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Dedhert.Jr (bicara | kontrib)
16.355 suntingan
Dedhert.Jr (bicara | kontrib)
16.355 suntingan
Membalikkan revisi 20778095 oleh Dedhert.Jr (bicara)
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Baris 1:
{{Periksa terjemahan|en|Pick's theorem}}
[[Berkas:Pick-theorem.svg|jmpl|{{color|red|{{math|''i'' {{=}} 7}}}}, {{color|green|{{math|''b'' {{=}} 8}}}}, {{math|''A'' {{=}} {{color|red|''i''}} + {{sfrac|{{color|green|''b''}}|2}} − 1 {{=}} 10}}]]
Dalam [[geometri]], '''teorema Pick''' merupakan sebuah rumus luas [[poligon sederhana]] dengan koordinat simpul berupa bilangan bulat dengan menjumlahkan titik-titik bilangan bulat dalam poligon dan batasnya. Hasil teorema ini dijelaskan pertama kali oleh [[Georg Alexander Pick]] pada tahun 1899.{{r<ref>[[Georg Alexander Pick|gs|steinhaus}}Pick, Georg]] (1899). [https://www.biodiversitylibrary.org/item/50207#page/327 "Geometrisches zur Zahlenlehre"]. ''Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag''. (Neue Folge). '''19''': 311–319. JFM [https://zbmath.org/?format=complete&q=an:33.0216.01 33.0216.01]. [http://citebank.org/node/47270 CiteBank:47270]</ref> Teorema ini dipopulerkan dalam bahasa Inggris oleh [[Hugo Steinhaus]] dalam bukunya berjudul ''Mathematical Snapshots'', edisi tahun 1950.<ref>[[Branko Grünbaum|Grünbaum, Branko]]; [[Geoffrey Colin Shephard|Shephard, G. C]]. (Februari 1993). "Pick's theorem". ''[[The American Mathematical Monthly]]''. '''100''' (2): 150–161. [[Pengenal objek digital|doi]]:[[doi:10.2307/2323771|10.2307/2323771]]. [[JSTOR]] [https://www.jstor.org/stable/2323771 2323771]. [[Mathematical Reviews|MR]] [https://www.ams.org/mathscinet-getitem?mr=1212401 1212401].</ref><ref>[[Hugo Steinhaus|Steinhaus, H.]] (1950). ''Mathematical Snapshots''. Oxford University Press. hlm. 76. [[Mathematical Reviews|MR]] [https://www.ams.org/mathscinet-getitem?mr=0036005 0036005].</ref> Teorema ini memiliki banyak bukti, dan teorema ini dapat dirampat ke rumus untuk jenis-jenis poligon tak sederhana.
 
== Rumus ==
Baris 22:
 
== Topik yang berkaitan ==
Ada beberapa topik dalam matematika yang mengaitkan luas daerah dengan jumlah titik kisi, di antaranya: [[teorema Blichfeldt]], yang mengatakan bahwa setiap bentuk yang dapat ditranslasikan memiliki setidaknya luas bentuk tersebut dalam titik kisi;<ref name="olds">{{cite book|last1=Olds|first1=C. D.|last2=Lax|first2=Anneli|last3=Davidoff|first3=Giuliana P.|year=2000|title=The Geometry of Numbers|title-link=The Geometry of Numbers|publisher=Mathematical Association of America, Washington, DC|isbn=0-88385-643-3|series=Anneli Lax New Mathematical Library|volume=41|pages=119–127|contribution=Chapter 9: A new principle in the geometry of numbers|mr=1817689|author1-link=Carl D. Olds|author2-link=Anneli Cahn Lax|author3-link=Giuliana Davidoff}}</ref> [[masalah lingkaran Gauss]] yang melibatkan batas galat antara luas lingkaran dengan jumlah titik kisi dalam lingkaran;<ref name="guy">{{cite book|last=Guy|first=Richard K.|year=2004|title=Unsolved Problems in Number Theory|location=New York|publisher=Springer-Verlag|isbn=0-387-20860-7|edition=3rd|series=Problem Books in Mathematics|pages=365–367|contribution=F1: Gauß's lattice point problem|doi=10.1007/978-0-387-26677-0|mr=2076335|author-link=Richard K. Guy}}</ref> serta masalah menghitung [[Titik bilangan bulat dalam polihedron cembung|jumlah titik bilangan bulat dalam polihedron cembung]] yang muncul dalam cabang-cabang matematika dan ilmu komputer.<ref name="barvinok">{{cite book|last=Barvinok|first=Alexander|year=2008|title=Integer Points In Polyhedra|location=Zürich|publisher=European Mathematical Society|isbn=978-3-03719-052-4|series=Zurich Lectures in Advanced Mathematics|doi=10.4171/052|mr=2455889|authorlink=Alexander Barvinok}}</ref> Dalam cabang terapan, [[planimeter titik]] merupakan perangkat berbasis transparansi yang mengestimasi luas bentuk dengan menghitung jumlah titik yang terdapat dalam bentuk tersebut.
 
