[fractal dimension]]=log4/log3 and [[topological dimension]]=1]]

Traditional geometry allowed dimensions 1 (a [[line (geometry)|line]] or curve), 2 (a [[Plane (mathematics)|plane]] or surface), and 3 (our ambient world conceived of as [[three-dimensional space]]). Furthermore, mathematicians and physicists have used [[higher dimension]]s for nearly two centuries.<ref name="Blacklock2018">{{cite book|author=Mark Blacklock|title=The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle|url=https://books.google.com/books?id=nrNSDwAAQBAJ|year=2018|publisher=Oxford University Press|isbn=978-0-19-875548-7|access-date=18 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227145318/https://books.google.com/books?id=nrNSDwAAQBAJ|url-status=live}}</ref> One example of a mathematical use for higher dimensions is the [[configuration space (physics)|configuration space]] of a physical system, which has a dimension equal to the system's [[degrees of freedom]]. For instance, the configuration of a screw can be described by five coordinates.<ref name="Joly1895">{{cite book|author=Charles Jasper Joly|title=Papers|url=https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62|year=1895|publisher=The Academy|pages=62–|access-date=18 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227195202/https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62|url-status=live}}</ref>

In [[general topology]], the concept of dimension has been extended from [[natural number]]s, to infinite dimension ([[Hilbert space]]s, for example) and positive [[real number]]s (in [[fractal geometry]]).<ref name="Temam2013">{{cite book|author=Roger Temam|title=Infinite-Dimensional Dynamical Systems in Mechanics and Physics|url=https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0645-3|page=367|access-date=18 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224015857/https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367|url-status=live}}</ref> In [[algebraic geometry]], the [[dimension of an algebraic variety]] has received a number of apparently different definitions, which are all equivalent in the most common cases.<ref name="JacobLam1994">{{cite book|author1=Bill Jacob|author2=Tsit-Yuen Lam|title=Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD Year, Berkeley, 1990–1991|url=https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111|year=1994|publisher=American Mathematical Soc.|isbn=978-0-8218-5154-8|page=111|access-date=18 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228124040/https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111|url-status=live}}</ref>

===Symmetry===
{{main |Symmetry}}

[[File:Order-3 heptakis heptagonal tiling.png|right|thumb|A [[Order-3 bisected heptagonal tiling|tiling]] of the [[Hyperbolic geometry|hyperbolic plane]]]]
The theme of [[symmetry]] in geometry is nearly as old as the science of geometry itself.<ref name="Stewart2008">{{cite book|author=Ian Stewart|author-link=Ian Stewart (mathematician)|title=Why Beauty Is Truth: A History of Symmetry|url=https://books.google.com/books?id=6akF1v7Ds3MC|date=2008|publisher=Basic Books|isbn=978-0-465-08237-7|page=14|access-date=23 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225201454/https://books.google.com/books?id=6akF1v7Ds3MC|url-status=live}}</ref> Symmetric shapes such as the [[circle]], [[regular polygon]]s and [[platonic solid]]s held deep significance for many ancient philosophers<ref name="Alexey2009">{{cite book|author=Stakhov Alexey|title=Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science|url=https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144|date=2009|publisher=World Scientific|isbn=978-981-4472-57-9|page=144|access-date=23 September 2019|archive-date=29 December 2019|archive-url=https://web.archive.org/web/20191229132952/https://books.google.com/books?id=3k7ICgAAQBAJ&pg=PA144|url-status=live}}</ref> and were investigated in detail before the time of Euclid.<ref name="Hartshorne2013" /> Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of [[Leonardo da Vinci]], [[M. C. Escher]], and others.<ref name="Hahn1998">{{cite book|author=Werner Hahn|title=Symmetry as a Developmental Principle in Nature and Art|url=https://books.google.com/books?id=wzhqDQAAQBAJ|year=1998|publisher=World Scientific|isbn=978-981-02-2363-2|access-date=23 September 2019|archive-date=1 January 2020|archive-url=https://web.archive.org/web/20200101082827/https://books.google.com/books?id=wzhqDQAAQBAJ|url-status=live}}</ref> In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. [[Felix Klein]]'s [[Erlangen program]] proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation [[group (mathematics)|group]], determines what geometry ''is''.<ref name="Cantwell2002">{{cite book|author=Brian J. Cantwell|title=Introduction to Symmetry Analysis|url=https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34|year=2002|publisher=Cambridge University Press|isbn=978-1-139-43171-2|page=34|access-date=23 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227012548/https://books.google.com/books?id=76RS2ZQ0UyUC&pg=PR34|url-status=live}}</ref> Symmetry in classical [[Euclidean geometry]] is represented by [[Congruence (geometry)|congruences]] and rigid motions, whereas in [[projective geometry]] an analogous role is played by [[collineation]]s, [[geometric transformation]]s that take straight lines into straight lines.<ref name="RosenfeldWiebe2013">{{cite book|author1=B. Rosenfeld|author2=Bill Wiebe|title=Geometry of Lie Groups|url=https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-5325-7|pages=158ff|access-date=23 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224193157/https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA158|url-status=live}}</ref> However it was in the new geometries of Bolyai and Lobachevsky, Riemann, [[William Kingdon Clifford|Clifford]] and Klein, and [[Sophus Lie]] that Klein's idea to 'define a geometry via its [[symmetry group]]' found its inspiration.<ref name="Pesic2007">{{cite book|author=Peter Pesic|title=Beyond Geometry: Classic Papers from Riemann to Einstein|url=https://books.google.com/books?id=Z67x6IOuOUAC|year=2007|publisher=Courier Corporation|isbn=978-0-486-45350-7|access-date=23 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183221/https://books.google.com/books?id=Z67x6IOuOUAC|url-status=live}}</ref> Both discrete and continuous symmetries play prominent roles in geometry, the former in [[topology]] and [[geometric group theory]],<ref name="Kaku2012">{{cite book|author=Michio Kaku|author-link=Michio Kaku|title=Strings, Conformal Fields, and Topology: An Introduction|url=https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-0397-8|page=151|access-date=23 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224015822/https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151|url-status=live}}</ref><ref name="BestvinaSageev2014">{{cite book|author1=Mladen Bestvina|author2=Michah Sageev|author3=Karen Vogtmann|author3-link=Karen Vogtmann|title=Geometric Group Theory|url=https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132|date=2014|publisher=American Mathematical Soc.|isbn=978-1-4704-1227-2|page=132|access-date=23 September 2019|archive-date=29 December 2019|archive-url=https://web.archive.org/web/20191229224624/https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132|url-status=live}}</ref> the latter in [[Lie theory]] and [[Riemannian geometry]].<ref name="Steeb1996">{{cite book|author=W-H. Steeb|title=Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra|url=https://books.google.com/books?id=rZBIDQAAQBAJ|year=1996|publisher=World Scientific Publishing Company|isbn=978-981-310-503-4|access-date=23 September 2019|archive-date=26 December 2019|archive-url=https://web.archive.org/web/20191226205450/https://books.google.com/books?id=rZBIDQAAQBAJ|url-status=live}}</ref><ref name="Misner2005">{{cite book|author=Charles W. Misner|author-link=Charles W. Misner|title=Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner|url=https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272|year=2005|publisher=Cambridge University Press|isbn=978-0-521-02139-5|page=272|access-date=23 September 2019|archive-date=26 December 2019|archive-url=https://web.archive.org/web/20191226063925/https://books.google.com/books?id=zpGZwmTJZIUC&pg=PA272|url-status=live}}</ref>

