کاربر:هيربد فودازى٢/صفحه تمرین ۲: تفاوت میان نسخه‌ها

از ویکی‌پدیا، دانشنامهٔ آزاد
نمایش تاریخچه به صورت تعاملی
→ تفاوت قدیمی‌ترتفاوت جدیدتر ←
محتوای حذف‌شده محتوای افزوده‌شده
دیداریویکی‌متن
مرتب‌سازی فهرست‌های گلوله‌ای با ابزار مرتب‌سازی فهرست
بدون خلاصۀ ویرایش
برچسب‌ها: واگردانی دستی ویرایش همراه ویرایش از وبگاه همراه ویرایش پیشرفتهٔ همراه
خط ۱: خط ۱:
* {{cite journal|doi=10.1080/10407782.2017.1372670 |first1=Ahmed T. |last1=Al-Sammarraie |first2=Kambiz |last2=Vafai |date=2017 |title=Heat transfer augmentation through convergence angles in a pipe |journal=Numerical Heat Transfer, Part A: Applications |volume=72 |issue=3 |page=197−214|s2cid=125509773}}
* {{citation
| last = Alexandrov | first = Victor
| arxiv = 0901.2989
| doi = 10.1007/s00022-011-0061-7
| issue = 1–2
| journal = Journal of Geometry
| mr = 2823098
| title = The Dehn invariants of the Bricard octahedra
| volume = 99
| year = 2010}}.
* {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7}} Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
* {{citation
| last1 = Bokowski | first1 = J.
| last2 = Guedes de Oliveira | first2 = A.
| doi = 10.1007/s004540010027
| issue = 2–3
| journal = [[Discrete and Computational Geometry]]
| mr = 1756651
| pages = 197–208
| title = On the generation of oriented matroids
| volume = 24
| year = 2000| doi-access = free
}}.
* {{citation
| last1 = Bruns | first1 = Winfried
| last2 = Gubeladze | first2 = Joseph
| contribution = Definition 1.1
| contribution-url = https://books.google.com/books?id=pbgg1pFxW8YC&pg=PA5
| doi = 10.1007/b105283
| isbn = 978-0-387-76355-2
| mr = 2508056
| page = 5
| publisher = Springer | location = Dordrecht
| series = Springer Monographs in Mathematics
| title = Polytopes, Rings, and ''K''-theory
| year = 2009| citeseerx = 10.1.1.693.2630
}}.
* {{citation
| last1 = Burgiel | first1 = H.
| last2 = Stanton | first2 = D.
| doi = 10.1007/s004540010030
| issue = 2–3
| journal = [[Discrete and Computational Geometry]]
| mr = 1758047
| pages = 241–255
| title = Realizations of regular abstract polyhedra of types {3,6} and {6,3}
| volume = 24
| year = 2000| doi-access = free
}}
* {{cite book|contribution=Regular and semiregular polyhedra|first=H. S. M.|last=Coxeter|author-link= Harold Scott MacDonald Coxeter|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geomtrical Imagination|url=https://archive.org/details/shapingspaceexpl00sene|edition=2nd|editor-first=Marjorie|editor-last=Senechal|editor-link=Marjorie Senechal|publisher=Springer|year=2013}}
* {{cite book |editor-first=H. S. M. |editor-last= Coxeter |editor2-first=Patrick |editor2-last=DuVal |editor3-first=H.T. |editor3-last=Flather |editor4-first=J.F. |editor4-last=Petrie |title=''The Fifty-nine Icosahedra'' |publisher=University of Toronto studies, mathematical series 6: 1–26 |year=1938 |isbn=978-1-899618-32-3}}
** Third edition (1999) Tarquin {{ISBN|978-1-899618-32-3}} {{mr|id=676126}}
* {{cite book | last=Coxeter | first=H.S.M. | title=Regular Polytopes | publisher=Dover Publications | series=Dover books on advanced mathematics | year=1973 | isbn=978-0-486-61480-9 | url=https://books.google.com/books?id=iWvXsVInpgMC | ref=harv | access-date=2018-11-25 | archive-url=https://www.webcitation.org/74iD18fl4?url=https://books.google.com/books?id=iWvXsVInpgMC | archive-date=16 دسامبر 2018}}
* {{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | author-link2=Michael S. Longuet-Higgins | last3=Miller | first3=J. C. P. | author-link3=J. C. P. Miller| title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=[[Philosophical Transactions of the Royal Society A]]| issn=0080-4614 | volume=246 |issue=916| doi=10.1098/rsta.1954.0003 |url=http://rsta.royalsocietypublishing.org/content/roypta/246/916/401.full.pdf}}
* {{Cite book |first=Harriet |last=Crawford|title=Sumer and the Sumerians|publisher= Cambridge University Press|location=New York|year= 1993|isbn= 0-521-38850-3}}
