Editing Golden ratio (section)

=== Citations ===
{{reflist|30em|refs=

<ref name="a001622">{{cite OEIS
|A001622 |2=Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2}}
</ref>

<ref name=Pacioli>
{{cite book
| last = Pacioli
| first = Luca
| title = [[De divina proportione]]
| publisher = Luca Paganinem de Paganinus de Brescia (Antonio Capella)
| year = 1509
| publication-place = Venice
}}
</ref>

<ref name="Elements 6.3">
{{cite book
| last = Euclid
| title = [[Euclid's Elements|Elements]]
| orig-year = c. 300 BCE
| chapter = Book 6, Definition 3
}}
</ref>

<ref name=goldencut>
{{cite book
| last = Summerson
| first = John
| year = 1963
| title = Heavenly Mansions and Other Essays on Architecture
| publication-place = New York
| publisher = W.W. Norton
| page = 37
| url = https://archive.org/details/heavenlymansions0000summ/page/37/
| quote = And the same applies in architecture, to the [[rectangle]]s representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
}}
</ref>

<ref name="strogatz nytimes">
{{Cite news
| first = Steven
| last = Strogatz
| author-link = Steven Strogatz
| title = Me, Myself, and Math: Proportion Control
| newspaper = [[The New York Times]]
| date = 2012-09-24
| url = http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/
}}
</ref>

<ref name=schielack>
{{Cite journal
| last = Schielack
| first = Vincent P.
| year = 1987
| title = The Fibonacci Sequence and the Golden Ratio
| journal = The Mathematics Teacher
| volume = 80 | issue = 5 | pages = 357–358 | doi = 10.5951/MT.80.5.0357
| jstor = 27965402
}} This source contains an elementary derivation of the golden ratio's value.
</ref>

<ref name=peters>
{{Cite journal
| last = Peters
| first = J. M. H.
| year = 1978
| title = An Approximate Relation between π and the Golden Ratio
| journal = The Mathematical Gazette
| volume = 62
| issue = 421
| pages = 197–198
| doi = 10.2307/3616690
| jstor = 3616690
| s2cid = 125919525
}}</ref>

<ref name="fitzpatrick elements">
{{cite book
| last = Euclid
| translator-last = Fitzpatrick
| translator-first = Richard
| year = 2007
| title = Euclid's Elements of Geometry
| isbn = 978-0615179841
| page = 156
| publisher = Lulu.com
}}
</ref>

<ref name=hemenway>
{{cite book
| last = Hemenway
| first = Priya
| title = Divine Proportion: Phi In Art, Nature, and Science
| url = https://archive.org/details/divineproportion0000heme/page/20/
| url-access = limited
| year = 2005
| publisher = Sterling
| location = New York
| pages = 20–21
| isbn = 9781402735226
}}
</ref>

<ref name=mackinnon>
{{cite journal
| last = Mackinnon
| first = Nick
| year = 1993
| title = The Portrait of Fra Luca Pacioli
| volume = 77
| issue = 479
| journal = The Mathematical Gazette
| pages = 130–219
| doi = 10.2307/3619717
| jstor = 3619717
| s2cid = 195006163
}}
</ref>

<ref name=baravalle>
{{cite journal
| last = Baravalle
| first = H. V.
| title = The geometry of the pentagon and the golden section
| journal = Mathematics Teacher
| volume = 41
| year = 1948
| pages = 22–31
| doi = 10.5951/MT.41.1.0022
}}
</ref>

<ref name=schreiber>
{{cite journal
| last = Schreiber
| first = Peter
| year = 1995
| title = A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'
| journal = [[Historia Mathematica]]
| volume = 22
| issue = 4
| pages = 422–424
| doi = 10.1006/hmat.1995.1033
| doi-access=free
}}
</ref>

<ref name=mactutor>
{{cite web
| last1 = O'Connor
| first1 = John J.
| last2 = Robertson
| first2 = Edmund F.
| authorlink2 = Edmund F. Robertson
| year = 2001
| title = The Golden Ratio
| work = [[MacTutor History of Mathematics archive]]
| url = http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html
| access-date = 2007-09-18
}}
</ref>