A different type of symmetry is the principle of [[Duality (projective geometry)|duality]] in [[projective geometry]], among other fields. This meta-phenomenon can roughly be described as follows: in any [[theorem]], exchange ''point'' with ''plane'', ''join'' with ''meet'', ''lies in'' with ''contains'', and the result is an equally true theorem.<ref name="Dowling1917">{{cite book|author=Linnaeus Wayland Dowling|title=Projective Geometry|url=https://archive.org/details/cu31924001523897|year=1917|publisher=McGraw-Hill book Company, Incorporated|page=[https://archive.org/details/cu31924001523897/page/n29 10]}}</ref> A similar and closely related form of duality exists between a [[vector space]] and its [[dual space]].<ref name="Gierz2006">{{cite book|author=G. Gierz|title=Bundles of Topological Vector Spaces and Their Duality|url=https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252|year=2006|publisher=Springer|isbn=978-3-540-39437-2|page=252|access-date=23 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227123430/https://books.google.com/books?id=2ml6CwAAQBAJ&pg=PA252|url-status=live}}</ref>

==Contemporary geometry==

===Euclidean geometry===
{{main|Euclidean geometry}}

[[Euclidean geometry]] is geometry in its classical sense.<ref name="ButtsBrown2012">{{cite book|author1=Robert E. Butts|author2=J.R. Brown|title=Constructivism and Science: Essays in Recent German Philosophy|url=https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127|year=2012|publisher=Springer Science & Business Media|isbn=978-94-009-0959-5|pages=127–|access-date=20 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183207/https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127|url-status=live}}</ref> As it models the space of the physical world, it is used in many scientific areas, such as [[mechanics]], [[astronomy]], [[crystallography]],<ref>{{cite book|title=Science|url=https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181|year=1886|publisher=Moses King|pages=181–|access-date=20 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227013042/https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181|url-status=live}}</ref> and many technical fields, such as [[engineering]],<ref name="Abbot2013">{{cite book|author=W. Abbot|title=Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students|url=https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6|year=2013|publisher=Springer Science & Business Media|isbn=978-94-017-2742-6|pages=6–|access-date=20 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225201450/https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6|url-status=live}}</ref> [[architecture]],<ref name="HerseyHersey2001">{{cite book|author1=George L. Hersey|title=Architecture and Geometry in the Age of the Baroque|url=https://books.google.com/books?id=F1Tl9ok-7_IC|year=2001|publisher=University of Chicago Press|isbn=978-0-226-32783-9|access-date=20 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225141623/https://books.google.com/books?id=F1Tl9ok-7_IC|url-status=live}}</ref> [[geodesy]],<ref name="VanícekKrakiwsky2015">{{cite book|author1=P. Vanícek|author2=E.J. Krakiwsky|title=Geodesy: The Concepts|url=https://books.google.com/books?id=1Mz-BAAAQBAJ|year=2015|publisher=Elsevier|isbn=978-1-4832-9079-9|page=23|access-date=20 September 2019|archive-date=31 December 2019|archive-url=https://web.archive.org/web/20191231233050/https://books.google.com/books?id=1Mz-BAAAQBAJ|url-status=live}}</ref> [[aerodynamics]],<ref name="CummingsMorton2015">{{cite book|author1=Russell M. Cummings|author2=Scott A. Morton|author3=William H. Mason|author4=David R. McDaniel|title=Applied Computational Aerodynamics|url=https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449|year=2015|publisher=Cambridge University Press|isbn=978-1-107-05374-8|page=449|access-date=20 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183207/https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449|url-status=live}}</ref> and [[navigation]].<ref name="Williams1998">{{cite book|author=Roy Williams|title=Geometry of Navigation|url=https://books.google.com/books?id=yNzf7OKGLxIC|year=1998|publisher=Horwood Pub.|isbn=978-1-898563-46-4|access-date=20 September 2019|archive-date=7 December 2019|archive-url=https://web.archive.org/web/20191207041213/https://books.google.com/books?id=yNzf7OKGLxIC|url-status=live}}</ref> The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as [[point (geometry)|points]], [[Line (geometry)|lines]], [[plane (mathematics)|planes]], [[angle]]s, [[triangle]]s, [[congruence (geometry)|congruence]], [[similarity (geometry)|similarity]], [[solid figure]]s, [[circle]]s, and [[analytic geometry]].<ref name="Schmidt, W. 2002">{{Cite journal |last1=Schmidt |first1=W. |last2=Houang |first2=R. |last3=Cogan |first3=Leland S. |date=2002 |title=A Coherent Curriculum: The Case of Mathematics. |url=https://www.nifdi.org/research/journal-of-di/volume-4-no-1-winter-2004/454-a-coherent-curriculum-the-case-of-mathematics/file.html |journal=The American Educator |language=en |volume=26 |issue=2 |pages=10–26 |s2cid=118964353}}</ref>

====Euclidean vectors====
{{main|Euclidean vector}}
Euclidean vectors are used for a myriad of applications in physics and engineering, such as [[position (geometry)|position]], [[displacement (geometry)|displacement]], [[deformation (physics)|deformation]], [[velocity]], [[acceleration]], [[force]], etc.

