Geometri: Perbedaan antara revisi

{{General geometry}}

{{other uses}}

[[Berkas:Hypercube.svg|190px|thumb|[[Tersseract]] atau [[Hiperkubus]] Salah satu bentuk geometri 4 Dimensi]]

 

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.

 

 

== Geometri awal ==

[[Berkas:Models of four platonic solids.JPG|jmpl|Model empat padatan Platonik]]

 

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''.

== Sejarah ==

{{Main|Sejarah geometri}}

[[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]]

 

 

 

[[Berkas:Woman teaching geometry.jpg|left|thumb|upright=.85|''Wanita mengajar geometri''. Ilustrasi di awal terjemahan abad pertengahan [[Euklides Element]], (c. 1310).]]

 

=== Dalam dua dimensi ===

==== 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'''.

 

{{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}}

 

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)}}

 

==== Jajar genjang ====

{{Main|Jajar genjang}}

[[Berkas:Jajaran genjang.JPG|jmpl|Jajar genjang{{br}}dengan alas '''<math>a</math>''' dan tinggi '''<math>t</math>''']]

 

==== Lingkaran ====

 

==== 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]]

 

 

 

=== Dalam tiga dimensi ===

== Konsep penting dalam geometri ==

 

 

=== Aksioma ===

[[Berkas:Parallel postulate en.svg|thumb|right|Ilustrasi [[postulat paralel]] Euclid]]

{{Lihat pula|Geometri Euklides|Aksioma}}

 

===Titik===

[[Berkas:Angle obtuse acute straight.svg|thumb|Sudut tajam (a), tumpul (b), dan lurus (c). Sudut lancip dan tumpul juga dikenal sebagai sudut miring.]]

 

 

 

===Kurva===

[[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''² + ''y''² + ''z''² − ''r''² {{=}} 0}}.)]]

 

 

===Manifold===

{{See also|Luas#Daftar rumus|Volume#Rumus volume}}

 

 

 

 

====Metrik dan ukuran====

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>

 

 

{{main|Kesesuaian (geometri)|Kesamaan (geometri)}}

 

 

 

 

 

===Dimensi===

 

[[Berkas:Von Koch curve.gif|thumb|[[Kepingan salju Koch]], dengan [[dimensi fraktal]]=log4/log3 dan [[dimensi topologi]]=1]]

 

 

===Simetri===

{{main |Simetri}}

 

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>

 

{{main|Geometri Euklides}}

 

 

[[Berkas:Hyperbolic triangle.svg|thumb|upright=1|right|[[Geometri diferensial]] menggunakan alat dari [[kalkulus]] untuk mempelajari masalah yang melibatkan kelengkungan.]]

{{main|Geometri diferensial}}

 

 

====Geometri non-Euklides====

{{main|Geometri non-Euklides}}

 

 

 

===Topologi===

{{main|Topologi}}

[[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 />

 

===Geometri aljabar===

{{main|Geometri aljabar}}

 

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]].

===Geometri diskrit===

{{main|Geometri diskrit}}

[[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]].-->

 

===Teori grup geometris===

{{main|Geometric group theory}}

[[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>

 

{{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]]

 

 

 

 

 

===Arsitektur===

{{main|Matematika dan arsitektur|Geometri arsitektur}}

 

 

===Fisika===

{{main|Fisika matematika}}

 

 

 

===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]].]]

 

 

 

==Lihat pula==

==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=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}}

 

==Bacaan lebih lanjut==

* {{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?}}

 

{{Sister project links|Geometry}}

{{Wikibooks}}

{{Cite EB1911 |wstitle=Geometry |volume=11 |pages=675–736 |short=1}}

* A [[v:Geometry|geometry]] course from [[v:|Wikiversity]]

* [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)

* [https://web.archive.org/web/20090321024112/http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm Dynamic Geometry Sketches (with some Student Explorations)]

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