<ref name="beutelspacher petri">
{{cite book
| last1 = Beutelspacher
| first1 = Albrecht
| last2 = Petri
| first2 = Bernhard
| year = 1996
| contribution = Fibonacci-Zahlen
| title = Der Goldene Schnitt
| series = Einblick in die Wissenschaft
| publisher = Vieweg+Teubner Verlag
| doi = 10.1007/978-3-322-85165-9_6
| language = de
| pages = 87–98
| isbn = 978-3-8154-2511-4
}}
</ref>

<ref name=fink>
{{cite book
| title = A Brief History of Mathematics
| last1 = Fink
| first1 = Karl
| translator-last1 = Beman
| translator-first1 = Wooster Woodruff
| translator-last2 = Smith
| translator-first2 = David Eugene
| translator-link2 = David Eugene Smith
| year = 1903
| publisher = Open Court
| location = Chicago
| edition = 2nd
| page = 223
| url = https://archive.org/details/bub_gb_3hkPAAAAIAAJ/page/n238
}} (Originally published as ''Geschichte der Elementar-Mathematik''.)
</ref>

<ref name=cook>
{{Cite book
| last = Cook
| first = Theodore Andrea
| author-link = Theodore Andrea Cook
| title = The Curves of Life
| year = 1914
| page = 420
| publisher = Constable
| location = London
| url = https://archive.org/details/cu31924028937179/page/n455
}}
</ref>

<ref name=barr>
{{cite magazine
|last=Barr
|first=Mark
|title=Parameters of beauty
|magazine=[[Architecture (magazine, 1900–1936)|Architecture]] (NY)
|volume=60
|page=325
|year=1929
}} Reprinted: {{cite magazine
| title = Parameters of beauty
| magazine = Think
| volume = 10–11
| publisher = [[IBM]]
| year = 1944
}}
</ref>

<ref name=gardner>
{{cite book
| last = Gardner
| first = Martin
| author-link = Martin Gardner
| title = The Colossal Book of Mathematics
| publisher = Norton
| year = 2001
| chapter = 7. Penrose Tiles
| pages=73–93
| chapter-url = https://archive.org/details/martingardnerthecolossalbookofmathematics/page/n89
}}
</ref>

<ref name=quasicrystals>
{{harvnb|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/203 203–209]}}
<br>
{{cite journal
| last1 = Gratias
| first1 = Denis
| author1-link = Denis Gratias
| last2 = Quiquandon
| first2 = Marianne
| author2-link = Marianne Quiquandon
| year = 2019
| title = Discovery of quasicrystals: The early days
| issue = 7–8
| journal = [[Comptes Rendus Physique]]
| pages = 803–816
| url = https://hal.archives-ouvertes.fr/hal-02379494
| volume = 20
| doi = 10.1016/j.crhy.2019.05.009
| bibcode = 2019CRPhy..20..803G
| s2cid = 182005594
| doi-access = free
}}
<br>
{{cite book
| last = Jaric
| first = Marko V.
| year = 1989
| title = Introduction to the Mathematics of Quasicrystals
| publisher = Academic Press
| page = x
| isbn = 9780120406029
| url = https://archive.org/details/introductiontoma0000unse_b9i6/page/n13/
| url-access = limited
| quote = Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by [[Nicolaas de Bruijn]], that provided the major influence on the new field.
}}
<br>
{{cite journal
| last1 = Goldman | first1 = Alan I.
| last2 = Anderegg | first2 = James W.
| last3 = Besser | first3 = Matthew F.
| last4 = Chang | first4 = Sheng-Liang
| last5 = Delaney | first5 = Drew W.
| last6 = Jenks | first6 = Cynthia J. | author6-link = Cynthia Jenks
| last7 = Kramer | first7 = Matthew J.
| last8 = Lograsso | first8 = Thomas A.
| last9 = Lynch | first9 = David W.
| last10 = McCallum | first10 = R. William
| last11 = Shield | first11 = Jeffrey E.
| last12 = Sordelet | first12 = Daniel J.
| last13 = Thiel | first13 = Patricia A. | author13-link = Patricia Thiel
| year = 1996
| volume = 84
| issue = 3
| journal = American Scientist
| jstor = 29775669
| pages = 230–241
| title = Quasicrystalline materials
| bibcode = 1996AmSci..84..230G
}}
</ref>