===Differential geometry===
[[File:Hyperbolic triangle.svg|thumb|upright=1|right|[[Differential geometry]] uses tools from [[calculus]] to study problems involving curvature.]]
{{main | Differential geometry}}
[[Differential geometry]] uses techniques of [[calculus]] and [[linear algebra]] to study problems in geometry.<ref name="Walschap2015">{{cite book|author=Gerard Walschap|title=Multivariable Calculus and Differential Geometry|url=https://books.google.com/books?id=cXPyCQAAQBAJ|year=2015|publisher=De Gruyter|isbn=978-3-11-036954-0|access-date=23 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227012551/https://books.google.com/books?id=cXPyCQAAQBAJ|url-status=live}}</ref> It has applications in [[physics]],<ref name="Flanders2012">{{cite book|author=Harley Flanders|title=Differential Forms with Applications to the Physical Sciences|url=https://books.google.com/books?id=U_GLN1eOKaMC|year=2012|publisher=Courier Corporation|isbn=978-0-486-13961-6|access-date=23 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183207/https://books.google.com/books?id=U_GLN1eOKaMC|url-status=live}}</ref> [[econometrics]],<ref name="MarriottSalmon2000">{{cite book|author1=Paul Marriott|author2=Mark Salmon|title=Applications of Differential Geometry to Econometrics|url=https://books.google.com/books?id=1Jjm4I5tqkUC|year=2000|publisher=Cambridge University Press|isbn=978-0-521-65116-5|access-date=23 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183207/https://books.google.com/books?id=1Jjm4I5tqkUC|url-status=live}}</ref> and [[bioinformatics]],<ref name="HePetoukhov2011">{{cite book|author1=Matthew He|author2=Sergey Petoukhov|title=Mathematics of Bioinformatics: Theory, Methods and Applications|url=https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-09952-0|page=106|access-date=23 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227163605/https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106|url-status=live}}</ref> among others.

In particular, differential geometry is of importance to [[mathematical physics]] due to [[Albert Einstein]]'s [[general relativity]] postulation that the [[universe]] is [[curvature|curved]].<ref name="Dirac2016">{{cite book|author=P.A.M. Dirac|title=General Theory of Relativity|url=https://books.google.com/books?id=qkWPDAAAQBAJ|year=2016|publisher=Princeton University Press|isbn=978-1-4008-8419-3|access-date=23 September 2019|archive-date=26 December 2019|archive-url=https://web.archive.org/web/20191226205400/https://books.google.com/books?id=qkWPDAAAQBAJ|url-status=live}}</ref> Differential geometry can either be ''intrinsic'' (meaning that the spaces it considers are [[smooth manifold]]s whose geometric structure is governed by a [[Riemannian metric]], which determines how distances are measured near each point) or ''extrinsic'' (where the object under study is a part of some ambient flat Euclidean space).<ref name="AyJost2017">{{cite book|author1=Nihat Ay|author2=Jürgen Jost|author3=Hông Vân Lê|author4=Lorenz Schwachhöfer|title=Information Geometry|url=https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185|year=2017|publisher=Springer|isbn=978-3-319-56478-4|page=185|access-date=23 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224015858/https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185|url-status=live}}</ref>

====Non-Euclidean geometry====
{{excerpt|Non-Euclidean geometry|templates=-General geometry,technical}}

===Topology===
{{main|Topology}}
[[File:Trefoil knot arb.png|thumb|right|A thickening of the [[trefoil knot]]]]
Topology is the field concerned with the properties of [[continuous mapping]]s,<ref name="Crossley2011">{{cite book|author=Martin D. Crossley|title=Essential Topology|url=https://books.google.com/books?id=QhCgVrLHlLgC|year=2011|publisher=Springer Science & Business Media|isbn=978-1-85233-782-7|access-date=24 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228094221/https://books.google.com/books?id=QhCgVrLHlLgC|url-status=live}}</ref> and can be considered a generalization of Euclidean geometry.<ref name="NashSen1988">{{cite book|author1=Charles Nash|author2=Siddhartha Sen|title=Topology and Geometry for Physicists|url=https://books.google.com/books?id=nnnNCgAAQBAJ|year=1988|publisher=Elsevier|isbn=978-0-08-057085-3|page=1|access-date=24 September 2019|archive-date=26 December 2019|archive-url=https://web.archive.org/web/20191226215609/https://books.google.com/books?id=nnnNCgAAQBAJ|url-status=live}}</ref> In practice, topology often means dealing with large-scale properties of spaces, such as [[connectedness]] and [[compact (topology)|compactness]].<ref name=Munkres />

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 [[homeomorphism]]s.<ref name="Martin1996">{{cite book|author=George E. Martin|title=Transformation Geometry: An Introduction to Symmetry|url=https://books.google.com/books?id=KW4EwONsQJgC|year=1996|publisher=Springer Science & Business Media|isbn=978-0-387-90636-2|access-date=24 September 2019|archive-date=22 December 2019|archive-url=https://web.archive.org/web/20191222024915/https://books.google.com/books?id=KW4EwONsQJgC|url-status=live}}</ref> This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include [[geometric topology]], [[differential topology]], [[algebraic topology]] and [[general topology]].<ref name="May1999">{{cite book|author=J. P. May|title=A Concise Course in Algebraic Topology|url=https://books.google.com/books?id=g8SG03R1bpgC|year=1999|publisher=University of Chicago Press|isbn=978-0-226-51183-2|access-date=24 September 2019|archive-date=23 December 2019|archive-url=https://web.archive.org/web/20191223144107/https://books.google.com/books?id=g8SG03R1bpgC|url-status=live}}</ref>

===Algebraic geometry===
{{main|Algebraic geometry}}
[[File:Calabi yau.jpg|thumb|Quintic [[Calabi–Yau manifold|Calabi–Yau threefold]]]]

