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{{Periksaterjemahan|en|Geometry}}
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[[Berkas:Teorema de desargues.svg|thumb|right|Ilustrasi [[teorema Desargues]], hasil penting dalam [[geometri Euclidean|Euclidean]] dan [[geometri proyektif]]]]
[[Berkas:Hypercube.svg|190px|thumb|[[Tersseract]] atau [[Hiperkubus]] Salah satu bentuk geometri 4 Dimensi]]
'''Geometri''' adalah cabang [[matematika]] yang bersangkutan dengan pertanyaan [[bentuk]]. Seorang ahli matematika yang bekerja di bidang geometri disebut ''ahli geometri''. Geometri muncul secara independen di sejumlah budaya awal sebagai ilmu pengetahuan praktis tentang [[panjang]], [[luas]], dan [[volume]], dengan unsur-unsur dari ilmu matematika formal yang muncul di Barat sedini [[Thales]] (abad 6 SM). Pada abad ke-3 SM geometri dimasukkan ke dalam bentuk aksiomatik oleh [[Euclid]], yang dibantu oleh geometri Euclid, menjadi standar selama berabad-abad. [[Archimedes]] mengembangkan teknik cerdik untuk menghitung luas dan isi, dalam banyak cara mengantisipasi [[kalkulus integral]] yang modern. Bidang [[astronomi]], terutama memetakan posisi bintang dan planet pada falak dan menggambarkan hubungan antara gerakan benda langit, menjabat sebagai sumber penting masalah geometrik selama satu berikutnya dan setengah milenium. Kedua geometri dan [[astronomi]] dianggap di dunia klasik untuk menjadi bagian dari [[Quadrivium]] tersebut, subset dari tujuh seni liberal dianggap penting untuk warga negara bebas untuk menguasai.
Pengenalan [[koordinat]] oleh [[René Descartes]] dan perkembangan bersamaan aljabar menandai tahap baru untuk geometri, karena tokoh geometris, seperti [[kurva pesawat]], sekarang bisa diwakili analitis, yakni dengan fungsi dan persamaan. Hal ini memainkan peran penting dalam munculnya kalkulus pada abad ke-17. Selanjutnya, teori perspektif menunjukkan bahwa ada lebih banyak geometri dari sekadar sifat metrik angka: perspektif adalah asal geometri proyektif. Subyek geometri selanjutnya diperkaya oleh studi struktur intrinsik benda geometris yang berasal dengan Euler dan [[Gauss]] dan menyebabkan penciptaan topologi dan geometri diferensial.
Dalam waktu Euclid tidak ada perbedaan yang jelas antara ruang fisik dan ruang geometris. Sejak penemuan abad ke-19 geometri non-Euclid, konsep ruang telah mengalami transformasi radikal, dan muncul pertanyaan: mana ruang geometris paling sesuai dengan ruang fisik? Dengan meningkatnya matematika formal dalam abad ke-20, juga 'ruang' (dan 'titik', 'garis', 'bidang') kehilangan isi intuitif, jadi hari ini kita harus membedakan antara ruang fisik, ruang geometris (di mana ' ruang ',' titik 'dll masih memiliki arti intuitif mereka) dan ruang abstrak. Geometri kontemporer menganggap manifold, ruang yang jauh lebih abstrak dari ruang Euclid yang kita kenal, yang mereka hanya sekitar menyerupai pada skala kecil. Ruang ini mungkin diberkahi dengan struktur tambahan, yang memungkinkan seseorang untuk berbicara tentang panjang. Geometri modern memiliki ikatan yang kuat dengan beberapa fisika, dicontohkan oleh hubungan antara geometri pseudo-Riemann dan relativitas umum. Salah satu teori fisika termuda, teori string, juga sangat geometris dalam rasa.
Sedangkan sifat visual geometri awalnya membuatnya lebih mudah diakses daripada bagian lain dari matematika, seperti aljabar atau teori bilangan, bahasa geometrik juga digunakan dalam konteks yang jauh dari tradisional, asal Euclidean nya (misalnya, dalam geometri fraktal dan geometri aljabar).
== Geometri awal ==
[[Berkas:Models of four platonic solids.JPG|jmpl|Model empat padatan Platonik]]
Catatan paling awal mengenai geometri dapat ditelusuri hingga ke zaman [[Mesir kuno]], peradaban [[Lembah Sungai Indus]] dan [[Babilonia]]. [[Peradaban|Peradaban-peradaban]] ini diketahui memiliki keahlian dalam [[drainase]] rawa, [[irigasi]], pengendalian [[banjir]] dan pendirian bangunan-bangunan besar. Kebanyakan geometri Mesir kuno dan Babilonia terbatas hanya pada perhitungan [[panjang]] ruas-ruas [[garis (geometri)|garis]], [[luas]], dan [[volume]].
Salah satu teori awal mengenai geometri dikatakan oleh [[Plato]] dalam dialog [[Timaeus]] (360SM) bahwa alam semesta terdiri dari 4 elemen: [[tanah]], [[air]], [[udara]] dan [[api]]. Hal tersebut tersebut dimaksud untuk menggambarkan kondisi material [[padat]], [[cair]], [[gas]] dan [[plasma]]. Hal ini mendasari bentuk-bentuk geometri: tetrahedron, [[kubus]](hexahedron), octahedron, dan icosahedron di mana masing-masing bentuk tersebut menggambarkan elemen [[api]], [[tanah]], [[udara]] dan [[air]]. Bentuk-bentuk ini yang lalu lebih dikenal dengan nama ''Platonic Solid''.
Ada penambahan bentuk kelima yaitu Dodecahedron, yang menurut Aristoteles untuk menggambarkan elemen kelima yaitu ''[[ether]]''.
== Sejarah ==
{{Main|Sejarah geometri}}
[[Berkas:Westerner and Arab practicing geometry 15th century manuscript.jpg|right|thumb|Salah satu [[Kelompok etnis di Eropa|Eropa]] dan [[Arab]] yang berlatih geometri pada abad ke-15]]
[[Berkas:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg|right|thumb|[[Gambar depan]] versi bahasa Inggris pertama Sir Henry Billingsley dari Euclid ''[[Element (matematika)|Elemen]]'', 1570]]
Permulaan geometri paling awal yang tercatat dapat ditelusuri ke [[Mesopotamia]] kuno dan [[Mesir Kuno|Mesir]] pada milenium ke-2 SM.<ref>J. Friberg, "Metode dan tradisi matematika Babilonia. Plimpton 322, Pythagoras tiga kali lipat, dan persamaan parameter segitiga Babilonia", ''Historia Mathematica'', 8, 1981, pp. 277–318.</ref><ref>{{Cite book | edition = 2 | publisher = [[Dover Publications]] | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = Ilmu Tepat di Zaman Kuno | origyear = 1957 | year = 1969 | isbn = 978-0-486-22332-2 | url = https://books.google.com/?id=JVhTtVA2zr8C|chapter=Chap. IV Matematika dan Astronomi Mesir|pages=71–96}}.</ref> Geometri pada awalnya adalah kumpulan prinsip yang ditemukan secara empiris mengenai panjang, sudut, luas, dan volume, yang dikembangkan untuk memenuhi beberapa kebutuhan praktis dalam [[survei]], dan [[konstruksi]]. Teks geometri paling awal yang diketahui adalah [[Matematika Mesir|Mesir]] ''[[Papirus Matematika Rhind|Papirus Rhind]]'' (2000–1800 SM) dan ''[[Papirus Matematika Moskow|Papirus Moskow]] '' (sekitar 1890 SM), [[Matematika Babilonia|Tablet tanah liat Babilonia]] seperti [[Plimpton 322]] (1900 SM). Contohnya, Papirus Moskow memberikan rumus untuk menghitung volume piramida terpotong, atau [[frustum]].<ref name="Boyer 1991 loc=Mesir p. 19">{{Harv|Boyer|1991|loc="Mesir" p. 19}}</ref> Tablet tanah liat (350-50 SM) menunjukkan bahwa astronom Babilonia menerapkan prosedur [[trapesium]] untuk menghitung posisi Jupiter dan [[Perpindahan (vektor)|gerakan]] dalam kecepatan waktu. Prosedur geometris tersebut mengantisipasi [[Kalkulator Oxford]], termasuk [[teorema kecepatan rata-rata]], pada abad ke 14.<ref>{{cite journal |last=Ossendrijver |first=Mathieu |date=29 Januari 2016 |title=Para astronom Babilonia kuno menghitung posisi Jupiter dari area di bawah grafik kecepatan waktu |journal=Ilmu |volume=351 |issue=6272 |pages=482–484 |doi=10.1126/science.aad8085 |pmid=26823423|bibcode=2016Sci...351..482O }}</ref> Di selatan Mesir, [[Nubia|Nubia kuno]] membangun sistem geometri termasuk versi awal [[jam matahari]].<ref>{{cite journal|title=Gnomons di Meroë dan Trigonometri Awal|first=Leo|last=Depuydt|date=1 Januari 1998|journal=The Journal of Egyptian Archaeology|volume=84|pages=171–180|doi=10.2307/3822211|jstor=3822211}}</ref><ref>{{cite web|url=http://www.archaeology.org/online/news/nubia.html|title=Neolithic Skywatchers|date=27 Mei 1998|first=Andrew|last=Slayman|website=Archaeology Magazine Archive|access-date=17 April 2011|archive-url=https://web.archive.org/web/20110605234044/http://www.archaeology.org/online/news/nubia.html|archive-date=5 Juni 2011|url-status=live}}</ref>
Pada abad ke 7 SM, [[Matematika Yunani|Yunani]] ahli matematika [[Thales of Miletus]] menggunakan geometri untuk menyelesaikan masalah seperti menghitung tinggi piramida dan jarak kapal. Hal tersebut dikreditkan dengan penggunaan pertama dari penalaran deduktif yang diterapkan pada geometri, dengan menurunkan empat akibat wajar dari [[Teorema Thales]].<ref name="Boyer 1991 loc=Ionia dan Pythagoras p. 43"/> Pythagoras mendirikan [[Pythagoras|Sekolah Pythagoras]], yang dikreditkan dengan bukti pertama dari [[Teorema Pythagoras]],<ref>Eves, Howard, Pengantar Sejarah Matematika, Saunders, 1990, {{ISBN|0-03-029558-0}}.</ref> Padahal pernyataan teorema tersebut memiliki sejarah yang panjang.<ref>{{cite journal|title=Penemuan Ketidakbandingan oleh Hippasus dari Metapontum|author=Kurt Von Fritz|journal=The Annals of Mathematics|year=1945}}</ref><ref>{{cite journal|title=Pentagram dan Penemuan Bilangan Irasional|journal=The Two-Year College Mathematics Journal|author=James R. Choike|year=1980}}</ref> [[Eudoxus dari Cnidus|Eudoxus]] (408–355 SM) mengembangkan [[metode]], yang memungkinkan perhitungan luas dan volume gambar lengkung,<ref>{{Harv|Boyer|1991|loc="Zaman Plato dan Aristoteles" p. 92}}</ref> serta teori rasio yang menghindari masalah [[besaran yang tidak dapat dibandingkan]], yang memungkinkan geometer berikutnya untuk membuat kemajuan yang signifikan. Sekitar 300 SM, geometri direvolusi oleh Euclid, yang '' [[Elemen Euklides|Elemen]] '', secara luas dianggap sebagai buku teks paling sukses dan berpengaruh sepanjang masa,<ref>{{Harv|Boyer|1991|loc="Euclid dari Alexandria" p. 119}}</ref> diperkenalkan [[ketelitian matematika]] melalui [[metode aksiomatik]] dan merupakan contoh paling awal dari format yang masih digunakan dalam matematika saat ini, bahwa definisi, aksioma, teorema, dan bukti. Meskipun sebagian besar konten '' Elemen '' sudah diketahui, Euclid mengatur menjadi satu kerangka kerja logis yang koheran.<ref name="Boyer 1991 loc=Euclid of Alexandria p. 104">{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 104}}</ref> ''Element'' diketahui oleh semua orang terpelajar di Barat hingga pertengahan abad ke 20 dan isinya masih diajarkan di kelas geometri hingga saat ini..<ref>Howard Eves, ''Pengantar Sejarah Matematika'', Saunders, 1990, {{ISBN|0-03-029558-0}} p. 141: "Tidak ada karya, kecuali [[Bible]], yang telah digunakan secara lebih luas...."</ref> [[Archimedes]] (c. 287–212 SM) dari [[Syracuse, Italy|Syracuse]] menggunakan [[metode|metode tersebut]] untuk menghitung [[luas]] di bawah busur dari [[parabola]] dengan [[Deret (matematika)|penjumlahan dari tak terhingga pada deret]], dan memberikan perkiraan yang sangat akurat dari [[Pi]].<ref>{{cite web | title = Sejarah kalkulus | 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 | accessdate = 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 | url-status = live }}</ref> Dia juga mempelajari [[Archimedes spiral|spiral]] yang menyandang namanya dan memperoleh rumus untuk [[volume]] dari [[permukaan revolusi]].