<ref name=constructions>
{{ cite book
| last = Martin
| first = George E.
| year = 1998
| title = Geometric Constructions
| title-link = Geometric Constructions
| series = Undergraduate Texts in Mathematics
| publisher = Springer
| pages = 13–14
| doi = 10.1007/978-1-4612-0629-3
| isbn = 978-1-4612-6845-1
}}
</ref>

<ref name=sizer>{{cite journal
 | last = Sizer | first = Walter S.
 | doi = 10.1080/0025570X.1986.11977215
 | issue = 1
 | journal = [[Mathematics Magazine]]
 | jstor = 2690013
 | mr = 828417
 | pages = 23–27
 | title = Continued roots
 | volume = 59
 | year = 1986}}</ref>

<ref name="Concrete Abstractions">
{{cite book
| last1 = Hailperin
| first1 = Max
| last2 = Kaiser
| first2 = Barbara K.
| last3 = Knight
| first3 = Karl W.
| year = 1999
| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
| publisher = Brooks/Cole
| page = 63
| url = https://archive.org/details/ost-computer-science-concreteabstractions/page/n78
}}
</ref>

<ref name=hardy>
{{cite book
| last1 = Hardy
| first1 = G. H.
| authorlink1 = G. H. Hardy
| last2 = Wright
| first2 = E. M.
| authorlink2 = E. M. Wright
| year = 1960
| orig-year = 1938
| chapter = §11.8. The measure of the closest approximations to an arbitrary irrational
| title = An Introduction to the Theory of Numbers
| edition = 4th
| publisher = Oxford University Press
| chapter-url = https://archive.org/details/introductiontoth0000hard/page/163
| pages = 163–164
| isbn = 978-0-19-853310-8
| chapter-url-access = limited
}}
</ref>

<ref name=tattersall>
{{Cite book
| last = Tattersall
| first = James Joseph
| year = 1999
| title = Elementary number theory in nine chapters
| publisher = Cambridge University Press
| page = 28
| url = https://archive.org/details/elementarynumber0000tatt/page/28/
| url-access = limited
}}
</ref>

<ref name=parker4d>
{{cite book
| last1 = Parker
| first1 = Matt
| date = 2014
| title = Things to Make and Do in the Fourth Dimension
| publisher = Farrar, Straus and Giroux
| page = 284
| isbn = 9780374275655
| url = https://archive.org/details/thingstomakedoin0000park_r9y6/page/284/
| url-access = limited
}}
</ref>

<ref name=phyllotaxis>{{cite journal
| last1 = King
| first1 = S.
| last2 = Beck
| first2 = F.
| last3 = Lüttge
| first3 = U.
| year = 2004
| title = On the mystery of the golden angle in phyllotaxis
| journal = Plant, Cell and Environment
| volume = 27
| issue = 6
| pages = 685–695
| doi = 10.1111/j.1365-3040.2004.01185.x
| bibcode = 2004PCEnv..27..685K
}}
</ref>

<ref name=triangleconstruction>
{{cite journal
| last1 = Odom
| first1 = George
| last2 = van de Craats
| first2 = Jan
| date = 1986
| title = E3007: The golden ratio from an equilateral triangle and its circumcircle
| department = Problems and solutions
| journal = The American Mathematical Monthly
| volume = 93
| issue = 7
| page = 572
| doi = 10.2307/2323047
| jstor = 2323047
}}
</ref>

<ref name="Liber mensurationum">
{{cite journal
| last = Busard
| first = Hubert L. L.
| date = 1968
| title = L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr
| journal = [[Journal des Savants]]
| volume = 1968
| issue = 2
| language = fr,la
| pages = 65–124
| url = https://www.persee.fr/doc/jds_0021-8103_1968_num_2_1_1175
| doi = 10.3406/jds.1968.1175
}} See problem 51, reproduced on p. 98
</ref>

<ref name=bruce>
{{cite journal
| last = Bruce
| first = Ian
| year = 1994
| title = Another instance of the golden right triangle
| journal = [[Fibonacci Quarterly]]
| volume = 32
| issue = 3
| pages = 232–233
| doi = 10.1080/00150517.1994.12429219
| url = https://www.mathstat.dal.ca/FQ/Scanned/32-3/bruce.pdf
}}
</ref>