Algebraic geometry is fundamentally the study by means of [[algebra]]ic methods of some geometrical shapes, called [[algebraic set]]s, and defined as common [[zero of a function|zeros]] of [[multivariate polynomial]]s.<ref name="AHartshorne2013">{{cite book|author=Robin Hartshorne|title=Algebraic Geometry|url=https://books.google.com/books?id=7z4mBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3849-0|access-date=24 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227163607/https://books.google.com/books?id=7z4mBQAAQBAJ|url-status=live}}</ref> Algebraic geometry became an autonomous subfield of geometry {{circa|1900}}, with a theorem called [[Hilbert's Nullstellensatz]] that establishes a strong correspondence between algebraic sets and [[ideal (ring theory)|ideal]]s of [[polynomial ring]]s. This led to a parallel development of algebraic geometry, and its algebraic counterpart, called [[commutative algebra]].<ref name="Dieudonne1985">{{cite book|author=Jean Dieudonné|translator=Judith D. Sally|title=History of Algebraic Geometry|url=https://books.google.com/books?id=_uhlf38jOrgC|year=1985|publisher=CRC Press|isbn=978-0-412-99371-8|access-date=24 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225090149/https://books.google.com/books?id=_uhlf38jOrgC|url-status=live}}</ref> From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by [[Alexander Grothendieck]] of [[scheme theory]], which allows using [[algebraic topology|topological methods]], including [[cohomology theory|cohomology theories]] in a purely algebraic context.<ref name="Dieudonne1985" /> Scheme theory allowed to solve many difficult problems not only in geometry, but also in [[number theory]]. [[Wiles' proof of Fermat's Last Theorem]] is a famous example of a long-standing problem of [[number theory]] whose solution uses scheme theory and its extensions such as [[stack (mathematics)|stack theory]]. One of seven [[Millennium Prize problems]], the [[Hodge conjecture]], is a question in algebraic geometry.<ref name="CarlsonCarlson2006">{{cite book|author1=James Carlson|author2=James A. Carlson|author3=Arthur Jaffe|author4=Andrew Wiles|title=The Millennium Prize Problems|url=https://books.google.com/books?id=7wJIPJ80RdUC|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3679-8|access-date=24 September 2019|archive-date=30 May 2016|archive-url=https://web.archive.org/web/20160530210333/https://books.google.com/books?id=7wJIPJ80RdUC|url-status=live}}</ref>

Algebraic geometry has applications in many areas, including [[cryptography]]<ref name="HoweLauter2017">{{cite book|author1=Everett W. Howe|author2=Kristin E. Lauter|author2-link=Kristin Lauter|author3=Judy L. Walker|author3-link=Judy L. Walker|title=Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016|url=https://books.google.com/books?id=bPM-DwAAQBAJ|year=2017|publisher=Springer|isbn=978-3-319-63931-4|access-date=24 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227195203/https://books.google.com/books?id=bPM-DwAAQBAJ|url-status=live}}</ref> and [[string theory]].<ref name="MarinoThaddeus2008">{{cite book|author1=Marcos Marino|author2=Michael Thaddeus|author3=Ravi Vakil|title=Enumerative Invariants in Algebraic Geometry and String Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6–11, 2005|url=https://books.google.com/books?id=mb1qCQAAQBAJ|year=2008|publisher=Springer|isbn=978-3-540-79814-9|access-date=24 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227041441/https://books.google.com/books?id=mb1qCQAAQBAJ|url-status=live}}</ref>

===Complex geometry===
{{Main|Complex geometry}}
[[Complex geometry]] studies the nature of geometric structures modelled on, or arising out of, the [[complex plane]].<ref>{{Cite book |last=Huybrechts |first=Daniel |url=https://books.google.com/books?id=eZPCfJlHkXMC |title=Complex geometry : an introduction |date=2005 |publisher=Springer |isbn=9783540266877 |location=Berlin |oclc=209857590 |access-date=10 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145147/https://books.google.com/books?id=eZPCfJlHkXMC |url-status=live }}</ref><ref>Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.</ref><ref>{{Cite book |last=Wells |first=R. O. Jr. |url=https://books.google.com/books?id=aZXAs9Vu14cC |title=Differential analysis on complex manifolds |date=2008 |publisher=Springer-Verlag |others=O. García-Prada |isbn=9780387738918 |edition=3rd |series=Graduate Texts in Mathematics |volume=65 |location=New York |doi=10.1007/978-0-387-73892-5 |oclc=233971394 |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145230/https://books.google.com/books?id=aZXAs9Vu14cC |url-status=live }}</ref> Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of [[several complex variables]], and has found applications to [[string theory]] and [[Mirror symmetry (string theory)|mirror symmetry]].<ref>
Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., ... & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.</ref>

Complex geometry first appeared as a distinct area of study in the work of [[Bernhard Riemann]] in his study of [[Riemann surface]]s.<ref>Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.
</ref><ref>Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.</ref><ref>{{Cite book |last=Donaldson |first=S. K.|author-link=Simon Donaldson |url=https://www.worldcat.org/oclc/861200296 |title=Riemann surfaces |date=2011 |isbn=978-0-19-154584-9 |location=Oxford |oclc=861200296 |access-date=9 September 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301145222/https://www.worldcat.org/title/861200296 |url-status=live |publisher=Oxford University Press }}</ref> Work in the spirit of Riemann was carried out by the [[Italian school of algebraic geometry]] in the early 1900s. Contemporary treatment of complex geometry began with the work of [[Jean-Pierre Serre]], who introduced the concept of [[sheaf (mathematics)|sheaves]] to the subject, and illuminated the relations between complex geometry and algebraic geometry.<ref>[[Jean-Pierre Serre|Serre, J. P.]] (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.</ref><ref>[[Jean-Pierre Serre|Serre, J. P.]] (1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42).</ref>
The primary objects of study in complex geometry are [[complex manifold]]s, [[complex algebraic varieties]], and [[complex analytic varieties]], and [[holomorphic vector bundles]] and [[coherent sheaves]] over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and [[Calabi–Yau manifold]]s, and these spaces find uses in string theory. In particular, [[worldsheet]]s of strings are modelled by Riemann surfaces, and [[superstring theory]] predicts that the extra 6 dimensions of 10 dimensional [[spacetime]] may be modelled by Calabi–Yau manifolds.