* {{cite book | last=Cromwell | first=P.R. | title=Polyhedra | publisher=Cambridge University Press | year=1999 | isbn=978-0-521-66405-9 | url=https://books.google.com/books?id=OJowej1QWpoC | ref=harv | access-date=2018-10-22 | archive-url=https://www.webcitation.org/74iCwK5i4?url=https://books.google.com/books?id=OJowej1QWpoC | archive-date=16 دسامبر 2018}}
* {{citation
| last = Cromwell | first = Peter R.
| isbn = 978-0-521-55432-9
| location = Cambridge
| mr = 1458063
| publisher = Cambridge University Press
| title = Polyhedra | title-link = Polyhedra (book)
| year = 1997}}.
* {{cite book | last=Cundy | first=H. Martyn | title=Mathematical models| publisher=Clarendon Press | year=1961 | isbn=0124167}}
* {{citation
| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
| last2 = Rollett | first2 = A.P.
| edition = 2nd
| location = Oxford
| mr = 0124167
| publisher = Clarendon Press
| title = Mathematical models
| title-link = Mathematical Models (Cundy and Rollett)
| year = 1961
}}
* {{citation|title=Computational Geometry: Algorithms and Applications|last1=de Berg|first1=M.|author1-link= Mark de Berg |last2=van Kreveld|first2=M.|author2-link= Marc van Kreveld |last3=Overmars|first3=M.|author3-link=Mark Overmars|last4=Schwarzkopf|first4=O.|author4-link=Otfried Cheong|edition=2nd|publisher=Springer|year=2000}}
* {{cite web|last=Eppstein|first=David|title=Twenty Proofs of Euler's Formula: V-E+F=2|url=http://www.ics.uci.edu/~eppstein/junkyard/euler/|access-date=3 June 2013}}
* {{citation
| last1 = F. Kern | first1 = William
| last2 = R Bland | first2 = James.
| title = Solid Mensuration with proofs
| year = 1938
}}
* {{citation
| last = Field | first =J.
| title = Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler
| journal = Archive for History of Exact Sciences
| volume = 50
| year = 1997
}}.
* {{cite book | last=French | first=K.L. | title=The Hidden Geometry of Life: The Science and Spirituality of Nature | publisher=Watkins Media Limited | series=Gateway series | year=2014 | isbn=978-1-78028-845-1 | url=https://books.google.com/books?id=ixPLCQAAQBAJ&pg=PA96 | ref=harv | access-date=2018-11-12 | archive-url=https://www.webcitation.org/74iD3hZUt?url=https://books.google.com/books?id=ixPLCQAAQBAJ | archive-date=16 دسامبر 2018}}
* {{citation
| last1 = Gaĭfullin | first1 = A. A.
| last2 = Ignashchenko | first2 = L. S.
| doi = 10.1134/S0371968518030068
| isbn = 5-7846-0147-4
| issue = Topologiya i Fizika
| journal = Trudy Matematicheskogo Instituta Imeni V. A. Steklova
| mr = 3894642
| pages = 143–160
| title = Dehn invariant and scissors congruence of flexible polyhedra
| volume = 302
| year = 2018}}.
* {{cite book | last=Gardner | first=Martin | chapter=Chapter 1: The Five Platonic Solids| title=The 2nd Scientific American book of mathematical puzzles & diversions | url=https://archive.org/details/2ndscientificame00gard | publisher=University of Chicago Press | publication-place=Chicago | year=1987 | isbn=0-226-28253-8 | oclc=15550017 | ref=harv}}
* {{citation|last=Ghomi|first=Mohammad|date=2018-01-01|title=Dürer's Unfolding Problem for Convex Polyhedra|journal=Notices of the American Mathematical Society|volume=65|issue=1|pages=25–27|doi=10.1090/noti1609|doi-access=free}}
* {{cite journal | title=A class of multi-symmetric polyhedra | first=Michael | last=Goldberg | journal= [[Tohoku Mathematical Journal]] | year=1937 |url=https://www.jstage.jst.go.jp/article/tmj1911/43/0/43_0_104/_article}}
* {{citation |last=Goldman |first=Ronald N.|author-link=Ron Goldman (mathematician) |editor-last=Arvo |editor-first=James |title=Graphic Gems Package: Graphics Gems II |publisher=Academic Press |year=1991 |pages=170–171 |chapter=Chapter IV.1: Area of planar polygons and volume of polyhedra}}
* {{citation
| last1 = Grünbaum
| first1 = B.
| author1-link = Branko Grünbaum
| last2 = Shephard
| first2 = G.C.
| author2-link = Geoffrey Colin Shephard
| doi = 10.1112/blms/1.3.257
| journal = [[Bulletin of the London Mathematical Society]]
| mr = 0250188
| pages = 257–300
| title = Convex polytopes
| url = http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf
| volume = 1
| issue = 3
| year = 1969
| access-date = 2017-02-21