<ref name=fletcher>
{{cite journal
| last = Fletcher
| first = Rachel
| year = 2006
| title = The golden section
| journal = Nexus Network Journal
| volume = 8
| issue = 1
| pages = 67–89
| doi = 10.1007/s00004-006-0004-z
| s2cid = 120991151
| doi-access = free
}}
</ref>

<ref name=loeb>
{{cite book
| last = Loeb
| first = Arthur
| year = 1992
| chapter = The Golden Triangle
| title = Concepts & Images: Visual Mathematics
| publisher = Birkhäuser
| pages = 179–192
| doi = 10.1007/978-1-4612-0343-8_20
| isbn = 978-1-4612-6716-4
}}
</ref>

<ref name=hexecontahedron>
{{cite journal
| last = Grünbaum
| first = Branko
| author-link = Branko Grünbaum
| year = 1996
| journal = Geombinatorics
| volume = 6
| issue = 1
| pages = 15–18
| title = A new rhombic hexecontahedron
| url = https://faculty.washington.edu/moishe/branko/BG207.New%20rhombic%20hexecontah.pdf
}}
</ref>

<ref name=quarter-circles>{{cite conference
 | last1 = Reitebuch | first1 = Ulrich
 | last2 = Skrodzki | first2 = Martin
 | last3 = Polthier | first3 = Konrad
 | editor1-last = Swart | editor1-first = David
 | editor2-last = Farris | editor2-first = Frank
 | editor3-last = Torrence | editor3-first = Eve
 | contribution = Approximating logarithmic spirals by quarter circles
 | contribution-url = https://archive.bridgesmathart.org/2021/bridges2021-95.html
 | isbn = 978-1-938664-39-7
 | location = Phoenix, Arizona
 | pages = 95–102
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture
 | year = 2021}}</ref>

<ref name=diedrichs>{{cite journal
 | last = Diedrichs | first = Danilo R.
 | date = February 2019
 | doi = 10.1017/mag.2019.7
 | issue = 556
 | journal = [[The Mathematical Gazette]]
 | pages = 52–64
 | title = Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature
 | volume = 103| s2cid = 127189159
 }}</ref>

<ref name=loeb-varney>{{cite book
 | last1 = Loeb | first1 = Arthur L.
 | last2 = Varney | first2 = William
 | editor1-last = Hargittai | editor1-first = István
 | editor2-last = Pickover | editor2-first = Clifford A.
 | contribution = Does the golden spiral exist, and if not, where is its center?
 | contribution-url = https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA47
 | date = March 1992
 | doi = 10.1142/9789814343084_0002
 | pages = 47–61
 | publisher = World Scientific
 | title = Spiral Symmetry| isbn = 978-981-02-0615-4
 }}</ref>

<ref name=BurgerStarbird>
{{cite book
| first1 = Edward B.
| last1 = Burger
| first2 = Michael P.
| last2 = Starbird
| year = 2005
| orig-year = 2000
| edition = 2nd
| title = The Heart of Mathematics: An Invitation to Effective Thinking
| publisher = Springer
| page = 382
| isbn = 9781931914413
| url = https://archive.org/details/heartofmathemati0000burg_z4x5/page/382/
}}
</ref>

<ref name="golden rhombohedra">
{{cite book
| last = Senechal
| first = Marjorie
| authorlink = Marjorie Senechal
| year = 2006
| contribution = Donald and the golden rhombohedra
| editor1-last = Davis
| editor1-first = Chandler
| editor2-last = Ellers
| editor2-first = Erich W.
| title = The Coxeter Legacy
| isbn = 0-8218-3722-2
| pages = 159–177
| publisher = American Mathematical Society
}}
</ref>

<ref name="Tilings and Patterns">
{{cite book
| last1 = Grünbaum
| first1 = Branko
| author1-link = Branko Grünbaum
| last2 = Shephard
| first2 = G. C.
| title = Tilings and Patterns
| location = New York
| publisher = W. H. Freeman
| year = 1987
| pages = 537–547
| isbn = 9780716711933
| url = https://archive.org/details/isbn_0716711931/page/537/
}}
</ref>