===Discrete geometry===
{{Main|Discrete geometry}}
[[File:Closepacking.svg|thumb|Discrete geometry includes the study of various [[sphere packing]]s.]]
[[Discrete geometry]] is a subject that has close connections with [[convex geometry]].<ref name="Matoušek2013">{{cite book|author=Jiří Matoušek|author-link=Jiří Matoušek (mathematician)|title=Lectures on Discrete Geometry|url=https://books.google.com/books?id=K0fhBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4613-0039-7|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227013417/https://books.google.com/books?id=K0fhBwAAQBAJ|url-status=live}}</ref><ref name="Zong2006">{{cite book|author=Chuanming Zong|title=The Cube – A Window to Convex and Discrete Geometry|url=https://books.google.com/books?id=Ola6htFUQ1IC|year=2006|publisher=Cambridge University Press|isbn=978-0-521-85535-8|access-date=25 September 2019|archive-date=23 December 2019|archive-url=https://web.archive.org/web/20191223150837/https://books.google.com/books?id=Ola6htFUQ1IC|url-status=live}}</ref><ref name="Gruber2007">{{cite book|author=Peter M. Gruber|title=Convex and Discrete Geometry|url=https://books.google.com/books?id=bSZKAAAAQBAJ|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-71133-9|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224175343/https://books.google.com/books?id=bSZKAAAAQBAJ|url-status=live}}</ref> It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of [[sphere packing]]s, [[triangulation (geometry)|triangulations]], the Kneser-Poulsen conjecture, etc.<ref name="DevadossO'Rourke2011">{{cite book|author1=Satyan L. Devadoss|author1-link=Satyan Devadoss|author2=Joseph O'Rourke|author2-link=Joseph O'Rourke (professor)|title=Discrete and Computational Geometry|url=https://books.google.com/books?id=InJL6iAaIQQC|year=2011|publisher=Princeton University Press|isbn=978-1-4008-3898-1|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227194659/https://books.google.com/books?id=InJL6iAaIQQC|url-status=live}}</ref><ref name="Bezdek2010">{{cite book|author=Károly Bezdek|author-link=Károly Bezdek|title=Classical Topics in Discrete Geometry|url=https://books.google.com/books?id=Tov0d9VMOfMC|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-0600-7|access-date=25 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228051643/https://books.google.com/books?id=Tov0d9VMOfMC|url-status=live}}</ref> It shares many methods and principles with [[combinatorics]].

===Computational geometry===
{{main|Computational geometry}}
[[Computational geometry]] deals with [[algorithm]]s and their [[implementation (computer science)|implementations]] for manipulating geometrical objects. Important problems historically have included the [[travelling salesman problem]], [[minimum spanning tree]]s, [[hidden-line removal]], and [[linear programming]].<ref name="PreparataShamos2012">{{cite book|author1=Franco P. Preparata|author1-link=Franco P. Preparata|author2=Michael I. Shamos|author2-link=Michael Ian Shamos|title=Computational Geometry: An Introduction|url=https://books.google.com/books?id=_p3eBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-1098-6|access-date=25 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228093733/https://books.google.com/books?id=_p3eBwAAQBAJ|url-status=live}}</ref>

Although being a young area of geometry, it has many applications in [[computer vision]], [[image processing]], [[computer-aided design]], [[medical imaging]], etc.<ref name="GuYau2008">{{cite book|author1=Xianfeng David Gu|author2=Shing-Tung Yau|title=Computational Conformal Geometry|url=https://books.google.com/books?id=4FDvAAAAMAAJ|year=2008|publisher=International Press|isbn=978-1-57146-171-1|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224054942/https://books.google.com/books?id=4FDvAAAAMAAJ|url-status=live}}</ref>

===Geometric group theory===
{{main|Geometric group theory}}
[[Image:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b'']]
[[Geometric group theory]] studies [[Group action|group actions]] on objects that are regarded as geometric (significantly, isometric actions on [[Metric space|metric spaces]]) to study [[finitely generated group]]s, often involving large-scale geometric techniques.<ref name="Löh2017">{{cite book|author=Clara Löh|title=Geometric Group Theory: An Introduction|url=https://books.google.com/books?id=1AxEDwAAQBAJ|year=2017|publisher=Springer|isbn=978-3-319-72254-2|access-date=25 September 2019|archive-date=29 December 2019|archive-url=https://web.archive.org/web/20191229132923/https://books.google.com/books?id=1AxEDwAAQBAJ|url-status=live}}</ref> It is closely connected to [[low-dimensional topology]], such as in [[Grigori Perelman]]'s proof of the [[Geometrization conjecture]], which included the proof of the [[Poincaré conjecture]], a [[Millennium Prize Problems|Millennium Prize Problem]].<ref name="MorganTian2014">{{cite book|author1=John Morgan|author2=Gang Tian|title=The Geometrization Conjecture|url=https://books.google.com/books?id=Qv2cAwAAQBAJ|year=2014|publisher=American Mathematical Soc.|isbn=978-0-8218-5201-9|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224030537/https://books.google.com/books?id=Qv2cAwAAQBAJ|url-status=live}}</ref>