| archive-url = https://web.archive.org/web/20170222114014/http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf
| archive-date = 2017-02-22
| url-status = dead
}}
* {{citation
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| editor1-last = Bisztriczky | editor1-first = Tibor
| editor2-last = Schneider | editor2-first = Peter McMullen;Rolf
| editor3-last = Weiss | editor3-first = A.
| contribution = Polyhedra with hollow faces
| doi = 10.1007/978-94-011-0924-6_3
| isbn = 978-94-010-4398-4
| location = Dordrecht
| mr = 1322057
| pages = 43–70
| publisher = Kluwer Acad. Publ.
| title = Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational
| year = 1994}}.
* {{citation
| last = Grünbaum
| first = Branko
| journal = Geombinatorics 4
| title = Uniform tilings of 3-space
| year = 1994
| pages = 49 - 56
}}.
* {{citation
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| contribution = Acoptic polyhedra
| doi = 10.1090/conm/223/03137
| mr = 1661382
| pages = 163–199
| publisher = American Mathematical Society | location = Providence, Rhode Island
| series = Contemporary Mathematics
| title = Advances in discrete and computational geometry (South Hadley, MA, 1996)
| url = https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
| volume = 223
| year = 1999| isbn = 978-0-8218-0674-6
}}
* {{citation
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| editor1-last = Aronov | editor1-first = Boris | editor1-link = Boris Aronov
| editor2-last = Basu | editor2-first = Saugata
| editor3-last = Pach | editor3-first = János | editor3-link = János Pach
| editor4-last = Sharir | editor4-first = Micha | editor4-link = Micha Sharir
| contribution = Are your polyhedra the same as my polyhedra?
| contribution-url = http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf
| doi = 10.1007/978-3-642-55566-4_21
| mr = 2038487
| pages = 461–488
| publisher = Springer|location= Berlin
| series = Algorithms and Combinatorics
| title = Discrete and Computational Geometry: The Goodman–Pollack Festschrift
| volume = 25
| year = 2003| isbn = 978-3-642-62442-1 | citeseerx = 10.1.1.102.755 }}.
* {{citation
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| doi = 10.1007/978-1-4613-0019-9
| edition = 2nd
| isbn = 978-0-387-00424-2
| location = New York
| mr = 1976856
| page = 26
| publisher = Springer-Verlag
| ref = Grünbaum-Convex-Polytopes<!-- Don't interfere with harv links to the other Grünbaum 2003 reference -->
| series = Graduate Texts in Mathematics
| title = Convex Polytopes
| title-link = Convex Polytopes
| volume = 221
| year = 2003}}.
*{{citation
| last = Grünbaum | first = Branko
| doi = 10.4171/EM/120
| issue = 3
| journal = Elemente der Mathematik
| mr = 2520469
| pages = 89–101
| title = An enduring error
| volume = 64
| year = 2009| doi-access = free
}}. Reprinted in {{citation|title=The Best Writing on Mathematics 2010|editor-first=Mircea|editor-last=Pitici|publisher=Princeton University Press|year=2011|pages=18–31}}.
* {{cite book | last=Haeckel | first=E. | title=Art Forms in Nature | publisher=Dover Publications | series=Dover Pictorial Archive | year=2012 | isbn=978-0-486-15532-6 | url=https://books.google.com/books?id=E1nCAgAAQBAJ | ref=harv | access-date=2018-11-12 | archive-url=https://www.webcitation.org/74iD9CaTh?url=https://books.google.com/books?id=E1nCAgAAQBAJ | archive-date=16 دسامبر 2018}}
* {{citation
| last1 = Harris | first1 = J. W.
| last2 = Stocker | first2 = H.
| isbn = 978-0-387-94746-4
| location = New York
|publisher = Springer
| title = "Wedge". §4.5.2 in ''Handbook of Mathematics and Computational Science''}}.
* {{SpringerEOM|first=M.|last=Hazewinkel|title=Dehn invariant|id=Dehn_invariant&oldid=35803}}
* {{cite book | last=Henle | first=M. | title=A Combinatorial Introduction to Topology | publisher=Dover | series=Dover Books on Mathematics Series | year=1994 | isbn=978-0-486-67966-2 | url=https://books.google.com/books?id=S0RJYkBxowEC&pg=PA9 | ref=harv | access-date=2018-10-22 | archive-url=https://www.webcitation.org/74iD9tMla?url=https://books.google.com/books?id=S0RJYkBxowEC | archive-date=16 دسامبر 2018}}