<ref name=pentaplexity>
{{cite magazine
| first = Roger
| last = Penrose
| author-link = Roger Penrose
| year = 1978
| title = Pentaplexity
| magazine = [[Eureka (University of Cambridge magazine)|Eureka]]
| volume =  39
| pages = 32
| url = https://archive.org/details/eureka-39/page/16/
}} ([https://www.archim.org.uk/eureka/archive/Eureka-39.pdf#page=16 original PDF])
</ref>

<ref name=borromean>
{{cite conference
| last1 = Gunn
| first1 = Charles
| last2 = Sullivan
| first2 = John M.
| author2-link = John M. Sullivan (mathematician)
| year = 2008
| title = The Borromean Rings: A video about the New IMU logo
| url = https://archive.bridgesmathart.org/2008/bridges2008-63.html
| editor1-last = Sarhangi
| editor1-first = Reza
| editor2-last = Séquin
| editor2-first = Carlo H.
| editor2-link = Carlo H. Séquin
| book-title = Proceedings of [[The Bridges Organization|Bridges]] 2008
| conference = Leeuwarden, the Netherlands
| pages = 63–70
| publisher = Tarquin Publications
}}; Video at {{cite web
| title = The Borromean Rings: A new logo for the IMU
| website = International Mathematical Union
| url = http://torus.math.uiuc.edu/jms/Videos/imu/
| archive-url = https://web.archive.org/web/20210308065039/http://torus.math.uiuc.edu/jms/Videos/imu/
| archive-date = 2021-03-08
}}
</ref>

<ref name=hume>
{{cite journal
| last = Hume
| first = Alfred
| year = 1900
| title = Some propositions on the regular dodecahedron
| journal = [[The American Mathematical Monthly]]
| volume = 7
| issue = 12
| pages = 293–295
| doi = 10.2307/2969130
| jstor = 2969130
}}
</ref>

<ref name="59 Icosahedra">
{{Cite book
| last1 = Coxeter
| first1 = H.S.M.
| author1-link = Harold Scott MacDonald Coxeter
| last2 = du Val
| first2 = Patrick
| author2-link = Patrick du Val
| last3 = Flather
| first3 = H.T.
| last4 = Petrie
| author4-link = John Flinders Petrie
| first4 = J.F.
| year = 1938
| title = The Fifty-Nine Icosahedra
| title-link = The Fifty-Nine Icosahedra
| publisher = University of Toronto Studies
| volume = 6
| page = 4
| quote = Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.
}}
</ref>

<ref name=muller>
{{cite book
| last1 = Muller
| first1 = J. M.
| date = 2006
| title = Elementary functions : algorithms and implementation
| edition = 2nd
| publisher = Birkhäuser
| location = Boston
| isbn = 978-0817643720
| page = 93
}}
</ref>

<ref name=ycruncher>
{{Cite web
| first =  Alexander J.
| last = Yee
| date = 2021-03-13
| title = Records Set by y-cruncher
| website = numberword.org
| url = http://www.numberworld.org/y-cruncher/records.html
}} Two independent computations done by Clifford Spielman.
</ref>

<ref name=horocycle>
[http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html Horocycles exinscrits : une propriété hyperbolique remarquable], cabri.net, retrieved 2009-07-21.
</ref>

<ref name=rrcf>{{cite journal
| last1 = Berndt
| first1 = Bruce C.
| last2 = Chan
| first2 = Heng Huat
| last3 = Huang
| first3 = Sen-Shan
| last4 = Kang
| first4 = Soon-Yi
| last5 = Sohn
| first5 = Jaebum
| last6 = Son
| first6 = Seung Hwan
| year = 1999
| title = The Rogers–Ramanujan Continued Fraction
| journal = Journal of Computational and Applied Mathematics
| volume = 105
| number = 1–2
| pages = 9–24
| url = https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf
| doi = 10.1016/S0377-0427(99)00033-3
| access-date = 2022-11-29
| archive-date = 2022-10-06
| archive-url = https://web.archive.org/web/20221006212209/https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf
| url-status = dead
}}</ref>

