geometry

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}}){{cite web |title=Geometry - Formulas, Examples {{!}} Plane and Solid Geometry |url=https://cuemath.com/geometry |access-date=2023-08-31 |website=Cuemath |language=en}} is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.{{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}} Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a 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 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 Earth geodesy and to navigate the oceans since antiquity.}} which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.{{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}}

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.{{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}} 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, 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 manifolds 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, 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

File:Westerner and Arab practicing geometry 15th century manuscript.jpg and an Arab practicing geometry in the 15th century]]

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.{{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 }}{{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}}. 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 Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ({{Circa|1890 BC}}), and the 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.{{Harv|Boyer|1991|loc="Egypt" p. 19}} Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries.{{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}} South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.{{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}}{{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}}

In the 7th century BC, the 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.{{Harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 43}} Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, {{ISBN|0-03-029558-0}}. though the statement of the theorem has a long history.{{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 }}{{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 }} Eudoxus (408–{{Circa|355 BC}}) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 92}} 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 Elements, widely considered the most successful and influential textbook of all time,{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 119}} 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.{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 104}} 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.Howard Eves, 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...." 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 summation of an infinite series, and gave remarkably accurate approximations of pi.{{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}} He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.

File:Woman teaching geometry.jpg, ({{Circa|1310}}).]]

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 Sulba Sutras.{{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 }}

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 $(a,b,c)$ with the property: $a^2+b^2=c^2$ . Thus, $3^2+4^2=5^2$ , $8^2+15^2=17^2$ , $12^2+35^2=37^2$ etc.}} which are particular cases of Diophantine equations.{{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."

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."{{Harv|Hayashi|2005|p=371}} Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.

Brahmagupta wrote his astronomical work Brahmasphutasiddhanta 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).{{Harv|Hayashi|2003|pp=121–122}} 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 triangles (i.e. triangles with rational sides and rational areas).

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.{{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}}{{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."". Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.{{MacTutor Biography|id=Al-Mahani|title=Al-Mahani|mode=cs1}} Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.{{MacTutor Biography|id=Thabit|title=Al-Sabi Thabit ibn Qurra al-Harrani|mode=cs1}} Omar Khayyam (1048–1131) found geometric solutions to cubic equations.{{MacTutor Biography|id=Khayyam|title=Omar Khayyam|mode=cs1}} The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, 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.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."}}

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).{{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}} This was a necessary precursor to the development of calculus and a precise quantitative science of physics.{{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}} The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).{{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}} Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.{{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}}

Two developments in geometry in the 19th century changed the way it had been studied previously.{{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}} These were the discovery of 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 systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.{{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}}

Main concepts

The following are some of the most important concepts in geometry.{{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}}

=Axioms=

File:Parallel postulate en.svg]]

Euclid took an abstract approach to geometry in his Elements,{{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}} one of the most influential books ever written.{{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}} Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes.{{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}} 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.{{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}} 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{{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}} 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.{{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}}

=Spaces and subspaces=

==Points==

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",Euclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press {{ISBN|1-888009-18-7}}. or in synthetic geometry. In modern mathematics, they are generally defined as elements of a set called 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.{{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}}{{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}} One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

==Lines==

Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". 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,{{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}} 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.{{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 }} In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.{{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}}

==Planes==

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; 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 topological surface without reference to distances or angles;{{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}} it can be studied as an affine space, where collinearity and ratios can be studied but not distances;{{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 }} it can be studied as the complex plane using techniques of complex analysis;{{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 }} and so on.

==Curves==

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

In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.

==Surfaces==

File:Sphere wireframe.svg

A surface is a two-dimensional object, such as a sphere or paraboloid.Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." {{ISBN|978-0-321-57056-7}}. In differential geometry{{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 }} and topology, surfaces are described by two-dimensional 'patches' (or neighborhoods) that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.{{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-Verlag |isbn=978-3-540-63293-1 |zbl=0945.14001}}

==Solids==

File:Blue ball.png, a ball is the volume bounded by a sphere.]]

A solid is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere.

==Manifolds==

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 that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.

Manifolds are used extensively in physics, including in general relativity and string theory.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}}.

=Angles=

File:Angle obtuse acute straight.svg

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. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.{{SpringerEOM|id=Angle&oldid=13323|title=Angle|year=2001|last=Sidorov|first=L.A.|mode=cs1}}

The size of an angle is formalized as an angular measure.

In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.{{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 }}

In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.Stewart, James (2012). Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning. {{ISBN|978-0-538-49790-9}}{{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}}.

=Measures: length, area, and volume{{anchor|Measures}}=

Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.{{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}}

In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.{{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}}

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. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral{{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}} or the Lebesgue integral.{{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}}

Other geometrical measures include the curvature and compactness.

==Metrics and measures==

File:Chinese pythagoras.jpg for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 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 metrics.Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, {{ISBN|0-8218-2129-6}}. 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 metrics of general relativity.{{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)}}

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 sets, where the measures follow rules similar to those of classical area and volume.{{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}}

=Congruence and similarity=

Congruence and similarity are concepts that describe when two shapes have similar characteristics.{{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}} 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.{{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}} Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms.

Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.{{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}}

=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 and 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, parabolas and other curves, or mechanical devices, were found.

=Rotation and orientation=

The geometrical concepts of rotation and orientation define part of the placement of objects embedded in the plane or in space.