[[Berkas:Woman teaching geometry.jpg|left|thumb|upright=.85|''Wanita mengajar geometri''. Ilustrasi di awal terjemahan abad pertengahan [[Euklides Element]], (c. 1310).]]
<!--[[Indian mathematics|Indian]] mathematicians also made many important contributions in geometry. The ''[[Satapatha Brahmana]]'' (3rd century BC) contains rules for ritual geometric constructions that are similar to the ''[[Shulba Sutras|Sulba Sutras]]''.<ref name="Staal 1999">{{Citation | 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 }}
</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]],<ref>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.</ref> 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 is 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|Brāhma Sphuṭa Siddhānta}}]]'' 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>R. Rashed (1994), ''The development of Arabic mathematics: between arithmetic and algebra'', p. 35 [[London]]</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}}</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"/> [[Omar Khayyám]] (1048–1131) found geometric solutions to [[cubic equation]]s.<ref>{{MacTutor Biography|id=Khayyam|title=Omar Khayyam}}</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 early results in [[hyperbolic geometry]], and along with their alternative postulates, such as [[Playfair's axiom]], these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including [[Witelo]] (c. 1230–c. 1314), [[Gersonides]] (1288–1344), [[Alfonso]], [[John Wallis]], and [[Giovanni Girolamo Saccheri]].<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: {{quote|"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|title=History of Analytic Geometry|url=https://books.google.com/books?id=2T4i5fXZbOYC|date=2012|publisher=Courier Corporation|isbn=978-0-486-15451-0}}</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}}</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|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}}</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}}</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}}</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" 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}}</ref>-->
== Geometri aljabar ==
{{Main|Geometri aljabar}}
[[Berkas:Togliatti surface.png|thumb|[[Permukaan Togliatti]] ini adalah [[permukaan aljabar]] derajat lima. Gambar tersebut mewakili sebagian dari [[Lokus (matematika)|lokus]] aslinya.]]
'''Geometri aljabar''' merupakan cabang [[matematika]] yang mempelajari akar dari suatu [[Polinomial|suku banyak]]. Dalam kajian modern, digunakan berbagai alat dari [[aljabar abstrak]] seperti aljabar komutatif dan [[teori kategori]]. Studi geometri aljabar dilakukan dengan mengonstruksi suatu objek matematika (misalnya, skema dan sheaf) lalu kemudian meninjau hubungannya dengan struktur yang sudah dikenal. Berbagai alat ini dibuat untuk membantu memahami permasalahan mendasar terkait geometri.<ref>{{Cite book|title=Foundations of Algebraic Geometry|last=Vakil|first=Ravi|date=2017|publisher=|isbn=|location=|pages=|url-status=live}}</ref>
Salah satu objek fundamental dalam studi geometri aljabar adalah varietas aljabarik yang merupakan manifestasi geometris dari akar suatu sistem suku banyak. Dari struktur ini, dapat dikaji berbagai kurva aljabarik seperti [[garis]], [[parabola]], [[elips]], kurva eliptik dan lain-lain.
Geometri aljabar merupakan salah satu topik sentral dalam matematika dengan berbagai topik terkait seperti analisis kompleks, [[topologi]], [[teori bilangan]], [[teori kategori]], dan lain-lain.
== Geometri dalam dimensi ==
=== Dalam dua dimensi ===
{{Lihat pula|Dua dimensi}}Geometri dalam dua dimensi adalah suatu bentuk yang berupa dua dimensi, yang berarti bangunan tersebut hanya melibatkan panjang dan lebar.<ref>{{Cite web|title=What is 2 Dimensional? - Definition, Facts & Example|url=https://www.splashlearn.com/math-vocabulary/geometry/2-dimensional|website=www.splashlearn.com|language=en|access-date=2021-12-29|archive-date=2023-03-24|archive-url=https://web.archive.org/web/20230324200506/https://www.splashlearn.com/math-vocabulary/geometry/2-dimensional|dead-url=no}}</ref>
==== Persegi ====
{{Main|Persegi}}
'''Persegi''' adalah bangun datar [[dua dimensi]] yang dibentuk oleh empat buah [[rusuk]] '''<math>(a)</math>''' yang sama panjang dan memiliki empat buah [[sudut]] yang kesemuanya adalah [[sudut siku-siku]]. Bangun ini disebut juga sebagai '''bujur sangkar'''.
==== Persegi panjang ====
{{Main|Persegi panjang}}
'''Persegi panjang''' adalah bangun datar [[dua dimensi]] yang dibentuk oleh dua pasang [[sisi]] yang masing-masing sama panjang dan [[sejajar]] dengan pasangannya, dan memiliki empat buah [[sudut]] yang kesemuanya adalah [[sudut siku-siku]].
==== Segitiga ====
{{Main|Segitiga}}
Sebuah '''segitiga''' adalah [[poligon]] dengan tiga [[Tepi (geometri)|ujung]] dan tiga simpul. Ini adalah salah satu [[bentuk]] dasar dalam geometri. Segitiga dengan simpul A, B, dan C dilambangkan <math>\triangle ABC</math>.
Dalam [[geometri Euclidean]], setiap tiga titik, ketika non-[[collinear]], menentukan segitiga unik dan sekaligus, sebuah [[Ilmu ukur bidang|bidang]] unik (yaitu [[ruang Euclidean]] dua dimensi). Dengan kata lain, hanya ada satu bidang yang mengandung segitiga itu, dan setiap segitiga terkandung dalam beberapa bidang. Jika seluruh geometri hanya [[bidang Euclidean]], hanya ada satu bidang dan semua segitiga terkandung di dalamnya; namun, dalam ruang Euclidean berdimensi lebih tinggi, ini tidak lagi benar.
==== Trapesium ====
{{Main|Trapesium (geometri)}}
'''Trapesium''' adalah bangun datar [[dua dimensi]] yang dibentuk oleh empat buah [[rusuk]] yang dua di antaranya saling [[sejajar]] namun tidak sama panjang.Trapesium termasuk jenis [[bangun datar]] [[segi empat]] yang mempunyai ciri khusus.
==== Jajar genjang ====
{{Main|Jajar genjang}}
[[Berkas:Jajaran genjang.JPG|jmpl|Jajar genjang{{br}}dengan alas '''<math>a</math>''' dan tinggi '''<math>t</math>''']]
'''Jajar genjang''' atau '''jajaran genjang''' ({{lang-en|parallelogram}}) adalah bangun datar [[dua dimensi]] yang dibentuk oleh dua pasang [[rusuk (geometri)|rusuk]] yang masing-masing sama panjang dan [[sejajar]] dengan pasangannya, dan memiliki dua pasang [[sudut]] yang masing-masing sama besar dengan sudut di hadapannya. Jajar genjang termasuk turunan segiempat yang mempunyai ciri khusus. Jajar genjang dengan empat rusuk yang sama panjang disebut [[belah ketupat]].
==== Lingkaran ====
{{Main|Lingkaran}}
'''Lingkaran''' adalah [[bentuk]] yang terdiri dari semua titik dalam [[Bidang (geometri)|bidang]] yang berjarak tertentu dari titik tertentu, pusat; ekuivalennya adalah kurva yang dilacak oleh titik yang bergerak dalam bidang sehingga jaraknya dari titik tertentu adalah [[Konstan (matematika)|konstan]]. Jarak antara titik mana pun dari lingkaran dan pusat disebut jari-jari. Artikel ini adalah tentang lingkaran dalam geometri Euclidean, dan, khususnya, bidang Euclidean, kecuali jika dinyatakan sebaliknya.