Group actions on their [[Cayley graph|Cayley graphs]] are foundational examples of isometric group actions. Other major topics include [[quasi-isometry|quasi-isometries]], [[Gromov-hyperbolic group]]s and their generalizations ([[Relatively hyperbolic group|relatively]] and [[Acylindrically hyperbolic group|acylindrically hyperbolic groups]]), [[Free group|free groups]] and [[Out(Fn)|their automorphisms]], [[Bass–Serre theory|groups acting on trees]], various notions of nonpositive curvature for groups ([[CAT(0) group|CAT(0) groups]], [[Dehn function|Dehn functions]], [[Automatic group|automaticity]]...), [[right angled Artin group]]s, and topics close to [[combinatorial group theory]] such as [[small cancellation theory]] and algorithmic problems (e.g. the [[Word problem for groups|word]], [[Conjugacy problem|conjugacy]], and [[Group isomorphism problem|isomorphism problems]]). Other group-theoretic topics like [[Mapping class group of a surface|mapping class groups]], [[Kazhdan's property (T)|property (T)]], [[Solvable group|solvability]], [[Amenable group|amenability]] and [[Lattice (discrete subgroup)|lattices in Lie groups]] are sometimes regarded as strongly geometric as well.<ref name="Löh2017" /><ref name="Wise2012">{{cite book |author=Daniel T. Wise |url=https://books.google.com/books?id=GsTW5oQhRPkC |title=From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry |publisher=American Mathematical Soc. |year=2012 |isbn=978-0-8218-8800-1 |access-date=25 September 2019 |archive-url=https://web.archive.org/web/20191228115647/https://books.google.com/books?id=GsTW5oQhRPkC |archive-date=28 December 2019 |url-status=live}}</ref><ref>{{Cite book |url=https://academic.oup.com/princeton-scholarship-online/book/13467?login=false |title=Office Hours with a Geometric Group Theorist |date=2017-07-11 |publisher=Princeton University Press |isbn=978-0-691-15866-2 |editor-last=Margalit |editor-first=Dan |language=en |doi=10.23943/princeton/9780691158662.001.0001 |editor-last2=Clay |editor-first2=Matt}}</ref><ref>{{Cite book |last1=Druţu |first1=Cornelia|author1-link=Cornelia Druțu |last2=Kapovich |first2=Michael|author2-link=Michael Kapovich |date=2018-03-28 |title=Geometric Group Theory |url=https://www.ams.org/books/coll/063/ |access-date=2024-12-08 |publisher=American Mathematical Society |series=Colloquium Publications |volume=63 |doi=10.1090/coll/063 |isbn=978-1-4704-1104-6 |language=en}}</ref>

===Convex geometry===
{{main|Convex geometry}}

[[Convex geometry]] investigates [[convex set|convex]] shapes in the Euclidean space and its more abstract analogues, often using techniques of [[real analysis]] and [[discrete mathematics]].<ref name="Meurant2014">{{cite book|author=Gerard Meurant|title=Handbook of Convex Geometry|url=https://books.google.com/books?id=M2viBQAAQBAJ|year=2014|publisher=Elsevier Science|isbn=978-0-08-093439-6|access-date=24 September 2019|archive-date=1 September 2021|archive-url=https://web.archive.org/web/20210901183208/https://books.google.com/books?id=M2viBQAAQBAJ|url-status=live}}</ref> It has close connections to [[convex analysis]], [[optimization]] and [[functional analysis]] and important applications in [[number theory]].

Convex geometry dates back to antiquity.<ref name="Meurant2014"/> [[Archimedes]] gave the first known precise definition of convexity. The [[isoperimetric problem]], a recurring concept in convex geometry, was studied by the Greeks as well, including [[Zenodorus (mathematician)|Zenodorus]]. Archimedes, [[Plato]], [[Euclid]], and later [[Kepler]] and [[Coxeter]] all studied [[convex polytope]]s and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, [[Gaussian curvature]], [[algorithms]], [[tiling (geometry)|tilings]] and [[lattice (group)|lattice]]s.

==Applications==
Geometry has found applications in many fields, some of which are described below.

===Art===
{{main |Mathematics and art}}
[[File:Fes Medersa Bou Inania Mosaique2.jpg|thumb|Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations]]
Mathematics and art are related in a variety of ways. For instance, the theory of [[perspective (graphical)|perspective]] showed that there is more to geometry than just the metric properties of figures: perspective is the origin of [[projective geometry]].<ref name="Richter-Gebert2011">{{cite book|author=Jürgen Richter-Gebert|title=Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry|url=https://books.google.com/books?id=F_NP8Kub2XYC|year=2011|publisher=Springer Science & Business Media|isbn=978-3-642-17286-1|access-date=25 September 2019|archive-date=29 December 2019|archive-url=https://web.archive.org/web/20191229224621/https://books.google.com/books?id=F_NP8Kub2XYC|url-status=live}}</ref>

Artists have long used concepts of [[proportionality (mathematics)|proportion]] in design. [[Vitruvius]] developed a complicated theory of ''ideal proportions'' for the human figure.<ref name="Elam2001">{{cite book|author=Kimberly Elam|title=Geometry of Design: Studies in Proportion and Composition|url=https://books.google.com/books?id=JXIEz2XYnp8C|year=2001|publisher=Princeton Architectural Press|isbn=978-1-56898-249-6|access-date=25 September 2019|archive-date=31 December 2019|archive-url=https://web.archive.org/web/20191231135913/https://books.google.com/books?id=JXIEz2XYnp8C|url-status=live}}</ref> These concepts have been used and adapted by artists from [[Michelangelo]] to modern comic book artists.<ref name="Guigar2004">{{cite book|author=Brad J. Guigar|title=The Everything Cartooning Book: Create Unique And Inspired Cartoons For Fun And Profit|url=https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82|year=2004|publisher=Adams Media|isbn=978-1-4405-2305-2|pages=82–|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227032210/https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82|url-status=live}}</ref>

The [[golden ratio]] is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.<ref name="Livio2008">{{cite book|author=Mario Livio|title=The Golden Ratio: The Story of PHI, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C&pg=PA166|year=2008|publisher=Crown/Archetype|isbn=978-0-307-48552-6|page=166|access-date=25 September 2019|archive-date=30 December 2019|archive-url=https://web.archive.org/web/20191230093236/https://books.google.com/books?id=bUARfgWRH14C&pg=PA166|url-status=live}}</ref>

[[Tiling (geometry)|Tilings]], or tessellations, have been used in art throughout history. [[Islamic art]] makes frequent use of tessellations, as did the art of [[M. C. Escher]].<ref name="EmmerSchattschneider2007">{{cite book|author1=Michele Emmer|author2=Doris Schattschneider|author2-link=Doris Schattschneider|title=M. C. Escher's Legacy: A Centennial Celebration|url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107|year=2007|publisher=Springer|isbn=978-3-540-28849-7|page=107|access-date=25 September 2019|archive-date=22 December 2019|archive-url=https://web.archive.org/web/20191222200130/https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107|url-status=live}}</ref> Escher's work also made use of [[hyperbolic geometry]].