* {{citation
| last = Hopf | first = Henning
| title = Classics in Hydrocarbon Chemistry
| publisher = Wiley VCH
| year = 2000
}}.
* {{citation
| last=Hopkins | first=Brian
|title= Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles}}
* {{cite journal | last=Johnson | first=Norman W. | author-link=Norman Johnson (mathematician) | title=Convex Solids with Regular Faces | journal=Canadian Journal of Mathematics | volume=18 | year=1966 | zbl=0132.14603 | issn=0008-414X | doi=10.4153/cjm-1966-021-8}}
* {{cite book|title=Nanostructured materials for solar energy conversion|editor=Sōga, Tetsuo |chapter=Fullerene Thin Films as Photovoltaic Material|author=Katz, E. A.|year=2006 |isbn=978-0-444-52844-5|publisher=Elsevier|pages= 361–443|chapter-url=https://books.google.com/books?id=GmQR1tuk5IgC&pg=PA361|ref=Katz}}
* {{cite book|title=Nanostructured materials for solar energy conversion|editor=Sōga, Tetsuo |chapter=Fullerene Thin Films as Photovoltaic Material|author=Katz, E. A.|year=2006 |isbn=978-0-444-52844-5|publisher=Elsevier|pages= 361–443|chapter-url=https://books.google.com/books?id=GmQR1tuk5IgC&pg=PA361|ref=Katz}}
* {{cite book | last=Kotrč | first=Ronald F. | title=THE DODECAHEDRON IN PLATO'S "TIMAEUS" | publisher=J.D. Sauerländers Verlag | journal=Rheinisches Museum für Philologie | year=1981 | 7= | url=https://www.jstor.org/stable/41245051?seq=1#page_scan_tab_contents | ref=harv | access-date=2018-10-22 | archive-url=https://www.webcitation.org/74iD9cVWk?url=https://www.jstor.org/stable/41245051?seq=1#page_scan_tab_contents | archive-date=16 دسامبر 2018}}
* {{Cite journal |title=The Topology and Combinatorics of Soccer Balls |first1=Dieter |author1-link=Dieter Kotschick |last1=Kotschick |journal=American Scientist |year=2006 |volume=94 |issue=4 |doi=10.1511/2006.60.350}}
* {{Citation
| last=Lakatos
| first=Imre.
| author-link=Imre Lakatos
| title=Proofs and Refutations
| publisher=[[Cambridge Technology Press]]
| year=1976}}
* {{cite book | last=Lewars | first=Errol G. | title=Modeling Marvels | publisher=Springer Netherlands | publication-place=Dordrecht | year=2008 | isbn=978-1-4020-6972-7 | doi=10.1007/978-1-4020-6973-4 | ref=harv}}
* {{cite book | last=MacLean | first=K.J.M. | title=A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra | publisher=Loving Healing Press | series=Geometric explorations series | year=2007 | isbn=978-1-932690-99-6 | url=https://books.google.com/books?id=vINuAwAAQBAJ | ref=harv | access-date=2018-11-10 | archive-url=https://www.webcitation.org/74iCuWAgX?url=https://books.google.com/books?id=vINuAwAAQBAJ | archive-date=16 دسامبر 2018}}
* {{Cite book|url=https://books.google.com/books?id=nNVsHwAACAAJ |author=Mark Lehner |title= The Complete Pyramids: Solving the Ancient Mysteries |publisher=Thames & Hudson|isbn=978-0-500-28547-3|date=2008}}
* {{SpringerEOM|title=Polyhedron, abstract|id=Polyhedron,_abstract&oldid=25452|first=S.V.|last=Matveev}}
* {{SpringerEOM|title=Polyhedron, abstract|id=Polyhedron,_abstract&oldid=25452|first=S.V.|last=Matveev}}
* {{citation|title=Solid Geometry|first=Joseph P.|last=McCormack|publisher=D. Appleton-Century Company|year=1931}}
* {{citation
| last1 = Meserve | first1 = B. E.
| last2 = Pingry | first2 = R. E.
| title = Some Notes on the Prismoidal Formula
|publisher=The Mathematics Teacher
| volume = 45
| year = 1952}}.
* {{cite journal |vauthors=Mettenleiter TC, Klupp BG, Granzow H | title = Herpesvirus assembly: a tale of two membranes | journal = Curr. Opin. Microbiol. | volume = 9 | issue = 4 | year = 2006 | pmid = 16814597 | doi = 10.1016/j.mib.2006.06.013}}
* {{citation|url=http://www.openproblemgarden.org/op/d_urers_conjecture|title=Dürer's conjecture|work=Open Problem Garden|first=D. |last=Moskovich|date=June 4, 2012}}
* {{cite book | last=Nayudu | first=M.V. | title=Plant Viruses | publisher=Tata McGraw-Hill | year=2008 | isbn=978-0-07-065660-4 | url=https://books.google.com/books?id=w11opH8eqY8C&pg=PA50 | ref=harv | access-date=2018-11-12 | archive-url=https://www.webcitation.org/74iCpOLjD?url=https://books.google.com/books?id=w11opH8eqY8C | archive-date=16 دسامبر 2018}}