Secara khusus, sebuah lingkaran adalah [[kurva]] tertutup sederhana yang membagi pesawat menjadi dua wilayah: interior dan eksterior. Dalam penggunaan sehari-hari, istilah "lingkaran" dapat digunakan secara bergantian untuk merujuk pada batas gambar, atau keseluruhan gambar termasuk bagian dalamnya; dalam penggunaan teknis yang ketat, lingkaran hanyalah batas dan seluruh gambar disebut [[Cakram (matematika)|cakram]].
Lingkaran juga dapat didefinisikan sebagai jenis elips khusus di mana dua fokus bertepatan dan eksentrisitasnya adalah 0, atau bentuk dua dimensi yang melingkupi area per satuan perimeter kuadrat, menggunakan kalkulus variasi.
==== Elips ====
[[Berkas:Ellipse-conic.svg|thumb|Elips (merah) diperoleh sebagai persimpangan kerucut dengan bidang miring.]]
[[Berkas:Ellipse-def0.svg|300px|thumb|Elips: notasi]]
[[Berkas:Ellipse-var.svg|thumb|Elips: contoh dengan eksentrisitas yang meningkat]]
'''Elips''' atau '''oval yang beraturan''' adalah gambar yang menyerupai [[lingkaran]] yang telah dipanjangkan ke satu arah. Elips adalah salah satu contoh dari [[irisan kerucut]] dan dapat didefinisikan sebagai [[Lokus (matematika)|lokus]] dari semua titik, dalam satu bidang, yang memiliki jumlah jarak yang sama dari dua titik tetap yang telah ditentukan sebelumnya (disebut '''[[Fokus (matematika)|fokus]]''').
Dalam bahasa Indonesia, elips atau oval yang beraturan juga sering dikenal istilah sepadan, yakni ''bulat lonjong'' (atau ''lonjong''<ref>{{Cite web|title=Arti kata lonjong - Kamus Besar Bahasa Indonesia (KBBI) Online|url=https://kbbi.web.id/lonjong|website=kbbi.web.id|access-date=2021-12-29|archive-date=2023-07-31|archive-url=https://web.archive.org/web/20230731131945/https://kbbi.web.id/lonjong|dead-url=no}}</ref> saja)'', bulat bujur<ref name=":0">{{Cite web|title=Arti kata bulat - Kamus Besar Bahasa Indonesia (KBBI) Online|url=https://kbbi.web.id/bulat|website=kbbi.web.id|access-date=2021-12-29|archive-date=2023-06-11|archive-url=https://web.archive.org/web/20230611014225/https://www.kbbi.web.id/bulat|dead-url=no}}</ref>'', dan ''bulat panjang''.<ref name=":0"/>
=== Dalam tiga dimensi ===
{{Lihat pula|Tiga dimensi}}
=== Dalam empat dimensi ===
{{Lihat pula|Empat dimensi}}
== Konsep penting dalam geometri ==
Berikut ini adalah beberapa konsep terpenting dalam geometri.<ref name="Tabak 2014 xiv"/><ref name="Schmidt, W. 2002"/><ref name="Kline1990">{{cite book|author=Morris Kline|title=Pemikiran Matematika Dari Zaman Kuno ke Modern: Volume 3|url=https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010|date=Maret 1990|publisher=Oxford University Press, USA|isbn=978-0-19-506137-6|pages=1010–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145216/https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1010|dead-url=no}}</ref>
=== Aksioma ===
[[Berkas:Parallel postulate en.svg|thumb|right|Ilustrasi [[postulat paralel]] Euclid]]
{{Lihat pula|Geometri Euklides|Aksioma}}
[[Euclid]] mengambil pendekatan abstrak untuk geometri di [[Elemen Euklides|Elements]],<ref name="Katz2000">{{cite book|author=Victor J. Katz|title=Menggunakan Sejarah untuk Mengajar Matematika: Perspektif Internasional|url=https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45|date=21 September 2000|publisher=Cambridge University Press|isbn=978-0-88385-163-0|pages=45–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145221/https://books.google.com/books?id=CbZ_YsdCmP0C&pg=PA45|dead-url=no}}</ref> salah satu buku paling berpengaruh yang pernah ditulis.<ref name="Berlinski2014">{{cite book|author=David Berlinski|title=Raja Ruang Tak Terbatas: Euclid dan Elemen-elemennya|url=https://archive.org/details/kingofinfinitesp00davi|url-access=registration|date=8 April 2014|publisher=Basic Books|isbn=978-0-465-03863-3}}</ref> Euklides memperkenalkan [[aksioma]], atau [[postulat]] tertentu, yang mengekspresikan sifat utama atau bukti dengan sendirinya dari titik, garis, dan bidang.<ref name="Hartshorne2013">{{cite book|author=Robin Hartshorne|title=Geometri: Euclid and Beyond|url=https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29|date=11 November 2013|publisher=Springer Science & Business Media|isbn=978-0-387-22676-7|pages=29–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145229/https://books.google.com/books?id=C5fSBwAAQBAJ&pg=PA29|dead-url=no}}</ref> Untuk melanjutkan untuk secara ketat menyimpulkan properti lain dengan penalaran matematika. Ciri khas pendekatan geometri Euclid adalah ketelitiannya, dan kemudian dikenal sebagai geometri ''aksiomatik'' atau ''[[geometri sintetik|sintetik]]''.<ref name="HerbstFujita2017">{{cite book|author1=Pat Herbst|author2=Taro Fujita|author3=Stefan Halverscheid|author4=Michael Weiss|title=Pembelajaran dan Pengajaran Geometri di Sekolah Menengah: Perspektif Modeling|url=https://books.google.com/books?id=6DAlDwAAQBAJ&pg=PA20|date=16 March 2017|publisher=Taylor & Francis|isbn=978-1-351-97353-3|pages=20–}}</ref> Pada awal abad ke-19, penemuan [[geometri non-Euclidean]] oleh [[Nikolai Ivanovich Lobachevsky]] (1792–1856), [[János Bolyai]] (1802–1860), [[Carl Friedrich Gauss]] (1777–1855) dan yang lainnya<ref name="Yaglom2012">{{cite book|author=I.M. Yaglom|title=Geometri Non-Euclidean Sederhana dan Dasar Fisiknya: Catatan Dasar Geometri Galilea dan Prinsip Relativitas Galilea|url=https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4612-6135-3|pages=6–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145221/https://books.google.com/books?id=FyToBwAAQBAJ&pg=PR6|dead-url=no}}</ref> menyebabkan kebangkitan minat dalam disiplin tersebut pada abad ke-20, [[David Hilbert]] (1862–1943) menggunakan penalaran aksiomatik dalam upaya untuk memberikan dasar geometri modern.<ref name="Holme2010">{{cite book|author=Audun Holme|title=Geometri: Warisan Budaya Kami|url=https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254|date=23 September 2010|publisher=Springer Science & Business Media|isbn=978-3-642-14441-7|pages=254–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145144/https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA254|dead-url=no}}</ref>
===Titik===
{{Main|Titik (geometri)}}
Titik yang dianggap sebagai objek fundamental dalam geometri Euclidean. Mereka telah didefinisikan dalam berbagai cara, termasuk definisi Euclid sebagai 'yang tidak memiliki bagian'<ref name=EuclidAll>''Elemen Euclid - Semua tiga belas buku dalam satu volume'', Berdasarkan terjemahan Heath, Green Lion Press {{ISBN|1-888009-18-7}}.</ref> dan melalui penggunaan aljabar atau set bersarang.<ref>{{cite journal |last= Clark|first=Bowman L. |date= Jan 1985|title= Individu dan Titik geometri|journal= Notre Dame Journal of Formal Logic|volume= 26|issue=1 |pages= 61–75|doi= 10.1305/ndjfl/1093870761|doi-access= free}}</ref> Banyak bidang geometri, seperti geometri analitik, geometri diferensial, dan topologi, semua objek dianggap dibangun dari titik. Namun demikian, ada beberapa studi geometri tanpa mengacu pada titik.<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=Buku Pegangan geometri insiden: bangunan dan fondasi|publisher=North-Holland|pages=1015–1031}}</ref>
===Garis===
{{main|Garis (geometri)}}
[[Euclid]] mendeskripsikan sebuah garis sebagai "panjang tanpa lebar" yang "terletak sama terhadap titik-titik pada dirinya sendiri".<ref name=EuclidAll /> Dalam matematika modern, mengingat banyaknya geometri, konsep garis terkait erat dengan cara menggambarkan geometri. Misalnya, dalam [[geometri analitik]], garis pada bidang sering didefinisikan sebagai himpunan titik yang koordinatnya memenuhi [[persamaan linier]] tertentu,<ref>{{cite book|author=[[John Casey (mathematician)|John Casey]]|year=1885|url= https://archive.org/details/cu31924001520455|title=Geometri Analitik Bagian Titik, Garis, Lingkaran, dan Kerucut}}</ref> tetapi dalam pengaturan yang lebih abstrak, seperti [[geometri kejadian]], garis mungkin merupakan objek independen, berbeda dari kumpulan titik yang terletak di atasnya.<ref>Buekenhout, Francis (1995), ''Buku Pegangan Geometri Insiden: Bangunan dan Fondasi'', Elsevier B.V.</ref> Dalam geometri diferensial, [[geodesik]] adalah generalisasi gagasan garis menjadi [[ruang melengkung]].<ref>{{cite web|url=https://www.oxforddictionaries.com/definition/english/geodesic|title=geodesik - definisi geodesik dalam bahasa Inggris dari kamus Oxford|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=live}}</ref>
===Bidang===
{{main|Bidang (geometri)}}
[[Bidang (geometri)|Bidang]] adalah permukaan datar dua dimensi yang memanjang jauh tak terhingga.<ref name= EuclidAll /> Bidang digunakan di setiap bidang geometri. Contohnya, bidang dapat dipelajari sebagai [[Permukaan (topologi)|permukaan topologi]] tanpa mengacu pada jarak atau sudut;<ref name=Munkres>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref> dapat dipelajari sebagai [[ruang affine]], di mana collinearity dan rasio dapat dipelajari tetapi bukan jarak;<ref>Szmielew, Wanda. 'Dari affine ke geometri Euclidean: Pendekatan aksiomatik.' Springer, 1983.</ref> itu dapat dipelajari sebagai [[bidang kompleks]] menggunakan teknik [[analisis kompleks]];<ref>Ahlfors, Lars V. ''Analisis kompleks: pengantar teori fungsi analitik dari satu variabel kompleks.'' New York, London (1953).</ref> dan seterusnya.