<ref name=duffin>
{{cite journal
| last = Duffin
| first = Richard J.
| author-link = Richard Duffin
| year = 1978
| title = Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers
| journal = [[Pacific Journal of Mathematics]]
| volume = 74
| issue = 1
| pages = 47–56
| url = https://projecteuclid.org/euclid.pjm/1102810434
| doi = 10.2140/pjm.1978.74.47
| doi-access = free
}}
</ref>

<ref name=modulor>Le Corbusier, ''The Modulor'', {{nobr|p. 25}}, as cited in
{{cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| page = 316
| publisher = Taylor & Francis
| doi = 10.4324/9780203477465
| isbn = 9781135811112
}}</ref>

<ref name=modulor2>Le Corbusier, ''The Modulor'', {{nobr|p. 35}}, as cited in
{{cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| page = 320
| publisher = Taylor & Francis
| doi = 10.4324/9780203477465
| isbn = 9781135811112
| quote = Both the paintings and the architectural designs make use of the golden section
}}</ref>

<ref name=Frings>
{{cite journal
| last = Frings
| first = Marcus
| year = 2002
| title = The Golden Section in Architectural Theory
| journal = Nexus Network Journal
| volume = 4
| number = 1
| pages =  9–32
| doi = 10.1007/s00004-001-0002-0
| s2cid = 123500957
| url = http://belveduto.de/text-nnj.htm
| doi-access = free
}}
</ref>

<ref name=urwin>
{{cite book
| last = Urwin
| first = Simon
| year = 2003
| title = Analysing Architecture
| edition = 2nd
| publisher = Routledge
| pages = 154–155
| url = https://archive.org/details/analysingarchite0000unwi/page/154/
| url-access = limited
}}
</ref>

<ref name=devlin>{{cite web
| first = Keith
| last = Devlin
| year = 2007
| title = The Myth That Will Not Go Away
| access-date = September 26, 2013
| url = http://www.maa.org/external_archive/devlin/devlin_05_07.html
| quote = Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
| archive-date = November 12, 2020
| archive-url = https://web.archive.org/web/20201112034625/https://www.maa.org/external_archive/devlin/devlin_05_07.html
| url-status = dead
}}</ref>

<ref name="livio plus">
{{cite web
| last = Livio
| first = Mario
| year = 2002
| title = The golden ratio and aesthetics
| url = https://plus.maths.org/content/os/issue22/features/golden/index
| ref = none
| website = Plus Magazine
| access-date = November 26, 2018
}}
</ref>

<ref name=simanek>
{{cite web
| first = Donald E.
| last =  Simanek
| title = Fibonacci Flim-Flam
| url = http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
| access-date = April 9, 2013
| url-status = dead
| archive-url = https://web.archive.org/web/20100109045556/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
| archive-date = January 9, 2010
}}
</ref>

<ref name=dalidimension>
{{cite video
| people = Salvador Dalí
| date = 2008
| title = The Dali Dimension: Decoding the Mind of a Genius
| format = DVD
| publisher = Media 3.14-TVC-FGSD-IRL-AVRO
| url = http://www.dalidimension.com/eng/index.html
}}
</ref>

<ref name="hunt gilkey">
{{cite book
| last1 = Hunt
| first1 = Carla Herndon
| last2 = Gilkey
| first2 = Susan Nicodemus
| year = 1998
| title = Teaching Mathematics in the Block
| pages = 44, 47
| publisher = Eye On Education
| isbn = 1-883001-51-X
}}
</ref>

<ref name=olariu>
{{cite arXiv
| last = Olariu
| first = Agata
| year = 1999
| title = Golden Section and the Art of Painting
| eprint = physics/9908036
}}
</ref>

<ref name=tosto>
{{cite book
| last = Tosto
| first = Pablo
| year = 1969
| title = La composición áurea en las artes plásticas
| language = es
| trans-title = The golden composition in the plastic arts
| publisher = Hachette
| pages = 134–144
}}
</ref>

<ref name=tschichold>
{{cite book
| last = Tschichold
| first = Jan
| author-link = Jan Tschichold
| year = 1991
| title = The Form of the Book
| publisher = Hartley & Marks
| isbn = 0-88179-116-4
| page = 43 Fig 4
| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n59/
| quote = Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.
}}
</ref>