=Dimension=

File:Von Koch curve.gif, with fractal dimension=log4/log3 and topological dimension=1]]

Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries.{{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}} One example of a mathematical use for higher dimensions is the 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.{{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}}

In general topology, the concept of dimension has been extended from natural numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).{{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}} 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.{{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}}

=Symmetry=

File:Order-3 heptakis heptagonal tiling.png of the hyperbolic plane]]

The theme of symmetry in geometry is nearly as old as the science of geometry itself.{{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}} Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers{{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}} and were investigated in detail before the time of Euclid. 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.{{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}} 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, determines what geometry is.{{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}} Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines.{{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}} However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration.{{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}} Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,{{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}}{{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}} the latter in Lie theory and Riemannian geometry.{{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}}{{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}}

A different type of symmetry is the principle of 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.{{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]}} A similar and closely related form of duality exists between a vector space and its dual space.{{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}}

Contemporary geometry

=Euclidean geometry=

Euclidean geometry is geometry in its classical sense.{{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}} As it models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography,{{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}} and many technical fields, such as engineering,{{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}} architecture,{{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}} geodesy,{{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}} aerodynamics,{{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}} and navigation.{{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}} The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.{{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}}

==Euclidean vectors==

Euclidean vectors are used for a myriad of applications in physics and engineering, such as position, displacement, deformation, velocity, acceleration, force, etc.

=Differential geometry=

File:Hyperbolic triangle.svg uses tools from calculus to study problems involving curvature.]]

Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.{{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}} It has applications in physics,{{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}} econometrics,{{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}} and bioinformatics,{{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}} among others.

In particular, differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that the universe is curved.{{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}} Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds 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).{{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}}

==Non-Euclidean geometry==

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

=Topology=

File:Trefoil knot arb.png]]

Topology is the field concerned with the properties of continuous mappings,{{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}} and can be considered a generalization of Euclidean geometry.{{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}} In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.

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 homeomorphisms.{{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}} 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.{{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}}

=Algebraic geometry=

File:Calabi yau.jpg]]

Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets, and defined as common zeros of multivariate polynomials.{{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}} 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 ideals of polynomial rings. This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra.{{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}} 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 topological methods, including cohomology theories in a purely algebraic context. 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 theory. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.{{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}}

Algebraic geometry has applications in many areas, including cryptography{{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}} and string theory.{{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}}

=Complex geometry=

Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane.{{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 }}Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.{{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 }} 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.

Hori, K., Thomas, R., Katz, S., Vafa, C., Pandharipande, R., Klemm, A., ... & Zaslow, E. (2003). Mirror symmetry (Vol. 1). American Mathematical Soc.

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.

Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.{{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 }} 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 sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.Serre, J. P. (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197–278.Serre, J. P. (1956). Géométrie algébrique et géométrie analytique. In Annales de l'Institut Fourier (vol. 6, pp. 1–42).

The primary objects of study in complex geometry are complex manifolds, 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 manifolds, and these spaces find uses in string theory. In particular, worldsheets 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=

File:Closepacking.svgs.]]

Discrete geometry is a subject that has close connections with convex geometry.{{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}}{{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}}{{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}} 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 packings, triangulations, the Kneser-Poulsen conjecture, etc.{{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}}{{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}} It shares many methods and principles with combinatorics.

=Computational geometry=

Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.{{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}}

Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.{{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}}

=Geometric group theory=

Image:Cayley graph of F2.svg on two generators a and b]]

Geometric group theory studies group actions on objects that are regarded as geometric (significantly, isometric actions on metric spaces) to study finitely generated groups, often involving large-scale geometric techniques.{{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}} 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 Problem.{{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}}

Group actions on their Cayley graphs are foundational examples of isometric group actions. Other major topics include quasi-isometries, Gromov-hyperbolic groups and their generalizations (relatively and acylindrically hyperbolic groups), free groups and their automorphisms, groups acting on trees, various notions of nonpositive curvature for groups (CAT(0) groups, Dehn functions, automaticity...), right angled Artin groups, and topics close to combinatorial group theory such as small cancellation theory and algorithmic problems (e.g. the word, conjugacy, and isomorphism problems). Other group-theoretic topics like mapping class groups, property (T), solvability, amenability and lattices in Lie groups are sometimes regarded as strongly geometric as well.{{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}}{{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}}{{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}}

=Convex geometry=

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics.{{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}} It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.

Convex geometry dates back to antiquity. 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. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes 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, tilings and lattices.

Applications

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

=Art=

File:Fes Medersa Bou Inania Mosaique2.jpg

Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.{{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}}

Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.{{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}} These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.{{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}}

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.{{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}}

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.{{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}} 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.{{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]}}{{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}}

=Architecture=

Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.{{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}}{{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}} Applications of geometry to architecture include the use of projective geometry to create forced perspective,{{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}} the use of conic sections in constructing domes and similar objects, the use of tessellations, and the use of symmetry.

=Physics=

The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.{{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}}

Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.{{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}} String theory makes use of several variants of geometry,{{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}} as does quantum information theory.{{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}}

=Other fields of mathematics=

File:Square root of 2 triangle.svg lengths.]]

Calculus was strongly influenced by geometry. 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 curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.{{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}}{{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}}

Another important area of application is number theory.{{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}} In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.{{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]}} 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.{{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}}

Notes

References

=Sources=

{{cite book |last=Boyer |first=C.B. |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second edition, revised by 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}}

External links

{{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 geometry course from 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