===Sudut===
{{main|Sudut}}
[[Euclid]] mendefinisikan bidang [[sudut]] sebagai kemiringan satu sama lain, dalam bidang, dari dua garis yang saling bertemu, dan tidak terletak lurus satu sama lain.<ref name= EuclidAll /> Dalam istilah modern, sudut adalah sosok yang dibentuk oleh dua [[Sinar (geometri)|sinar]], disebut ''sisi'' dari sudut, berbagi titik akhir yang sama, disebut ''[[Simpul (geometri)|simpul]]'' dari sudut.<ref>{{SpringerEOM|id=Angle&oldid=13323|title=Angle|year=2001|last=Sidorov|first=L.A.}}</ref>
[[Berkas:Angle obtuse acute straight.svg|thumb|Sudut tajam (a), tumpul (b), dan lurus (c). Sudut lancip dan tumpul juga dikenal sebagai sudut miring.]]
Dalam [[geometri Euklides]], sudut digunakan untuk mempelajari [[poligon]] dan [[segitiga]], serta membentuk sebuah objek belajar dengan sendirinya.<ref name= EuclidAll /> Studi tentang sudut segitiga atau [[Sudut dalam dan luar|sudut dalam]] sebuah [[lingkaran satuan]] membentuk dasar dari [[trigonometri]].<ref>Gelʹfand, Izrailʹ Moiseevič, dan Mark Saul. "Trigonometri." 'Trigonometri'. Birkhäuser Boston, 2001. 1–20.</ref>
Dalam [[geometri diferensial]] dan [[kalkulus]], sudut antara [[kurva bidang]] atau [[kurva ruang]] atau [[Permukaan (geometri)|permukaan]] dapat dihitung menggunakan [[Turunan (kalkulus)|turunan]].<ref name="Stewart">[[James Stewart (matematikawan)|Stewart, James]] (2012). ''Kalkulus: Transendental Awal'', 7th ed., Brooks Cole Cengage Learning. {{ISBN|978-0-538-49790-9}}</ref><ref>{{citation |last=Jost |first=Jürgen |title=Analisis Geometri dan Geometri Riemannian |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-540-42627-1}}.</ref>-->
===Kurva===
{{main|Kurva (geometri)}}
[[Kurva (geometri)|Kurva]] adalah objek 1 dimensi yang bisa lurus (seperti garis) atau tidak; kurva dalam ruang 2 dimensi disebut [[kurva bidang]] dan kurva dalam ruang 3 dimensi disebut.<ref>Baker, Henry Frederick. Prinsip geometri. Vol. 2. CUP Archive, 1954.</ref>
Dalam topologi, kurva didefinisikan dari fungsi pada interval bilangan real ke ruang lain.<ref name = Munkres /> Dalam geometri diferensial, definisi yang sama digunakan, tetapi fungsi penentu harus dapat terdiferensiasi <ref name = Carmo /> Studi geometri aljabar [[kurva aljabar]], yang didefinisikan sebagai [[varietas aljabar]] dari [[Dimensi Variasi Aljabar|dimensi]] satu.<ref name = mumford />
===Permukaan===
{{main|Permukaan (matematika)}}
[[Berkas:Sphere wireframe.svg|thumb|upright=0.85|Bola adalah permukaan yang dapat didefinisikan secara parametrik (dengan {{nowrap|''x'' {{=}} ''r'' sin ''θ'' cos ''φ'',}} {{nowrap|''y'' {{=}} ''r'' sin ''θ'' sin ''φ'',}} {{nowrap|''z'' {{=}} ''r'' cos ''θ'')}} atau secara implisit (by {{nowrap|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> − ''r''<sup>2</sup> {{=}} 0}}.)]]
[[Permukaan (matematika)|Permukaan]] adalah objek dua dimensi, seperti bola atau parabola.<ref>Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." {{ISBN|978-0321570567}}.</ref> Dalam [[geometri diferensial]]<ref name=Carmo>Do Carmo, Manfredo Perdigao, dan Manfredo Perdigao Do Carmo. Geometri diferensial dari kurva dan permukaan. Vol. 2. Englewood Cliffs: Prentice-hall, 1976.</ref> dan [[topologi]],<ref name=Munkres /> permukaan dijelaskan oleh 'tambalan' dua dimensi (atau [[Lingkungan (topologi)|lingkungan]]) yang dirangkai oleh [[diffeomorphism]] atau [[homeomorphism]], masing-masing. Dalam geometri aljabar, permukaan dijelaskan oleh [[persamaan polinomial]].<ref name=mumford>{{cite book |last=Mumford |first=David |authorlink=David Mumford |title=Buku Merah Varietas dan Skema Termasuk Ceramah Michigan tentang Kurva dan Jacobian Mereka |url=https://archive.org/details/redbookofvarieti0002mumf |edition=2nd |year=1999 |publisher=[[Springer Science+Business Media|Springer-Verlag]] |isbn=978-3-540-63293-1 |zbl=0945.14001}}</ref>-->
===Manifold===
{{main|Manifold}}
[[manifold]] adalah generalisasi dari konsep kurva dan permukaan. Dalam [[topologi]], monifold adalah [[ruang topologi]] di mana setiap titik memiliki [[Lingkungan (topologi)|lingkungan]] yaitu [[homeomorfisme|homeomorfik]] ke ruang Euklides.<ref name = Munkres /> Dalam [[geometri diferensial]], [[monifold terdiferensiasi]] adalah ruang di mana setiap tetangga [[diffeomorphism|diffeomorphic]] terhadap dimensi pada ruang Euklides.<ref name=Carmo/>
Manifold digunakan secara luas dalam fisika, termasuk dalam [[relativitas umum]] dan [[teori string]].<ref>Yau, Shing-Tung; Nadis, Steve (2010). Bentuk Ruang Dalam: Teori String dan Geometri Dimensi Tersembunyi Alam Semesta. Buku Dasar. {{ISBN|978-0-465-02023-2}}.</ref>
===Panjang, luas, dan volume===
{{main|Panjang|Luas|Volume}}
{{See also|Luas#Daftar rumus|Volume#Rumus volume}}
[[Panjang]], [[luas]], dan [[volume]] mendeskripsikan ukuran atau luas suatu objek masing-masing dalam satu dimensi, dua dimensi, dan tiga dimensi.<ref name="Treese2018">{{cite book|author=Steven A. Treese|title=Sejarah dan Pengukuran Basis dan Unit Turunan|url=https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101|date=17 May 2018|publisher=Springer International Publishing|isbn=978-3-319-77577-7|pages=101–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145145/https://books.google.com/books?id=bi1bDwAAQBAJ&pg=PA101|dead-url=no}}</ref>
Dalam [[geometri Euklides]] dan [[geometri analitik]], panjang [[ruas garis]] sering kali dapat dihitung dengan [[Teorema Pythagoras]].<ref name="Cannon2017">{{cite book|author=James W. Cannon|title=Geometri Panjang, Luas, dan Volume|url=https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11|date=16 November 2017|publisher=American Mathematical Soc.|isbn=978-1-4704-3714-5|pages=11|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145145/https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA11|dead-url=no}}</ref>
Luas dan volume dapat didefinisikan sebagai besaran fundamental yang terpisah dari panjang, atau dapat dijelaskan dan dihitung dalam istilah panjang dalam bidang atau ruang 3 dimensi.<ref name="Treese2018"/> Matematikawan telah menemukan banyak [[Luas#Daftar rumus|rumus untuk luas]] dan [[Volume#Rumus volume|rumus untuk volume]] dari berbagai objek geometri. Dalam [[kalkulus]], luas dan volume dapat didefinisikan dalam [[integral]] s, seperti [[integral Riemann]]<ref name="Strang1991">{{cite book|author=Gilbert Strang|title=Kalkulus|url=https://books.google.com/books?id=OisInC1zvEMC|date=1 January 1991|publisher=SIAM|isbn=978-0-9614088-2-4|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145146/https://books.google.com/books?id=OisInC1zvEMC|dead-url=no}}</ref> atau [[Integral Lebesgue]].<ref name="Bear2002">{{cite book|author=H. S. Bear|title=Primer Integrasi Lebesgue|url=https://books.google.com/books?id=__AmiGnEEewC|year=2002|publisher=Academic Press|isbn=978-0-12-083971-1|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145156/https://books.google.com/books?id=__AmiGnEEewC|dead-url=no}}</ref>-->
====Metrik dan ukuran====
{{main|Metrik (matematika)|Ukur (matematika)}}
[[Berkas:Chinese pythagoras.jpg|thumb|right|Pemeriksaan visual [[Teorema Pythagoras]] untuk (3, 4, 5) [[segitiga]] seperti pada [[Zhoubi Suanjing]] 500–200 SM. Teorema Pythagoras adalah konsekuensi dari [[metrik Euklides]].]]