== Rujukan ==
{{div col|col width=30em}}
{{reflist|refs=
<references />
{{div col end}}
 
<ref name=az>{{cite book | last1 = Aigner | first1 = Martin | author1-link = Martin Aigner | last2 = Ziegler | first2 = Günter M. | author2-link = Günter M. Ziegler | contribution = Three applications of Euler’s formula: Pick's theorem | doi = 10.1007/978-3-662-57265-8 | edition = 6th | isbn = 978-3-662-57265-8 | pages = 93–94 | publisher = Springer | title = Proofs from THE BOOK | title-link = Proofs from THE BOOK | year = 2018}}</ref>
 
<ref name=ball>{{cite book | last = Ball | first = Keith | author-link = Keith Martin Ball | contribution = Chapter 2: Counting Dots | isbn = 0-691-11321-1 | mr = 2015451 | pages = 25–40 | publisher = Princeton University Press, Princeton, NJ | title = Strange Curves, Counting Rabbits, and Other Mathematical Explorations | year = 2003}}</ref>
 
<ref name=barvinok>{{cite book
| last = Barvinok | first = Alexander | authorlink = Alexander Barvinok
| doi = 10.4171/052
| isbn = 978-3-03719-052-4
| mr = 2455889
| publisher = European Mathematical Society | location = Zürich
| series = Zurich Lectures in Advanced Mathematics
| title = Integer Points In Polyhedra
| year = 2008}}</ref>
 
<ref name=bcrt>{{cite journal
| last1 = Brandolini | first1 = L.
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| last3 = Robins | first3 = S.
| last4 = Travaglini | first4 = G.
| doi = 10.1080/00029890.2021.1839241
| issue = 1
| journal = [[The American Mathematical Monthly]]
| mr = 4200451
| pages = 41–49
| title = Pick's theorem and convergence of multiple Fourier series
| volume = 128
| year = 2021| s2cid = 231624428
}}</ref>
 
 
<ref name=bellhouse>{{cite journal
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| doi = 10.2307/2530419
| issue = 2
| journal = Biometrics
| jstor = 2530419
| mr = 673040
| pages = 303–312
| title = Area estimation by point-counting techniques
| volume = 37
| year = 1981}}</ref>
 
<ref name=braden>{{cite journal|author=Bart Braden|title=The Surveyor's Area Formula|journal=The College Mathematics Journal|volume=17|issue=4|year=1986|pages=326–337|url=http://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma063.pdf|doi=10.2307/2686282|jstor=2686282}}</ref>
 
<ref name=bruarc>{{cite journal
| last1 = Bruckheimer | first1 = Maxim
| last2 = Arcavi | first2 = Abraham
| doi = 10.1007/BF03024792
| issue = 4
| journal = [[The Mathematical Intelligencer]]
| mr = 1365013
| pages = 64–67
| title = Farey series and Pick's area theorem
| volume = 17
| year = 1995}}</ref>
 