<ref name=tschichold2>
{{cite book
| last = Tschichold
| first = Jan
| year = 1991
| title = The Form of the Book
| publisher = Hartley & Marks
| isbn = 0-88179-116-4
| pages = 27–28
| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n43/
}}
</ref>

<ref name=miscellany>
{{Cite journal
| last = Jones
| first = Ronald
| title = The golden section: A most remarkable measure
| year = 1971
| journal = The Structurist
| volume = 11
| pages = 44–52
| quote = Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
}}
{{pb}}
{{cite book
| last = Johnson
| first = Art
| year = 1999
| title = Famous problems and their mathematicians
| publisher = Teacher Ideas Press
| page = 45
| isbn = 9781563084461
| url = https://archive.org/details/famousproblemsth0000john/page/45
| url-access = limited
| quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
}}
{{pb}}
{{cite book
| last1 = Stakhov
| first1 = Alexey P.
| author1-link = Alexey Stakhov
| last2 = Olsen
| first2 = Scott
| year = 2009
| title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science
| chapter = §1.4.1 A Golden Rectangle with a Side Ratio of ''τ''
| publisher = World Scientific
| pages = 20–21
| quote = A credit card has a form of the golden rectangle
| chapter-url = https://archive.org/details/alexy-stakhov-the-mathematics-of-harmony-from-euclid-to-contemporary-mathematics/page/21/
}}
{{pb}}
{{cite book
| last = Cox
| first = Simon
| year = 2004
| title = Cracking the Da Vinci Code
| publisher = Barnes & Noble
| page = 62
| isbn = 978-1-84317-103-4
| url = https://archive.org/details/crackingdavincic00coxs/page/62/
| quote = The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
}}
</ref>

<ref name=lendvai>
{{cite book
| last = Lendvai
| first = Ernő
| year = 1971
| title = Béla Bartók: An Analysis of His Music
| publication-place = London
| publisher = Kahn and Averill
}}
</ref>

<ref name=Smith>
{{cite book
| last = Smith
| first = Peter F.
| year = 2003
| title = The Dynamics of Delight: Architecture and Aesthetics
| url = https://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83
| publisher = Routledge
| page = 83
| isbn = 9780415300100
}}
</ref>

<ref name=howat>
{{cite book
| last= Howat
| first = Roy
| year = 1983
| title = Debussy in Proportion: A Musical Analysis
| chapter = 1. Proportional structure and the Golden Section
| chapter-url = https://archive.org/details/debussyinproport0000howa/page/1
| publisher = Cambridge University Press
| pages = 1–10
}}
</ref>

<ref name=trezise>
{{cite book
| last = Trezise
| first = Simon
| year = 1994
| title = Debussy: La Mer
| publisher = Cambridge University Press
| page = 53
| isbn = 9780521446563
| url = https://books.google.com/books?id=THD1nge_UzcC&pg=PA53
}}
</ref>

<ref name=833cents>
{{cite journal
| last = Mongoven
| first = Casey
| year = 2010
| title = A style of music characterized by Fibonacci and the golden ratio
| url = https://www.mat.ucsb.edu/Publications/mongoven_CongressusNumerantium2010.pdf
| journal = Congressus Numerantium
| volume = 201
| pages = 127–138
}}
{{pb}}
{{cite journal
| last = Hasegawa
| first = Robert
| year = 2011
| title = ''Gegenstrebige Harmonik'' in the Music of Hans Zender
| journal = Perspectives of New Music
| volume = 49
| issue = 1
| doi = 10.1353/pnm.2011.0000
| publisher = Project Muse
| jstor = 10.7757/persnewmusi.49.1.0207
| pages = 207–234
}}
{{pb}}
{{cite conference
| last = Smethurst
| first = Reilly
| year = 2016
| title = Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal
| url = https://archive.bridgesmathart.org/2016/bridges2016-519.html
| editor1-last = Torrence
| editor1-first = Eve
| editor2-last = Torrence
| editor2-first = Bruce
| editor3-last = Séquin
| editor3-first = Carlo
| editor4-last = McKenna
| editor4-first = Douglas
| editor5-last = Fenyvesi
| editor5-first = Kristóf
| editor6-last = Sarhangi
| editor6-first = Reza
| display-editors = 1
| book-title = Proceedings of [[The Bridges Organization|Bridges]] 2016
| conference = Jyväskylä, Finland
| pages = 519–522
| publisher = Tessellations Publishing