Konsep panjang atau jarak dapat digeneralisasikan, yang mengarah ke gagasan [[ruang metrik|metrik]].<ref>Dmitri Burago, [[Yuri Dmitrievich Burago|Yu D Burago]], Sergei Ivanov, ''Kursus dalam Geometri Metrik'', American Mathematical Society, 2001, {{ISBN|0-8218-2129-6}}.</ref> Misalnya, [[metrik Euclidean]] mengukur jarak antar titik di [[bidang Euclidean]], sedangkan [[metrik hiperbolik]] mengukur jarak di [[bidang hiperbolik]]. Contoh penting lainnya dari metrik termasuk [[metrik Lorentz]] dari [[relativitas khusus]] dan semi [[metrik Riemannian]] dari [[relativitas umum]].<ref>{{Citation|last=Wald|first=Robert M.|authorlink=Robert Wald|title=Relativitas umum|publisher=University of Chicago Press|date=1984|isbn=978-0-226-87033-5|title-link=General Relativity (buku)}}</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|title=An Introduction to Measure Theory|url=https://books.google.com/books?id=HoGDAwAAQBAJ|date=14 September 2011|publisher=American Mathematical Soc.|isbn=978-0-8218-6919-2|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145156/https://books.google.com/books?id=HoGDAwAAQBAJ|dead-url=no}}</ref>
===Kekongruenan dan keserupaan===
{{main|Kesesuaian (geometri)|Kesamaan (geometri)}}
[[Kesamaan (geometri)|Kesesuaian]] dan [[Kesamaan (geometri)|kesamaan]] adalah konsep yang mendeskripsikan jika dua bentuk memiliki karakteristik yang serupa.<ref name="Libeskind2008">{{cite book|author=Shlomo Libeskind|title=Euklides dan Geometri Transformasional: Penyelidikan Deduktif|url=https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255|date=12 February 2008|publisher=Jones & Bartlett Learning|isbn=978-0-7637-4366-6|page=255|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145232/https://books.google.com/books?id=et6WMlkQlFcC&pg=PA255|dead-url=no}}</ref> Dalam geometri Euclidean, kesamaan digunakan untuk mendeskripsikan objek yang memiliki bentuk yang sama, sedangkan congruence digunakan untuk mendeskripsikan objek yang memiliki ukuran dan bentuk yang sama.<ref name="Freitag2013">{{cite book|author=Mark A. Freitag|title=Matematika untuk Guru Sekolah Dasar: Pendekatan Proses|url=https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614|date=1 January 2013|publisher=Cengage Learning|isbn=978-0-618-61008-2|page=614|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145159/https://books.google.com/books?id=G4BVGFiVKG0C&pg=PA614|dead-url=no}}</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.-->
Kesamaan dan kesamaan digeneralisasikan dalam [[geometri transformasi]], yang mempelajari properti objek geometris yang dipertahankan oleh berbagai jenis transformasi.<ref name="Martin2012">{{cite book|author=George E. Martin|title=Transformasi Geometri: Pengantar Simetri|url=https://books.google.com/books?id=gevlBwAAQBAJ|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4612-5680-9|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145158/https://books.google.com/books?id=gevlBwAAQBAJ|dead-url=no}}</ref>-->
===Lukisan dengan jangka dan mistar===
{{Main|Lukisan jangka dan mistar}}
Geometer klasik memberikan perhatian khusus untuk membangun objek geometris yang telah dijelaskan dengan cara lain. Secara klasik, satu-satunya instrumen yang diperbolehkan dalam konstruksi geometris adalah [[Jangka|kompas]] dan [[penggaris|penggaris lurus]]. Selain itu, setiap konstruksi harus diselesaikan dalam jumlah langkah yang terbatas. Namun, beberapa masalah ternyata sulit atau tidak mungkin diselesaikan dengan cara ini sendiri, dan konstruksi cerdik menggunakan parabola dan kurva lainnya, serta perangkat mekanis.
===Dimensi===
{{main|Dimensi}}
[[Berkas:Von Koch curve.gif|thumb|[[Kepingan salju Koch]], dengan [[dimensi fraktal]]=log4/log3 dan [[dimensi topologi]]=1]]
Dimana geometri tradisional mengizinkan dimensi 1 (a [[garis (geometri)|garis]]), 2 (a [[Bidang (matematika)|bidang]]) dan 3 (dunia ambien kita dipahami sebagai [[ruang tiga dimensi)]]), matematikawan dan fisikawan telah menggunakan [[dimensi yang lebih tinggi]] selama hampir dua abad.<ref name="Blacklock2018">{{cite book|author=Mark Blacklock|title=Munculnya Dimensi Keempat: Pemikiran Spasial yang Lebih Tinggi di 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=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145202/https://books.google.com/books?id=nrNSDwAAQBAJ|dead-url=no}}</ref> Salah satu contoh penggunaan matematika untuk dimensi yang lebih tinggi adalah [[ruang konfigurasi (fisika)|ruang konfigurasi]] dari sistem fisik, yang memiliki dimensi yang sama dengan [[derajat bebas]]. Misalnya, konfigurasi sekrup dapat digambarkan dengan lima koordinat.<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=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145206/https://books.google.com/books?id=cOTuAAAAMAAJ&pg=PA62|dead-url=no}}</ref>
Dalam [[topologi umum]], konsep dimensi telah diperpanjang dari [[bilangan asli]], menjadi dimensi tak hingga ([[ruang Hilbert]] s, misalnya) dan positif [[bilangan real]] (dalam [[geometri fraktal]]).<ref name="Temam2013">{{cite book|author=Roger Temam|title=Sistem Dinamika Dimensi Tak Terbatas dalam Mekanika dan Fisika|url=https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367|date=11 December 2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0645-3|page=367|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145146/https://books.google.com/books?id=OB_vBwAAQBAJ&pg=PA367|dead-url=no}}</ref> Dalam [[geometri aljabar]], [[dimensi variasi aljabar]] telah menerima sejumlah definisi yang tampaknya berbeda, yang semuanya setara dalam kasus yang paling umum.<ref name="JacobLam1994">{{cite book|author1=Bill Jacob|author2=Tsit-Yuen Lam|title=Kemajuan Terbaru dalam Geometri Aljabar Nyata dan Bentuk Kuadrat: Prosiding Tahun RAGSQUAD, 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=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145147/https://books.google.com/books?id=mHwcCAAAQBAJ&pg=PA111|dead-url=no}}</ref>
===Simetri===
{{main |Simetri}}
<!--[[Berkas: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|title=Why Beauty Is Truth: A History of Symmetry|url=https://books.google.com/books?id=6akF1v7Ds3MC|date=29 April 2008|publisher=Basic Books|isbn=978-0-465-08237-7|page=14}}</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=11 September 2009|publisher=World Scientific|isbn=978-981-4472-57-9|page=144}}</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 [[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}}</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|date=23 September 2002|publisher=Cambridge University Press|isbn=978-1-139-43171-2|page=34}}</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|date=9 March 2013|publisher=Springer Science & Business Media|isbn=978-1-4757-5325-7|pages=158ff}}</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" /> 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|title=Strings, Conformal Fields, and Topology: An Introduction|url=https://books.google.com/books?id=pM8FCAAAQBAJ&pg=PA151|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4684-0397-8|page=151}}</ref><ref name="BestvinaSageev2014">{{cite book|author1=Mladen Bestvina|author2=Michah Sageev|author3=Karen Vogtmann|title=Geometric Group Theory|url=https://books.google.com/books?id=RGz1BQAAQBAJ&pg=PA132|date=24 December 2014|publisher=American Mathematical Soc.|isbn=978-1-4704-1227-2|page=132}}</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|date=30 September 1996|publisher=World Scientific Publishing Company|isbn=978-981-310-503-4}}</ref><ref name="Misner2005">{{cite book|author=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|date=20 October 2005|publisher=Cambridge University Press|isbn=978-0-521-02139-5|pages=272}}</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|date=15 November 2006|publisher=Springer|isbn=978-3-540-39437-2|page=252}}</ref>-->
==Geometri kompentasi==
===Geometri Euklides===
{{main|Geometri Euklides}}
[[Geometri Euklides]] adalah geometri dalam pengertian klasiknya.<ref name="ButtsBrown2012">{{cite book|author1=Robert E. Butts|author2=J.R. Brown|title=Konstruktivisme dan Sains: Esai dalam Filsafat Jerman Terbaru|url=https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-94-009-0959-5|pages=127–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145201/https://books.google.com/books?id=vzTqCAAAQBAJ&pg=PA127|dead-url=no}}</ref> Karena memodelkan ruang dunia fisik, ia menggunakan di banyak bidang ilmiah, seperti [[mekanika]], [[astronomi]], [[kristalografi]],<ref>{{cite book|title=Science|url=https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181|year=1886|publisher=Moses King|pages=181–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145202/https://books.google.com/books?id=xfNRAQAAMAAJ&pg=PA181|dead-url=no}}</ref> dan banyak bidang teknis, seperti [[teknik]],<ref name="Abbot2013">{{cite book|author=W. Abbot|title=Geometri Praktis dan Grafis Teknik: Buku Ajar untuk Mahasiswa Teknik dan Lainnya|url=https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6|date=11 November 2013|publisher=Springer Science & Business Media|isbn=978-94-017-2742-6|pages=6–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145205/https://books.google.com/books?id=1LDsCAAAQBAJ&pg=PP6|dead-url=no}}</ref> [[Arsitektur]],<ref name="HerseyHersey2001">{{cite book|author1=George L. Hersey|title=Arsitektur dan Geometri di Zaman Barok|url=https://books.google.com/books?id=F1Tl9ok-7_IC|date=March 2001|publisher=University of Chicago Press|isbn=978-0-226-32783-9|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145217/https://books.google.com/books?id=F1Tl9ok-7_IC|dead-url=no}}</ref> [[geodesi]],<ref name="VanícekKrakiwsky2015">{{cite book|author1=P. Vanícek|author2=E.J. Krakiwsky|title=Geodesi: Konsep|url=https://books.google.com/books?id=1Mz-BAAAQBAJ|date=3 June 2015|publisher=Elsevier|isbn=978-1-4832-9079-9|page=23|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145205/https://books.google.com/books?id=1Mz-BAAAQBAJ|dead-url=no}}</ref> [[aerodinamika]],<ref name="CummingsMorton2015">{{cite book|author1=Russell M. Cummings|author2=Scott A. Morton|author3=William H. Mason|author4=David R. McDaniel|title=Aerodinamika Komputasi Terapan|url=https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449|date=27 April 2015|publisher=Cambridge University Press|isbn=978-1-107-05374-8|page=449|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145206/https://books.google.com/books?id=gwzUBwAAQBAJ&pg=PA449|dead-url=no}}</ref> and [[navigasi]].<ref name="Williams1998">{{cite book|author=Roy Williams|title=Geometri Navigasi|url=https://books.google.com/books?id=yNzf7OKGLxIC|year=1998|publisher=Horwood Pub.|isbn=978-1-898563-46-4|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145150/https://books.google.com/books?id=yNzf7OKGLxIC|dead-url=no}}</ref> Kurikulum pendidikan wajib dari sebagian besar negara mencakup studi tentang konsep Euklides seperti [[titik (geometri)|titik]], [[Garis (geometri)|garis]], [[bidang (matematika)|bidang]], [[sudut]], [[segitiga]], [[kesesuaian (geometri)|kongruensi]], [[kesamaan (geometri)|kesamaan]].<ref name="Schmidt, W. 2002">Schmidt, W., Houang, R., & Cogan, L. (2002). "Kurikulum yang koheren". ''Pendidik Amerika'', 26(2), 1–18.</ref>
===Geometri diferensial===
[[Berkas:Hyperbolic triangle.svg|thumb|upright=1|right|[[Geometri diferensial]] menggunakan alat dari [[kalkulus]] untuk mempelajari masalah yang melibatkan kelengkungan.]]