* {{citation
| last = Needham | first = Joseph
| title = Science and Civilization in China
| location = Taipei
| publisher = Caves Books Ltd.
| year = 1986
}}
* {{citation|title=Computational Geometry in C|journal=Computers in Physics|volume=9|issue=1|first=Joseph|last=O'Rourke|author-link=Joseph O'Rourke (professor)|year=1993|bibcode=1995ComPh...9...55O|doi=10.1063/1.4823371}}
* {{cite news| url=http://news.bbc.co.uk/1/hi/world/africa/3641516.stm | work=BBC News | title=Sudan's past uncovered | date=2004-09-09 | access-date=2010-04-12 | first=Lawrence | last=Pollard}}
* {{cite book | last=Pottmann | first=Helmut | title=Architectural geometry | publisher=Bentley Institute Press | publication-place=Exton, Pa | year=2007 | isbn=978-1-934493-04-5 | oclc=180177477 | ref=harv}}
* {{cite book | last=Pottmann | first=Helmut | last2=Asperl | first2=Andreas | last3=Hofer | first3=Michael | last4=Kilian | first4=Axel | last5=Bentley | first5=Daril | title=Architectural geometry | publisher=Bentley Institute Press | year=2007 | isbn=1-934493-04-X | oclc=180177477 | ref=harv}}
* {{citation
| last = Richeson | first = David S. | author-link = David Richeson
| isbn = 978-0-691-12677-7
| location = Princeton, NJ
| mr = 2440945
| publisher = Princeton University Press
| title = Euler's Gem: The polyhedron formula and the birth of topology
| title-link = Euler's Gem
| year = 2008}}.
* {{cite web | first=Michael | last=Ritter | date=2003 | url=http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/connections/connections_dating_pyramids.html |title=Dating the Pyramids |url-status=dead |archive-url=https://web.archive.org/web/20080511163647/http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/connections/connections_dating_pyramids.html |archive-date=11 May 2008| access-date=15 May 2008}}
* {{Cite book |title=Art, Mathematics and Architecture for Humanistic Renaissance: the Platonic Solids|last=Sala|first=Nicoletta |year=2004|publisher=University of Italian Switzerland, Academy of Architecture, Switzerland | ref=harv | url = http://math.unipa.it/~grim/SiSala2.PDF}}
* {{cite journal | last=Sarhangi | first=Reza | title=Illustrating Abu al-Wafā' Būzjānī: Flat Images, Spherical Constructions | journal=Iranian Studies | publisher=Informa UK Limited | volume=41 | issue=4 | year=2008 | issn=0021-0862 | doi=10.1080/00210860802246184 | pages=511–523 | ref=harv}}
* {{cite book | last=Senechal | first=M. | title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination | publisher=Springer New York | series=EBSCOhost ebooks online | year=2013 | isbn=978-0-387-92714-5 | url=https://books.google.com/books?id=kZtCAAAAQBAJ | ref=harv | access-date=2018-11-10 | archive-url=https://www.webcitation.org/74iCm8uGF?url=https://books.google.com/books?id=kZtCAAAAQBAJ | archive-date=16 دسامبر 2018}}
* {{citation| last = Shephard | first = G. C. | author-link = Geoffrey Colin Shephard| issue = 3| journal = Mathematical Proceedings of the Cambridge Philosophical Society| mr = 0390915| pages = 389–403| title = Convex polytopes with convex nets| volume = 78| year = 1975 |doi=10.1017/s0305004100051860| bibcode = 1975MPCPS..78..389S}}
* {{cite book |editor-first=F. Arthur |editor-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivic |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |isbn=978-0-471-01003-6 |url=https://archive.org/details/kaleidoscopessel0000coxe |url-access=registration}}