[[Cézanne]] advanced the theory that all images can be built up from the [[sphere]], the [[cone]], and the [[cylinder]]. This is still used in art theory today, although the exact list of shapes varies from author to author.<ref name="CapitoloSchwab2004">{{cite book|author1=Robert Capitolo|author2=Ken Schwab|title=Drawing Course 101|url=https://archive.org/details/drawingcourse1010000capi|url-access=registration|year=2004|publisher=Sterling Publishing Company, Inc.|isbn=978-1-4027-0383-6|page=[https://archive.org/details/drawingcourse1010000capi/page/22 22]}}</ref><ref name="Gelineau2011">{{cite book|author=Phyllis Gelineau|title=Integrating the Arts Across the Elementary School Curriculum|url=https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55|year=2011|publisher=Cengage Learning|isbn=978-1-111-30126-2|page=55|access-date=25 September 2019|archive-date=7 December 2019|archive-url=https://web.archive.org/web/20191207041800/https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55|url-status=live}}</ref>

===Architecture===
{{main|Mathematics and architecture|Architectural geometry}}

Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.<ref name="CeccatoHesselgren2016">{{cite book|author1=Cristiano Ceccato|author2=Lars Hesselgren|author3=Mark Pauly|author4=Helmut Pottmann, Johannes Wallner|title=Advances in Architectural Geometry 2010|url=https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6|year=2016|publisher=Birkhäuser|isbn=978-3-99043-371-3|page=6|access-date=25 September 2019|archive-date=25 December 2019|archive-url=https://web.archive.org/web/20191225201452/https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6|url-status=live}}</ref><ref name="Pottmann2007">{{cite book|author=Helmut Pottmann|title=Architectural geometry|url=https://books.google.com/books?id=bIceAQAAIAAJ|year=2007|publisher=Bentley Institute Press|isbn=978-1-934493-04-5|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224030536/https://books.google.com/books?id=bIceAQAAIAAJ|url-status=live}}</ref> Applications of geometry to architecture include the use of [[projective geometry]] to create [[forced perspective]],<ref name="MoffettFazio2003">{{cite book|author1=Marian Moffett|author2=Michael W. Fazio|author3=Lawrence Wodehouse|title=A World History of Architecture|url=https://books.google.com/books?id=IFMohetegAcC&pg=PT371|year=2003|publisher=Laurence King Publishing|isbn=978-1-85669-371-4|page=371|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227145458/https://books.google.com/books?id=IFMohetegAcC&pg=PT371|url-status=live}}</ref> the use of [[conic section]]s in constructing domes and similar objects,<ref name="HerseyHersey2001" /> the use of [[tessellations]],<ref name="HerseyHersey2001"/> and the use of symmetry.<ref name="HerseyHersey2001"/>

===Physics===
{{main|Mathematical physics}}

The field of [[astronomy]], especially as it relates to mapping the positions of [[star]]s and [[planet]]s on the [[celestial sphere]] and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.<ref name="GreenGreen1985">{{cite book|author1=Robin M. Green|author2=Robin Michael Green|title=Spherical Astronomy|url=https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1|year=1985|publisher=Cambridge University Press|isbn=978-0-521-31779-5|page=1|access-date=25 September 2019|archive-date=21 December 2019|archive-url=https://web.archive.org/web/20191221211420/https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1|url-status=live}}</ref>

[[Riemannian geometry]] and [[pseudo-Riemannian]] geometry are used in [[general relativity]].<ref name="Alekseevskiĭ2008">{{cite book|author=Dmitriĭ Vladimirovich Alekseevskiĭ|title=Recent Developments in Pseudo-Riemannian Geometry|url=https://books.google.com/books?id=K6-TgxMKu4QC|year=2008|publisher=European Mathematical Society|isbn=978-3-03719-051-7|access-date=25 September 2019|archive-date=28 December 2019|archive-url=https://web.archive.org/web/20191228115649/https://books.google.com/books?id=K6-TgxMKu4QC|url-status=live}}</ref> [[String theory]] makes use of several variants of geometry,<ref name="YauNadis2010">{{cite book|author1=Shing-Tung Yau|author2=Steve Nadis|title=The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions|url=https://books.google.com/books?id=M40Ytp8Os_gC|year=2010|publisher=Basic Books|isbn=978-0-465-02266-3|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224015855/https://books.google.com/books?id=M40Ytp8Os_gC|url-status=live}}</ref> as does [[quantum information theory]].<ref>{{cite book|last1=Bengtsson |first1=Ingemar |last2=Życzkowski |first2=Karol |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |publisher=[[Cambridge University Press]] |edition=2nd |year=2017 |isbn=978-1-107-02625-4 |oclc=1004572791}}</ref>

===Other fields of mathematics===
[[File:Square root of 2 triangle.svg|thumb|right|The Pythagoreans discovered that the sides of a triangle could have [[Commensurability (mathematics)|incommensurable]] lengths.]]

[[Calculus]] was strongly influenced by geometry.<ref name="Boyer2012"/> For instance, the introduction of [[coordinates]] by [[René Descartes]] and the concurrent developments of [[algebra]] marked a new stage for geometry, since geometric figures such as [[plane curve]]s could now be represented [[analytic geometry|analytically]] in the form of functions and equations. This played a key role in the emergence of [[infinitesimal calculus]] in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.<ref name="FlandersPrice2014">{{cite book|author1=Harley Flanders|author2=Justin J. Price|title=Calculus with Analytic Geometry|url=https://books.google.com/books?id=5abiBQAAQBAJ|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6240-6|access-date=25 September 2019|archive-date=24 December 2019|archive-url=https://web.archive.org/web/20191224175037/https://books.google.com/books?id=5abiBQAAQBAJ|url-status=live}}</ref><ref name="RogawskiAdams2015">{{cite book|author1=Jon Rogawski|author2=Colin Adams|title=Calculus|url=https://books.google.com/books?id=OWeZBgAAQBAJ|year=2015|publisher=W. H. Freeman|isbn=978-1-4641-7499-5|access-date=25 September 2019|archive-date=1 January 2020|archive-url=https://web.archive.org/web/20200101083409/https://books.google.com/books?id=OWeZBgAAQBAJ|url-status=live}}</ref>