{{main|Geometri diferensial}}
[[Geometri Diferensial]] menggunakan teknik [[kalkulus]] dan [[aljabar linier]] untuk mempelajari masalah dalam geometri.<ref name="Walschap2015">{{cite book|author=Gerard Walschap|title=Kalkulus Multivariabel dan Geometri Diferensial|url=https://books.google.com/books?id=cXPyCQAAQBAJ|date=1 July 2015|publisher=De Gruyter|isbn=978-3-11-036954-0|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145212/https://books.google.com/books?id=cXPyCQAAQBAJ|dead-url=no}}</ref> Hal tersebut memiliki aplikasi dalam [[fisika]],<ref name="Flanders2012">{{cite book|author=Harley Flanders|title=Bentuk Diferensial dengan Aplikasi untuk Ilmu Fisika|url=https://books.google.com/books?id=U_GLN1eOKaMC|date=26 April 2012|publisher=Courier Corporation|isbn=978-0-486-13961-6|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145212/https://books.google.com/books?id=U_GLN1eOKaMC|dead-url=no}}</ref> [[ekonometrik]],<ref name="MarriottSalmon2000">{{cite book|author1=Paul Marriott|author2=Mark Salmon|title=Aplikasi Geometri Diferensial ke Ekonometrika|url=https://books.google.com/books?id=1Jjm4I5tqkUC|date=31 Agustus 2000|publisher=Cambridge University Press|isbn=978-0-521-65116-5|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145157/https://books.google.com/books?id=1Jjm4I5tqkUC|dead-url=no}}</ref> dan [[bioinformatika]],<ref name="HePetoukhov2011">{{cite book|author1=Matthew He|author2=Sergey Petoukhov|title=Matematika Bioinformatika: Teori, Metode dan Aplikasi|url=https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106|date=16 March 2011|publisher=John Wiley & Sons|isbn=978-1-118-09952-0|page=106|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145147/https://books.google.com/books?id=Skov-LJ1mmQC&pg=PA106|dead-url=no}}</ref> diantara yang lain.
Khususnya, geometri diferensial penting bagi [[fisika matematika]] karena postulasi [[relativitas umum]] [[Albert Einstein]] bahwa [[alam semesta]] adalah [[kelengkungan|lengkung]].<ref name="Dirac2016">{{cite book|author=P.A.M. Dirac|title=Teori Relativitas Umum|url=https://books.google.com/books?id=qkWPDAAAQBAJ|date=10 August 2016|publisher=Princeton University Press|isbn=978-1-4008-8419-3|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145158/https://books.google.com/books?id=qkWPDAAAQBAJ|dead-url=no}}</ref> Geometri diferensial dapat berupa ''intrinsik'' (artinya ruang yang dianggapnya adalah [[lipatan halus]] yang struktur geometrisnya diatur oleh [[metrik Riemannian]], yang menentukan bagaimana jarak diukur di dekat setiap titik) atau ''ekstrinsik'' (di mana objek yang diteliti adalah bagian dari beberapa ruang Euclide datar ambien).<ref name="AyJost2017">{{cite book|author1=Nihat Ay|author2=Jürgen Jost|author3=Hông Vân Lê|author4=Lorenz Schwachhöfer|title=Geometri Informasi|url=https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185|date=25 August 2017|publisher=Springer|isbn=978-3-319-56478-4|page=185|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145148/https://books.google.com/books?id=pLsyDwAAQBAJ&pg=PA185|dead-url=no}}</ref>
====Geometri non-Euklides====
{{main|Geometri non-Euklides}}
Geometri Euklides bukanlah satu-satunya bentuk geometri historis yang dipelajari. [[Geometri bola]] telah lama digunakan oleh astronom, astrolog, dan navigator.<ref name="Rosenfeld2012">{{cite book|author=Boris A. Rosenfeld|title=Sejarah Geometri Non-Euclidean: Evolusi Konsep Ruang Geometri|url=https://books.google.com/books?id=3wzSBwAAQBAJ|date=8 September 2012|publisher=Springer Science & Business Media|isbn=978-1-4419-8680-1|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145148/https://books.google.com/books?id=3wzSBwAAQBAJ|dead-url=no}}</ref>
[[Immanuel Kant]] berpendapat bahwa hanya ada satu, ''mutlak'', geometri, yang diketahui benar ''a priori'' oleh fakultas pikiran batin: Geometri Euklides adalah [[sintetik a priori]].<ref>Kline (1972) "Pemikiran matematis dari zaman kuno hingga modern", Oxford University Press, p. 1032. Kant tidak menolak 'kemungkinan' logis (analitik a priori) dari geometri non-Euklides, lihat [[Jeremy Gray]], "Ide Ruang Euclidean, Non-Euklides, dan Relativistik", Oxford, 1989; p. 85. Beberapa menyiratkan bahwa, dalam terang ini, Kant sebenarnya telah ''meramalkan'' perkembangan geometri non-Euklides, lih. Leonard Nelson, "Filsafat dan Aksioma," Socratic Method and Critical Philosophy, Dover, 1965, p. 164.</ref> Pandangan ini pada awalnya agak ditantang oleh para pemikir seperti [[Saccheri]], kemudian akhirnya dibatalkan oleh penemuan revolusioner [[geometri non-Euklides]] dalam karya-karya Bolyai, Lobachevsky, dan Gauss (yang tidak pernah menerbitkan teorinya).<ref name="Sommerville1919">{{cite book|author=Duncan M'Laren Young Sommerville|title=Elemen Geometri Non-Euklides ...|url=https://books.google.com/books?id=6eASAQAAMAAJ&pg=PA15|year=1919|publisher=Open Court|pages=15ff|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145217/https://books.google.com/books?id=6eASAQAAMAAJ&pg=PA15|dead-url=no}}</ref> They demonstrated that ordinary [[Euclidean space]] is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by [[Riemann]] in his 1867 inauguration lecture ''Über die Hypothesen, welche der Geometrie zu Grunde liegen'' (''On the hypotheses on which geometry is based''),<ref>{{cite web|url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ |title=Ueber die Hypothesen, welche der Geometrie zu Grunde liegen |url-status=dead |archiveurl=https://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ |archivedate=18 March 2016 |df= }}</ref> hanya setelah kematiannya. Ide baru Riemann tentang ruang terbukti penting dalam [[teori relativitas umum]] [[Albert Einstein]]. [[Geometri Riemannian]], yang mempertimbangkan ruang yang sangat umum di mana pengertian panjang didefinisikan, adalah andalan geometri modern.<ref name="Pesic2007">{{cite book|author=Peter Pesic|title=Di luar Geometri: Makalah Klasik dari Riemann hingga Einstein|url=https://books.google.com/books?id=Z67x6IOuOUAC|date=1 January 2007|publisher=Courier Corporation|isbn=978-0-486-45350-7|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145149/https://books.google.com/books?id=Z67x6IOuOUAC|dead-url=no}}</ref>
===Topologi===
{{main|Topologi}}
<!--[[Berkas: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|date=11 February 2011|publisher=Springer Science & Business Media|isbn=978-1-85233-782-7}}</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|date=4 January 1988|publisher=Elsevier|isbn=978-0-08-057085-3|page = 1}}</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|date=20 December 1996|publisher=Springer Science & Business Media|isbn=978-0-387-90636-2}}</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|date=September 1999|publisher=University of Chicago Press|isbn=978-0-226-51183-2}}</ref>
===Geometri aljabar===
{{main|Geometri aljabar}}
<!--[[Berkas:Calabi yau.jpg|thumb|Quintic [[Calabi–Yau manifold|Calabi–Yau threefold]]]]
The field of [[algebraic geometry]] developed from the [[Cartesian geometry]] of [[co-ordinates]].<ref name="inc1905">{{cite book|title=The Encyclopedia Americana: A Universal Reference Library Comprising the Arts and Sciences, Literature, History, Biography, Geography, Commerce, Etc., of the World|url=https://books.google.com/books?id=EGEMAAAAYAAJ&pg=PT489|year=1905|publisher=Scientific American Compiling Department|pages=489–}}</ref> It underwent periodic periods of growth, accompanied by the creation and study of [[projective geometry]], [[birational geometry]], [[algebraic variety|algebraic varieties]], and [[commutative algebra]], among other topics.<ref name="Dieudonne1985">{{cite book|author=Suzanne C. Dieudonne|title=History Algebraic Geometry|url=https://books.google.com/books?id=_uhlf38jOrgC|date=30 May 1985|publisher=CRC Press|isbn=978-0-412-99371-8}}</ref> From the late 1950s through the mid-1970s it had undergone major foundational development, largely due to work of [[Jean-Pierre Serre]] and [[Alexander Grothendieck]].<ref name="Dieudonne1985" /> This led to the introduction of [[scheme (algebraic geometry)|schemes]] and greater emphasis on [[algebraic topology|topological]] methods, including various [[cohomology theory|cohomology theories]]. 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}}</ref> [[Wiles' proof of Fermat's Last Theorem]] uses advanced methods of algebraic geometry for solving a long-standing problem of [[number theory]].