** (Paper 10) {{cite journal |first=H.S.M. |last=Coxeter |title=Star Polytopes and the Schlafli Function f(α,β,γ) |journal=Elemente der Mathematik |volume=44 |issue=2 |pages=25–36 |year=1989 |url=https://eudml.org/doc/141447}}
** (Paper 22) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA251 pp. 251–278] {{cite journal |first=H.S.M. |last=Coxeter |title=Regular and Semi Regular Polytopes I |journal=Math. Zeit. |volume=46 |year=1940 |doi=10.1007/BF01181449 |zbl=0022.38305}} MR 2,10
** (Paper 23) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA279 pp. 279–312] {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes II |journal=Math. Zeit. |volume=188 |issue=4 |year=1985 |doi=10.1007/BF01161657 |zbl=0547.52005}}
** (Paper 24) [https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313 pp. 313–358] {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes III |journal=Math. Zeit. |volume=200 |issue=1 |year=1988 |doi=10.1007/BF01161745 |zbl=0633.52006}}
* {{cite news |last=Slackman |first=Michael |date=16 November 2007 |title=In the Shadow of a Long Past, Patiently Awaiting the Future |work=The New York Times|url=https://www.nytimes.com/2008/11/17/world/middleeast/17cairo.html |access-date=17 November 2008 |quote=Deep below the Egyptian desert, archaeologists have found evidence of yet another pyramid, this one constructed 4,300 years ago to store the remains of a pharaoh’s mother. That makes 138 pyramids discovered here so far, and officials say they expect to find more.}}
* {{cite news |last=Slackman |first=Michael |date=17 November 2008 |title=In the Shadow of a Long Past, Patiently Awaiting the Future |work=The New York Times |url=https://www.nytimes.com/2008/11/17/world/middleeast/17cairo.html |access-date=1 May 2010}}
* {{citation |title=An Etruscan dodecahedron|first=Amelia Carolina|last=Sparavigna|year=2012|arxiv=1205.0706}}
* {{citation
| last1=Steurer | first1=Walter
| last2=Deloudi | first2=Sofia
| title=Crystallography of Quasicrystals: Concepts, Methods and Structures
| year=2009
|url=https://books.google.com/books?id=nVx-tu596twC&pg=PA18
}}
* {{citation|first=B. M.|last=Stewart|title=Adventures Among the Toroids: A Study of Orientable Polyhedra with Regular Faces|title-link= Adventures Among the Toroids |edition=2nd|year=1980|isbn=978-0-686-11936-4|publisher=B. M. Stewart}}
* {{cite book | last=Tanna | first=S. | title=Amazing Math: Introduction to Platonic Solids | publisher=CreateSpace Independent Publishing Platform | year=2014 | isbn=978-1-5030-8485-8 | url=https://books.google.com/books?id=8-nVrQEACAAJ | ref=harv | access-date=2018-09-15 | archive-url=https://www.webcitation.org/74iCq0awA?url=https://books.google.com/books?id=8-nVrQEACAAJ | archive-date=16 دسامبر 2018}}
* {{citation
| last = Trigg | first = Charles W.
| issue = 1
| journal = Mathematics Magazine
| jstor = 2689647
| pages = 55–57
| title = An Infinite Class of Deltahedra
| volume = 51
| year = 1978| doi = 10.1080/0025570X.1978.11976675
}}.
* {{citation
| last=Webber | first=William T.
| doi=10.1023/A:1004997029852
| issue=1
| journal=[[Geometriae Dedicata]]
| mr=1468859
| title=Monohedral idemvalent polyhedra that are toroids
| volume=67
| year=1997| s2cid=117884274
}}
* {{Mathworld|urlname=EulerCharacteristic|title=Euler Characteristic|mode=cs2}}
* {{Mathworld|urlname=Polyhedron|title=Polyhedron|mode=cs2}}
* {{Mathworld|urlname=Prism|title=Prism|mode=cs2}}
* {{Mathworld|urlname=Pyramid|title=Pyramid|mode=cs2}}
* {{Mathworld|urlname=PyramidalFrustum|title=Pyramidal Frustum|mode=cs2}}
* {{Mathworld|urlname=RegularPolyhedron|title=Regular Polyhedron|mode=cs2}}
* {{Mathworld|urlname=ShephardsConjecture|title=Shephard's Conjecture|mode=cs2}}
* {{citation
| last = Wells | first = A.F.
| title = Structural Inorganic Chemistry
| edition = 5th
|publisher = Oxford Science Publications
| isbn = 0-19-855370-6
| year = 1984
}}.
* {{citation|first=Magnus J.|last=Wenninger|author-link=Magnus Wenninger|title=Polyhedron Models|publisher=Cambridge University Press|year=1971}}
* {{citation
| last=Whiteley | first=Walter | authorlink=Walter Whiteley
| issue=1
| journal=Structural Topology
| mr=621628
| title=Realizability of polyhedra
| url=http://www.iri.upc.edu/people/ros/StructuralTopology/ST1/st1-07-a4.pdf
| year=1979}}.
* {{citation
| last=Whittaker | first=Robert
|title=Vertex Symbol
|url=http://polyhedra.mathmos.net/entry/vertexsymbol.html
}}
* {{Citation
| last=Ziegler
| first=Günter M.
| author-link=Günter M. Ziegler
| title=Lectures on Polytopes
| publisher=[[Springer-Verlag]]
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| year=1995
| volume=152}}