<ref name=diaz-robins>{{cite journal
| last1 = Diaz | first1 = Ricardo
| last2 = Robins | first2 = Sinai
| doi = 10.2307/2975035
| issue = 5
| journal = [[The American Mathematical Monthly]]
| jstor = 2975035
| mr = 1327788
| pages = 431–437
| title = Pick's formula via the Weierstrass <math>\wp</math>-function
| volume = 102
| year = 1995}}</ref>
 
<ref name=discretely>{{cite book | last1 = Beck | first1 = Matthias | last2 = Robins | first2 = Sinai | contribution = 2.6 Pick's theorem | doi = 10.1007/978-1-4939-2969-6 | edition = 2nd | isbn = 978-1-4939-2968-9 | mr = 3410115 | pages = 40–43 | publisher = Springer | series = Undergraduate Texts in Mathematics | title = Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra | year = 2015}}</ref>
<ref name=br2>{{harvtxt|Beck|Robins|2015}}, 3.6 "From the discrete to the continuous volume of a polytope", pp. 76–77</ref>
 
<ref name=edward>{{cite book|last=Martin|first=George Edward|doi=10.1007/978-1-4612-5680-9|isbn=0-387-90636-3|mr=718119|at=Theorem 12.1, page 120|publisher=Springer-Verlag|series=Undergraduate Texts in Mathematics|title=Transformation geometry|url=https://books.google.com/books?id=gevlBwAAQBAJ&pg=PA120|year=1982}}</ref>
 
<ref name=ehrhart>{{cite journal | last1 = Diaz | first1 = Ricardo | last2 = Robins | first2 = Sinai | doi = 10.2307/2951842 | issue = 3 | journal = [[Annals of Mathematics]] | mr = 1454701 | pages = 503–518 | series = Second Series | title = The Ehrhart polynomial of a lattice polytope | volume = 145 | year = 1997| jstor = 2951842 }}</ref>
 
<ref name=equivalence>{{cite journal
| last1 = DeTemple | first1 = Duane
| last2 = Robertson | first2 = Jack M.
| date = March 1974
| doi = 10.5951/mt.67.3.0222
| issue = 3
| journal = [[The Mathematics Teacher]]
| jstor = 27959631
| mr = 0444503
| pages = 222–226
| title = The equivalence of Euler's and Pick's theorems
| volume = 67}}</ref>
 
<ref name=funkenbusch>{{cite journal
| last = Funkenbusch | first = W. W.
| department = Classroom Notes
| date = June–July 1974
| doi = 10.2307/2319224
| issue = 6
| journal = [[The American Mathematical Monthly]]
| jstor = 2319224
| mr = 1537447
| pages = 647–648
| title = From Euler's formula to Pick's formula using an edge theorem
| volume = 81}}</ref>
 
<ref name=gs>{{cite journal | last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard | date = February 1993 | doi = 10.2307/2323771 | issue = 2 | journal = [[The American Mathematical Monthly]] | jstor = 2323771 | pages = 150–161 | title = Pick's theorem | volume = 100 | mr = 1212401}}</ref>
 
<ref name=guy>{{cite book
| last = Guy | first = Richard K. | author-link = Richard K. Guy
| contribution = F1: Gauß's lattice point problem
| doi = 10.1007/978-0-387-26677-0
| edition = 3rd
| isbn = 0-387-20860-7
| mr = 2076335
| pages = 365–367
| publisher = Springer-Verlag | location = New York
| series = Problem Books in Mathematics
| title = Unsolved Problems in Number Theory
| year = 2004}}</ref>
 