In general, algebraic geometry studies geometry through the use of concepts in [[commutative algebra]] such as [[multivariate polynomial]]s.<ref name="AHartshorne2013">{{cite book|author=Robin Hartshorne|title=Algebraic Geometry|url=https://books.google.com/books?id=7z4mBQAAQBAJ|date=29 June 2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3849-0}}</ref> It has applications in many areas, including [[cryptography]]<ref name="HoweLauter2017">{{cite book|author1=Everett W. Howe|author2=Kristin E. Lauter|author3=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|date=15 November 2017|publisher=Springer|isbn=978-3-319-63931-4}}</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|date=15 August 2008|publisher=Springer|isbn=978-3-540-79814-9}}</ref>-->
===Geometri kompleks===
{{Main|Geometri kompleks}}
<!--[[Complex geometry]] studies the nature of geometric structures modelled on, or arising out of, the [[complex plane]].<ref>Huybrechts, D. (2006). Complex geometry: an introduction. Springer Science & Business Media.
</ref><ref>Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.</ref><ref>Wells, R. O. N., & García-Prada, O. (1980). Differential analysis on complex manifolds (Vol. 21980). New York: Springer.</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>Donaldson, S. (2011). Riemann surfaces. 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>Serre, J. P. (1955). Faisceaux algébriques cohérents. Annals of Mathematics, 197-278.</ref><ref>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.-->
===Geometri diskrit===
{{main|Geometri diskrit}}
<!--[[Berkas: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|title=Lectures on Discrete Geometry|url=https://books.google.com/books?id=K0fhBwAAQBAJ|date=1 December 2013|publisher=Springer Science & Business Media|isbn=978-1-4613-0039-7}}</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|date=2 February 2006|publisher=Cambridge University Press|isbn=978-0-521-85535-8}}</ref><ref name="Gruber2007">{{cite book|author=Peter M. Gruber|title=Convex and Discrete Geometry|url=https://books.google.com/books?id=bSZKAAAAQBAJ|date=17 May 2007|publisher=Springer Science & Business Media|isbn=978-3-540-71133-9}}</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|author2=Joseph O'Rourke|title=Discrete and Computational Geometry|url=https://books.google.com/books?id=InJL6iAaIQQC|date=11 April 2011|publisher=Princeton University Press|isbn=978-1-4008-3898-1}}</ref><ref name="Bezdek2010">{{cite book|author=Károly Bezdek|title=Classical Topics in Discrete Geometry|url=https://books.google.com/books?id=Tov0d9VMOfMC|date=23 June 2010|publisher=Springer Science & Business Media|isbn=978-1-4419-0600-7}}</ref> It shares many methods and principles with [[combinatorics]].-->
===Geometri komputasi===
{{main|Geometri komputasi}}
<!--[[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|author2=Michael I. Shamos|title=Computational Geometry: An Introduction|url=https://books.google.com/books?id=_p3eBwAAQBAJ|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4612-1098-6}}</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}}</ref>
===Teori grup geometris===
{{main|Geometric group theory}}
[[Gambar:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b'']]
[[Geometric group theory]] uses large-scale geometric techniques to study [[finitely generated group]]s.<ref name="Löh2017">{{cite book|author=Clara Löh|title=Geometric Group Theory: An Introduction|url=https://books.google.com/books?id=1AxEDwAAQBAJ|date=19 December 2017|publisher=Springer|isbn=978-3-319-72254-2}}</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|date=21 May 2014|publisher=American Mathematical Soc.|isbn=978-0-8218-5201-9}}</ref>
Geometric group theory often revolves around the [[Cayley graph]], which is a geometric representation of a group. Other important topics include [[quasi-isometry|quasi-isometries]], [[Gromov-hyperbolic group]]s, and [[right angled Artin group]]s.<ref name="Löh2017"/><ref name="Wise2012">{{cite book|author=Daniel T. Wise|title=From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry|url=https://books.google.com/books?id=GsTW5oQhRPkC|year=2012|publisher=American Mathematical Soc.|isbn=978-0-8218-8800-1}}</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|date=28 June 2014|publisher=Elsevier Science|isbn=978-0-08-093439-6}}</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.-->
==Aplikasi==
Geometri telah menemukan aplikasi di banyak bidang, beberapa di antaranya dijelaskan di bawah ini.
===Seni===
{{main|Matematika dan seni}}
[[Berkas:Fes Medersa Bou Inania Mosaique2.jpg|thumb|Bou Inania Madrasa, Fes, Maroko, ubin mosaik zellige membentuk tessellations geometris yang rumit]]
Matematika dan seni terkait dalam berbagai cara. Contohnya, teori [[Perspektif (grafis)|perspektif]] menunjukkan bahwa geometri lebih dari sekadar properti metrik dari sebuah figur.: perspektif adalah asal mula [[geometri proyektif]].<ref name="Richter-Gebert2011">{{cite book|author=Jürgen Richter-Gebert|title=Perspektif tentang Geometri Proyektif: Tur Terpandu Melalui Geometri Nyata dan Kompleks|url=https://books.google.com/books?id=F_NP8Kub2XYC|date=4 February 2011|publisher=Springer Science & Business Media|isbn=978-3-642-17286-1|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145147/https://books.google.com/books?id=F_NP8Kub2XYC|dead-url=no}}</ref>
Seniman telah lama menggunakan konsep [[Proporsionalitas (matematika)|proporsi]] dalam desain. [[Vitruvius]] mengembangkan teori rumit tentang ''proporsi ideal'' untuk sosok manusia.<ref name="Elam2001">{{cite book|author=Kimberly Elam|title=Geometri Desain: Studi dalam Proporsi dan Komposisi|url=https://books.google.com/books?id=JXIEz2XYnp8C|year=2001|publisher=Princeton Architectural Press|isbn=978-1-56898-249-6|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145149/https://books.google.com/books?id=JXIEz2XYnp8C|dead-url=no}}</ref> Konsep tersebut telah digunakan dan diadaptasi oleh seniman dari [[Michelangelo]] hingga seniman komik modern.<ref name="Guigar2004">{{cite book|author=Brad J. Guigar|title=The Everything Cartooning Book: Buat Kartun Unik Dan Terinspirasi Untuk Kesenangan Dan Untung|url=https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82|date=4 November 2004|publisher=Adams Media|isbn=978-1-4405-2305-2|pages=82–|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145157/https://books.google.com/books?id=7gftDQAAQBAJ&pg=PT82|dead-url=no}}</ref>
[[Rasio emas]] adalah proporsi tertentu yang memiliki peran kontroversial dalam seni. Sering diklaim sebagai rasio panjang yang paling estetis, sering dikatakan sebagai rasio panjang karya seni terkenal, meskipun contoh yang paling dapat diandalkan dan tidak ambigu dibuat dengan sengaja oleh seniman yang mengetahui legenda tersebut.<ref name="Livio2008">{{cite book|author=Mario Livio|title=Rasio Emas: Kisah PHI, Angka Paling Mengagumkan di Dunia|url=https://books.google.com/books?id=bUARfgWRH14C&pg=PA166|date=12 November 2008|publisher=Crown/Archetype|isbn=978-0-307-48552-6|page=166|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145159/https://books.google.com/books?id=bUARfgWRH14C&pg=PA166|dead-url=no}}</ref>
[[Ubin (geometri)|Ubin]], atau tessellations, telah digunakan dalam seni sepanjang sejarah. [[Seni Islam]] sering menggunakan tessellation, seperti halnya seni [[Escher]].<ref name="EmmerSchattschneider2007">{{cite book|author1=Michele Emmer|author2=Doris Schattschneider|title=M.C. Warisan Escher: Perayaan Seratus Tahun|url=https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107|date=8 Mei 2007|publisher=Springer|isbn=978-3-540-28849-7|page=107|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145200/https://books.google.com/books?id=5DDyBwAAQBAJ&pg=PA107|dead-url=no}}</ref> Karya Escher juga memanfaatkan [[geometri hiperbolik]].
[[Cézanne]] mengajukan teori bahwa semua gambar dapat dibangun dari [[bola]], [[kerucut]], dan [[Tabung (geometri)|tabung]]. Ini masih digunakan dalam teori seni hari ini, meskipun daftar pasti bentuk bervariasi dari penulis ke penulis.<ref name="CapitoloSchwab2004">{{cite book|author1=Robert Capitolo|author2=Ken Schwab|title=Kursus Menggambar 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=Mengintegrasikan Seni di Seluruh Kurikulum Sekolah Dasar|url=https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55|date=1 January 2011|publisher=Cengage Learning|isbn=978-1-111-30126-2|pages=55|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145200/https://books.google.com/books?id=1Ib0mUl_VhwC&pg=PA55|dead-url=no}}</ref>
===Arsitektur===
{{main|Matematika dan arsitektur|Geometri arsitektur}}
Geometri memiliki banyak aplikasi dalam arsitektur. Faktanya, telah dikatakan bahwa geometri merupakan inti dari desain arsitektur.<ref name="CeccatoHesselgren2016">{{cite book|author1=Cristiano Ceccato|author2=Lars Hesselgren|author3=Mark Pauly|author4=Helmut Pottmann, Johannes Wallner|title=Kemajuan dalam Geometri Arsitektur 2010|url=https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6|date=5 December 2016|publisher=Birkhäuser|isbn=978-3-99043-371-3|page=6|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145200/https://books.google.com/books?id=q45sDwAAQBAJ&pg=PA6|dead-url=no}}</ref><ref name="Pottmann2007">{{cite book|author=Helmut Pottmann|title=Geometri arsitektur|url=https://books.google.com/books?id=bIceAQAAIAAJ|year=2007|publisher=Bentley Institute Press|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145202/https://books.google.com/books?id=bIceAQAAIAAJ|dead-url=no}}</ref> Aplikasi geometri pada arsitektur mencakup penggunaan [[geometri proyektif]] untuk membuat [[perspektif paksa]],<ref name="MoffettFazio2003">{{cite book|author1=Marian Moffett|author2=Michael W. Fazio|author3=Lawrence Wodehouse|title=Sejarah Arsitektur Dunia|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=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145232/https://books.google.com/books?id=IFMohetegAcC&pg=PT371|dead-url=no}}</ref> penggunaan [[bagian berbentuk kerucut]] dalam membangun kubah dan benda serupa,<ref name="HerseyHersey2001" /> penggunaan [[tessellations]],<ref name="HerseyHersey2001"/> dan penggunaan simetri.<ref name="HerseyHersey2001"/>
===Fisika===
{{main|Fisika matematika}}
Bidang [[astronomi]], terutama yang berkaitan dengan pemetaan posisi [[bintang]] dan [[planet]] pada [[bola langit]] dan menjelaskan hubungan antara pergerakan benda-benda langit, telah menjadi sumber penting masalah geometris sepanjang sejarah.<ref name="GreenGreen1985">{{cite book|author1=Robin M. Green|author2=Robin Michael Green|title=Astronomi Bulat|url=https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1|date=31 October 1985|publisher=Cambridge University Press|isbn=978-0-521-31779-5|page=1|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145206/https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA1|dead-url=no}}</ref>
Geometri [[geometri Riemannian]] dan [[pseudo-Riemannian]] digunakan dalam [[relativitas umum]].<ref name="Alekseevskiĭ2008">{{cite book|author=Dmitriĭ Vladimirovich Alekseevskiĭ|title=Perkembangan Terbaru dalam Geometri Pseudo-Riemannian|url=https://books.google.com/books?id=K6-TgxMKu4QC|year=2008|publisher=Masyarakat Matematika Eropa|isbn=978-3-03719-051-7|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145210/https://books.google.com/books?id=K6-TgxMKu4QC|dead-url=no}}</ref> [[Teori string]] menggunakan beberapa varian geometri,<ref name="YauNadis2010">{{cite book|author1=Shing-Tung Yau|author2=Steve Nadis|title=Bentuk Ruang Dalam: Teori String dan Geometri Dimensi Tersembunyi Alam Semesta|url=https://books.google.com/books?id=M40Ytp8Os_gC|date=7 September 2010|publisher=Basic Books|isbn=978-0-465-02266-3|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145212/https://books.google.com/books?id=M40Ytp8Os_gC|dead-url=no}}</ref> seperti halnya [[teori informasi kuantum]].<ref>{{cite book|last1=Bengtsson |first1=Ingemar |last2=Życzkowski |first2=Karol |authorlink2=Karol Życzkowski |title=Geometri Status Kuantum: Pengantar Keterikatan Kuantum|publisher=[[Cambridge University Press]] |edition=2nd |year=2017 |isbn=9781107026254 |oclc=1004572791}}</ref>
===Bidang matematika lainnya===
[[Berkas:Square root of 2 triangle.svg|thumb|right|Pythagoras menemukan bahwa sisi-sisi segitiga bisa memiliki panjang [[Kesesuaian (matematika)|yang tak dapat dibandingkan]].]]