=== گسترده ===
=== گسترده ===
[[File:Dodecahedron flat.svg|thumb|240px|یکی از گسترده‌های دوازده وجهی منتظم.]]
[[File:Dodecahedron flat.svg|thumb|240px|یکی از گسترده‌های دوازده وجهی منتظم.]]

نسخهٔ ‏۱۵ سپتامبر ۲۰۲۱، ساعت ۰۵:۲۲

گسترده

یکی از گسترده‌های دوازده وجهی منتظم.

گسترده یک چند وجهی یک شکل مسطح است که از تعدادی چندضلعی در امتداد برخی طرفها تشکیل شده‌است، که می‌تواند در فضا خم شود تا بسته شود و یک چند وجهی تشکیل دهد. گسترده ابزاری کاربردی برای ساخت چندوجهی کاغذی است.[۱]

در سال ۱۹۷۵، جی.سی. شپارد پرسید که آیا هر چندوجهی محدب حداقل دارای یک گسترده ساده است.[۲] این سؤال که به عنوان حدس دورر یا مسئله گسترده دورر نیز شناخته می‌شود، بی پاسخ مانده‌است.[۳][۴][۵]

مسئله حل نشده در ریاضی:

آیا هر چندوجهی محدب یک گسترده ساده دارد؟

(مسائل حل‌نشده بیشتر در ریاضی)

ساختار ترکیبی و متریک

یک متوازی السطوح همیشه فارغ از طول اضلاع و زاویه‌هایی که تشکیل می‌دهند، ساختار ترکیبی یکسانی دارد.

ساختار ترکیبی چندوجهی مجموعه رئوس، ضلع‌ها و وجه‌های آن و روابط بین آنها است. ساختار متریک چندوجهی، ساختار چندوجهی به عنوان یک فضای متریک است، یعنی به عنوان فضایی با فاصله بین نقاط.