<ref name=minkowski>{{cite journal
| last1 = Ram Murty | first1 = M.
| last2 = Thain | first2 = Nithum
| doi = 10.1080/00029890.2007.11920465
| issue = 8
| journal = [[The American Mathematical Monthly]]
| jstor = 27642309
| mr = 2354443
| pages = 732–736
| title = Pick's theorem via Minkowski's theorem
| volume = 114
| year = 2007| s2cid = 38855683
}}</ref>
 
<ref name=olds>{{cite book
| last1 = Olds | first1 = C. D. | author1-link = Carl D. Olds
| last2 = Lax | first2 = Anneli | author2-link = Anneli Cahn Lax
| last3 = Davidoff | first3 = Giuliana P. | author3-link = Giuliana Davidoff
| contribution = Chapter 9: A new principle in the geometry of numbers
| isbn = 0-88385-643-3
| mr = 1817689
| pages = 119–127
| publisher = Mathematical Association of America, Washington, DC
| series = Anneli Lax New Mathematical Library
| title = The Geometry of Numbers
| title-link = The Geometry of Numbers
| volume = 41
| year = 2000}}</ref>
 
<ref name=pick>{{cite journal |last=Pick |first=Georg | author-link = Georg Alexander Pick |title=Geometrisches zur Zahlenlehre |journal=Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag |series=(Neue Folge) |year=1899 |volume=19 |pages=311–319 |url=https://www.biodiversitylibrary.org/item/50207#page/327 |jfm=33.0216.01 }} [http://citebank.org/node/47270 CiteBank:47270]</ref>
 
<ref name=reeve>{{cite journal | last = Reeve | first = J. E. | doi = 10.1112/plms/s3-7.1.378 | journal = [[Proceedings of the London Mathematical Society]] | mr = 0095452 | pages = 378–395 | series = Third Series | title = On the volume of lattice polyhedra | volume = 7 | year = 1957}}</ref>
 
<ref name=rosenholtz>{{cite journal
| last = Rosenholtz | first = Ira
| doi = 10.1080/0025570X.1979.11976797
| issue = 4
| journal = [[Mathematics Magazine]]
| jstor = 2689425
| mr = 1572312
| pages = 252–256
| title = Calculating surface areas from a blueprint
| volume = 52
| year = 1979}}</ref>
 
<ref name=sankri>{{cite journal
| last1 = Sankar | first1 = P. V.
| last2 = Krishnamurthy | first2 = E. V.
| date = August 1978
| doi = 10.1016/s0146-664x(78)80021-5
| issue = 1
| journal = Computer Graphics and Image Processing
| pages = 136–143
| title = On the compactness of subsets of digital pictures
| volume = 8}}</ref>
 
<ref name=steinhaus>{{cite book
| last = Steinhaus | first = H. | author-link = Hugo Steinhaus
| mr = 0036005
| page = 76
| publisher = Oxford University Press
| title = Mathematical Snapshots
| year = 1950}}</ref>
 
<ref name=trainin>{{cite journal |last=Trainin |first=J. |title=An elementary proof of Pick's theorem |journal=[[The Mathematical Gazette]] |volume=91 |issue=522 |date=November 2007 |pages=536–540|doi=10.1017/S0025557200182270 | jstor = 40378436|s2cid=124831432 }}</ref>
 
<ref name=varberg>{{cite journal
| last = Varberg | first = Dale E.
| doi = 10.2307/2323172
| issue = 8
| journal = [[The American Mathematical Monthly]]
| jstor = 2323172
| mr = 812105
| pages = 584–587
| title = Pick's theorem revisited
| volume = 92
| year = 1985}}</ref>
 
<ref name=wells>{{cite book | last = Wells | first = David | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | year = 1991 | contribution = Pick's theorem | pages = 183–184}}</ref>
 
<ref name=wiedijk>{{cite web|first=Freek|last=Wiedijk|url=https://www.cs.ru.nl/~freek/100/|title=Formalizing 100 Theorems|publisher=Radboud University Institute for Computing and Information Sciences|access-date=2021-07-10}}</ref>
 
}}
== Pranala luar ==
 
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