[[Kalkulus]] sangat dipengaruhi oleh geometri.<ref name="Boyer2012"/> Misalnya, pengenalan [[koordinat]] oleh [[René Descartes]] dan perkembangan bersamaan [[aljabar]] menandai tahapan baru untuk geometri, karena figur geometris seperti [[kurva bidang]] dari sekarang dapat direpresentasikan [[Geometri analitik|secara analitis]] dalam bentuk fungsi dan persamaan. Ini memainkan peran kunci dalam munculnya [[kalkulus sangat kecil]] pada abad ke-17. Geometri analitik terus menjadi andalan dalam kurikulum pra-kalkulus dan kalkulus.<ref name="FlandersPrice2014">{{cite book|author1=Harley Flanders|author2=Justin J. Price|title=Kalkulus dengan Geometri Analitik|url=https://books.google.com/books?id=5abiBQAAQBAJ|date=10 May 2014|publisher=Elsevier Science|isbn=978-1-4832-6240-6|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145212/https://books.google.com/books?id=5abiBQAAQBAJ|dead-url=no}}</ref><ref name="RogawskiAdams2015">{{cite book|author1=Jon Rogawski|author2=Colin Adams|title=Kalkulus|url=https://books.google.com/books?id=OWeZBgAAQBAJ|date=30 January 2015|publisher=W. H. Freeman|isbn=978-1-4641-7499-5|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145214/https://books.google.com/books?id=OWeZBgAAQBAJ|dead-url=no}}</ref>
Area aplikasi penting lainnya adalah [[teori bilangan]].<ref name="Lozano-Robledo2019">{{cite book|author=Álvaro Lozano-Robledo|title=Teori Bilangan dan Geometri: Pengantar Geometri Aritmatika|url=https://books.google.com/books?id=ESiODwAAQBAJ|date=21 March 2019|publisher=American Mathematical Soc.|isbn=978-1-4704-5016-8|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145213/https://books.google.com/books?id=ESiODwAAQBAJ|dead-url=no}}</ref> Di [[Yunani kuno]] [[Pythagoras]] menganggap peran angka dalam geometri. Namun, penemuan panjang yang tak dapat dibandingkan itu bertentangan dengan pandangan filosofis mereka.<ref name="Sangalli2009">{{cite book|author=Arturo Sangalli|title=Balas Dendam Pythagoras: Misteri Matematika|url=https://archive.org/details/pythagorasreveng0000sang|url-access=registration|date=10 May 2009|publisher=Princeton University Press|isbn=978-0-691-04955-7|page=[https://archive.org/details/pythagorasreveng0000sang/page/57 57]}}</ref> Sejak abad ke-19, geometri telah digunakan untuk menyelesaikan masalah dalam teori bilangan, misalnya melalui [[geometri bilangan]] atau, yang lebih baru, [[teori skema]], yang digunakan dalam [[bukti Wiles tentang Teorema Terakhir Fermat]].<ref name="CornellSilverman2013">{{cite book|author1=Gary Cornell|author2=Joseph H. Silverman|author3=Glenn Stevens|title=Bentuk Modular dan Teorema Terakhir Fermat|url=https://books.google.com/books?id=jD3TBwAAQBAJ|date=1 December 2013|publisher=Springer Science & Business Media|isbn=978-1-4612-1974-3|access-date=2020-08-25|archive-date=2023-03-01|archive-url=https://web.archive.org/web/20230301145214/https://books.google.com/books?id=jD3TBwAAQBAJ|dead-url=no}}</ref>
==Lihat pula==
===Daftar===
* [[Daftar geometer]]
** [[:Kategori:Geometer aljabar]]
** [[:Kategori:Geometer Diferensial]]
** [[:Kategori:Geometer]]
** [[:Kategori:Ahli topologi]]
* [[Daftar rumus dalam geometri dasar]]
* [[Daftar topik geometri]]
* [[Daftar publikasi penting dalam matematika#Geometri|Daftar publikasi penting dalam geometri]]
* [[Daftar topik matematika]]
===topik-topik terkait===
* [[Daftar topik Geometri]]
* [[Geometri deskriptif]]
* [[Geometri hingga]]
* ''[[Tanah Datar]]'', sebuah buku yang ditulis oleh [[Edwin Abbott]] tentang dua dan [[ruang tiga dimensi]], untuk memahami konsep empat dimensi
* [[Daftar perangkat lunak geometri interaktif]]
===Bidang lain===
* [[Geometri molekuler]]
==Catatan==
{{reflist|40em}}
==Sumber==
* {{cite book |last=Boyer |first=C.B. |authorlink=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 |origyear=1989 |isbn=978-0-471-54397-8 |url-access=registration |url=https://archive.org/details/historyofmathema00boye }}
* {{cite book| last=Cooke| first=Roger| authorlink=| year=2005| title=The History of Mathematics| url=https://archive.org/details/historyofmathema0000cook_o3g3| 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}}
* {{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}}
==Bacaan lebih lanjut==
* {{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|ISBN=978-981-4556-70-5|access-date=2020-08-25|archive-date=2023-02-09|archive-url=https://web.archive.org/web/20230209215011/https://www.worldscientific.com/worldscibooks/10.1142/8952|dead-url=no}}
* {{cite book|ref=none|author=[[Leonard Mlodinow]]|title=Euclid's Window – The Story of Geometry from Parallel Lines to Hyperspace|edition=UK|publisher=Allen Lane|year=1992}} {{ISBN?}}
==Pranala luar==
{{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''] {{Webarchive|url=https://web.archive.org/web/20221105050516/https://www.8foxes.com/ |date=2022-11-05 }}
* [http://mathforum.org/library/topics/geometry/ ''The Math Forum'' – Geometry] {{Webarchive|url=https://web.archive.org/web/20220128062957/http://mathforum.org/library/topics/geometry/ |date=2022-01-28 }}
** [http://mathforum.org/geometry/k12.geometry.html ''The Math Forum'' – K–12 Geometry] {{Webarchive|url=https://web.archive.org/web/20080415225526/http://mathforum.org/geometry/k12.geometry.html |date=2008-04-15 }}
** [http://mathforum.org/geometry/coll.geometry.html ''The Math Forum'' – College Geometry] {{Webarchive|url=https://web.archive.org/web/20080415055232/http://mathforum.org/geometry/coll.geometry.html |date=2008-04-15 }}
** [http://mathforum.org/advanced/geom.html ''The Math Forum'' – Advanced Geometry] {{Webarchive|url=https://web.archive.org/web/20080416182158/http://mathforum.org/advanced/geom.html |date=2008-04-16 }}
* [http://precedings.nature.com/documents/2153/version/1/ Nature Precedings – ''Pegs and Ropes Geometry at Stonehenge''] {{Webarchive|url=https://web.archive.org/web/20200226022808/http://precedings.nature.com/documents/2153/version/1 |date=2020-02-26 }}
* [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] {{Webarchive|url=https://web.archive.org/web/20080512012132/http://plato.stanford.edu/entries/geometry-finitism/ |date=2008-05-12 }} at the Stanford Encyclopedia of Philosophy
* [http://www.ics.uci.edu/~eppstein/junkyard/topic.html The Geometry Junkyard] {{Webarchive|url=https://web.archive.org/web/20080225234940/http://www.ics.uci.edu/~eppstein/junkyard/topic.html |date=2008-02-25 }}
* [http://www.mathopenref.com Interactive geometry reference with hundreds of applets] {{Webarchive|url=https://web.archive.org/web/20110208113043/http://mathopenref.com/ |date=2011-02-08 }}
* [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] {{Webarchive|url=https://web.archive.org/web/20230531130702/https://www.khanacademy.org/?video=ca-geometry--area--pythagorean-theorem#california-standards-test-geometry |date=2023-05-31 }} at [[Khan Academy]]
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