Another important area of application is [[number theory]].<ref name="Lozano-Robledo2019">{{cite book|author=Álvaro Lozano-Robledo|title=Number Theory and Geometry: An Introduction to Arithmetic Geometry|url=https://books.google.com/books?id=ESiODwAAQBAJ|year=2019|publisher=American Mathematical Soc.|isbn=978-1-4704-5016-8|access-date=25 September 2019|archive-date=27 December 2019|archive-url=https://web.archive.org/web/20191227145316/https://books.google.com/books?id=ESiODwAAQBAJ|url-status=live}}</ref> In [[ancient Greece]] the [[Pythagoreans]] considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.<ref name="Sangalli2009">{{cite book|author=Arturo Sangalli|title=Pythagoras' Revenge: A Mathematical Mystery|url=https://archive.org/details/pythagorasreveng0000sang|url-access=registration|year= 2009|publisher=Princeton University Press|isbn=978-0-691-04955-7|page=[https://archive.org/details/pythagorasreveng0000sang/page/57 57]}}</ref> Since the 19th century, geometry has been used for solving problems in number theory, for example through the [[geometry of numbers]] or, more recently, [[scheme theory]], which is used in [[Wiles's proof of Fermat's Last Theorem]].<ref name="CornellSilverman2013">{{cite book|author1=Gary Cornell|author2=Joseph H. Silverman|author3=Glenn Stevens|title=Modular Forms and Fermat's Last Theorem|url=https://books.google.com/books?id=jD3TBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-1974-3|access-date=25 September 2019|archive-date=30 December 2019|archive-url=https://web.archive.org/web/20191230181409/https://books.google.com/books?id=jD3TBwAAQBAJ|url-status=live}}</ref>

==See also==
{{portal|Mathematics}}
{{main category|Geometry}}

;Lists
* [[List of geometers]]
** [[:Category:Algebraic geometers]]
** [[:Category:Differential geometers]]
** [[:Category:Geometers]]
** [[:Category:Topologists]]
* [[List of formulas in elementary geometry]]
* [[List of geometry topics]]
* [[List of important publications in mathematics#Geometry|List of important publications in geometry]]
* [[Lists of mathematics topics]]

;Related topics
* [[Descriptive geometry]]
* ''[[Flatland]]'', a book written by [[Edwin Abbott Abbott]] about two- and [[three-dimensional space]], to understand the concept of four dimensions
* [[List of interactive geometry software]]

;Other applications
* [[Molecular geometry]]

==Notes==
{{notelist}}

==References==
{{reflist|30em}}

===Sources===
{{refbegin|30em}}
* {{cite book |last=Boyer |first=C.B. |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second edition, revised by [[Uta Merzbach|Uta C. Merzbach]] |location=New York |publisher=Wiley |year=1991 |orig-year=1989 |isbn=978-0-471-54397-8 |url-access=registration |url=https://archive.org/details/historyofmathema00boye }}
* {{cite book| last=Cooke| first=Roger| year=2005| title=The History of Mathematics| place=New York| publisher=Wiley-Interscience| isbn=978-0-471-44459-6}}
* {{cite book| last=Hayashi| first=Takao| chapter=Indian Mathematics| year=2003| editor-last=Grattan-Guinness| editor-first=Ivor| title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences| volume=1| pages=118–130| place=Baltimore, MD| publisher=The [[Johns Hopkins University Press]]| isbn=978-0-8018-7396-6}}
* {{cite book| last=Hayashi| first=Takao| year=2005| chapter=Indian Mathematics| pages=360–375| editor-last=Flood| editor-first=Gavin| title=The Blackwell Companion to Hinduism| place=Oxford| publisher=[[Basil Blackwell]]| isbn=978-1-4051-3251-0}}
{{refend}}

==Further reading==
{{refbegin|30em}}
* {{cite book|ref=none|author=[[Jay Kappraff]]|url= http://www.worldscientific.com/worldscibooks/10.1142/8952|title=A Participatory Approach to Modern Geometry|year=2014|publisher=World Scientific Publishing|doi=10.1142/8952|zbl=1364.00004 |isbn=978-981-4556-70-5}}
* {{cite book|author=Nikolai I. Lobachevsky|title=Pangeometry|others=translator and editor: A. Papadopoulos|series=Heritage of European Mathematics Series|volume=4|publisher=European Mathematical Society|year=2010}}
* {{cite book|ref=none|author=[[Leonard Mlodinow]]|title=Euclid's Window – The Story of Geometry from Parallel Lines to Hyperspace|year=2002|edition=UK|publisher=Allen Lane|isbn=978-0-7139-9634-0}}
{{refend}}

==External links==
{{Sister project links|Geometry}}
{{Wikibooks}}
{{Library resources box |by=no |onlinebooks=no |others=no |about=yes |label=Geometry}}
*{{Cite EB1911 |wstitle=Geometry |volume=11 |pages=675–736 |short=1}}
* A [[v:Geometry|geometry]] course from [[v:|Wikiversity]]
* [http://www.8foxes.com/ ''Unusual Geometry Problems'']
* [http://mathforum.org/library/topics/geometry/ ''The Math Forum'' – Geometry]
** [http://mathforum.org/geometry/k12.geometry.html ''The Math Forum'' – K–12 Geometry]
** [http://mathforum.org/geometry/coll.geometry.html ''The Math Forum'' – College Geometry]
** [http://mathforum.org/advanced/geom.html ''The Math Forum'' – Advanced Geometry]
* [http://precedings.nature.com/documents/2153/version/1/ Nature Precedings – ''Pegs and Ropes Geometry at Stonehenge'']
* [https://web.archive.org/web/20060906203141/http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html ''The Mathematical Atlas'' – Geometric Areas of Mathematics]
* [https://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "4000 Years of Geometry"], lecture by Robin Wilson given at [[Gresham College]], 3 October 2007 (available for MP3 and MP4 download as well as a text file)
** [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry] at the Stanford Encyclopedia of Philosophy
* [http://www.ics.uci.edu/~eppstein/junkyard/topic.html The Geometry Junkyard]
* [http://www.mathopenref.com Interactive geometry reference with hundreds of applets]
* [https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm Dynamic Geometry Sketches (with some Student Explorations)]
* [http://www.khanacademy.org/?video=ca-geometry--area--pythagorean-theorem#california-standards-test-geometry Geometry classes] at [[Khan Academy]]
{{Geometry}}
{{Areas of mathematics | state=collapsed}}

{{Authority control}}

[[Category:Geometry| ]]