چرخش حول محور، ساختارهای متریک و ترکیبی چند وجهی را بدون تغییر می‌گذارد. افزایش ابعاد ساختار متریک را تغییر می‌دهد (از آنجا که مثلاً فاصله بین دو رأس را تغییر می‌دهد) اما ساختار ترکیبی را بدون تغییر می‌گذارد. به‌طور کلی، ساختار ترکیبی از انعطاف‌پذیری بیشتری برخوردار است: به عنوان مثال، دو متوازی السطوح همیشه ساختار ترکیبی یکسانی دارند، اما لزوماً دارای ساختار متریک یکسانی نیستند.

مثلاً نمادی مانند که در آن F و E و V به ترتیب تعداد وجوه، اضلاع و رئوس هستند، توصیف خوبی از ساختار ترکیبی چندوجهی ارائه می‌دهد: با این حال توصیف کاملی نیست، زیرا شامل تمام اطلاعات مربوط به مجاورت‌ها نیست.

قبل

تناوب

مقالهٔ اصلی: تناوب (هندسه)

در چندوجهی‌ها، تناوب که بریدگی جزئی نیز نامیده می‌شود، عملی است که در اثر آن رئوس چندوجهی یکی در میان برداشته می‌شوند. از آنجا که در اثر تناوب تعداد اضلاع نیز همانند رئوس نصف می‌شود، پس فقط می‌تواند روی چندوجهی‌هایی که تعداد اضلاع همه وجوهشان زوج است صورت گیرد. جدول زیر برخی چندوجهی‌ها (راست) و تناوبشان (چپ) را نشان می‌دهد:[۶]


مکعب:
چهاروجهی منتظم

مکعب هشت وجهی بریده شده:
مکعب اسناب غیر یکنواخت

بیست دوازده وجهی بریده شده:
دوازده وجهی اسناب

هشت وجهی بریده شده:
بیست وجهی با تقارن پیریتووجهی

فرمول اویلر و تولد توپولوژی

در سال ۱۷۵۰ لئونارد اویلر برای اولین بار ضلع‌های یک چند وجهی را در نظر گرفت، به او اجازه داد فرمول چندوجهی خود را که مربوط به تعداد رئوس، ضلع‌ها و وجوه است کشف کند. این نشانه تولد توپولوژی بود. هنری پوانکاره ایده‌های اصلیش را در اواخر قرن نوزدهم توسعه داد. این امر باعث شد بسیاری از مسائل دیرینه دربارهٔ اینکه چندوجهی چیست، حل و فصل شوند.

ماکس بروکنر خلاصه ای از کارهای مربوط به چند وجهی، از جمله بسیاری از یافته‌های خود را در کتاب «Vielecke und Vielflache: Theorie und Geschichte»[الف] در سال ۱۹۰۰ به زبان آلمانی منتشر کرد، اما کمتر شناخته شد.

در همین حال، کشف ابعاد بالاتر منجر به ایده چندوجهی به عنوان یک نمونه سه بعدی از پلیتوپ عمومی تر شد.[۷]

  1. ↑ Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press
  2. ↑ Shephard, G. C. (1975), "Convex polytopes with convex nets", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (3): 389–403, Bibcode:1975MPCPS..78..389S, doi:10.1017/s0305004100051860, MR 0390915
  3. ↑ Weisstein, Eric W. "Shephard's Conjecture". MathWorld.
  4. ↑ Moskovich, D. (June 4, 2012), "Dürer's conjecture", Open Problem Garden
  5. ↑ Ghomi, Mohammad (2018-01-01), "Dürer's Unfolding Problem for Convex Polyhedra", Notices of the American Mathematical Society, 65 (1): 25–27, doi:10.1090/noti1609
  6. ↑ Coxeter, Regular polytopes, pp.  154–156 8.6 Partial truncation, or alternation
  7. ↑ Richeson, David S. ; Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press 2008.


خطای یادکرد: خطای یادکرد: برچسب <ref> برای گروهی به نام «persian-alpha» وجود دارد، اما برچسب <references group="persian-alpha"/> متناظر پیدا نشد. ().

برگرفته از «https://fa.wikipedia.org/w/index.php?title=کاربر:هيربد_فودازى٢/صفحه_تمرین_۲&oldid=33114350»