}}
</ref>

<ref name=padovan>
{{Cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| publisher = Taylor & Francis
| pages = 305–306
| doi = 10.4324/9780203477465
| isbn = 9781135811112
}}
{{pb}}
{{cite journal
| last = Padovan
| first = Richard
| year=2002
| title = Proportion: Science, Philosophy, Architecture
| journal = Nexus Network Journal
| volume = 4
| issue=1
| pages = 113–122
| doi=10.1007/s00004-001-0008-7
| doi-access=free
}}
</ref>

<ref name=zeising>
{{cite book
| first = Adolf
| last = Zeising
| author-link = Adolf Zeising
| title = Neue Lehre von den Proportionen des menschlichen Körpers
| trans-title = New doctrine of the proportions of the human body
| language = de
| year = 1854
| publisher = Weigel
| chapter = Einleitung [preface]
| pages = 1–10
| chapter-url = https://archive.org/details/neuelehrevondenp00zeis/page/n25/
}}
</ref>

<ref name=pommersheim>
{{cite book
| editor1-last = Pommersheim
| editor1-first = James E.
| editor2-last = Marks
| editor2-first = Tim K.
| editor3-last = Flapan
| editor3-first = Erica L.
| editor3-link = Erica Flapan
| year = 2010
| title = Number Theory: A Lively Introduction with Proofs, Applications, and Stories
| publisher = Wiley
| page = 82
}}
</ref>

<ref name=ising>
{{cite journal
| last1 = Coldea
| first1 = R.
| last2 = Tennant
| first2 = D.A.
| last3 = Wheeler
| first3 = E.M.
| last4 = Wawrzynksa
| first4 = E.
| last5 = Prabhakaran
| first5 = D.
| last6 = Telling
| first6 = M.
| last7 = Habicht
| first7 = K.
| last8 = Smeibidl
| first8 = P.
| last9 = Keifer
| first9 = K.
| year = 2010
| title = Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry
| journal = Science
| volume = 327
| issue = 5962
| pages = 177–180
| doi = 10.1126/science.1180085
| pmid = 20056884
| arxiv = 1103.3694
| bibcode = 2010Sci...327..177C
| s2cid = 206522808
}}
</ref>

<ref name=disco>{{cite web
| url = https://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/
| title = A Disco Ball in Space
| publisher = NASA
| date = 2001-10-09
| access-date = 2007-04-16
| archive-date = 2020-12-22
| archive-url = https://web.archive.org/web/20201222224745/https://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/
| url-status = dead
}}</ref>

<ref name=pheasant>
{{cite book
| last = Pheasant
| first = Stephen
| year = 1986
| title = Bodyspace
| publisher = Taylor & Francis
| isbn = 9780850663402
| url = https://archive.org/details/bodyspaceanthrop0000phea/
| url-access = limited
}}
</ref>

<ref name=vanLaack>
{{cite book
| last = van Laack
| first = Walter
| title = A Better History Of Our World: Volume 1 The Universe
| location = Aachen
| publisher = van Laach
| year = 2001
}}
</ref>

<ref name=dunlap>
{{cite book
| last = Dunlap
| first = Richard A.
| year = 1997
| title = The Golden Ratio and Fibonacci Numbers
| publisher = World Scientific
| page = [https://books.google.com/books?id=qBftCgAAQBAJ&pg=PA130 130]
}}
</ref>

<ref name=falbo>
{{cite journal
| last = Falbo
| first = Clement
| date = March 2005
| title = The golden ratio—a contrary viewpoint
| journal = The College Mathematics Journal
| volume = 36
| issue = 2
| pages = 123–134
| doi = 10.1080/07468342.2005.11922119
| s2cid = 14816926
}}
</ref>

<ref name=miller>
{{cite journal
| last = Miller
| first = William
| year = 1996
| title = Pentagons and Golden Triangles
| journal = Mathematics in School
| volume = 25
| issue = 4
| pages = 2–4
| jstor = 30216571
}}
</ref>

<ref name=robinson>
{{cite web
| last1 = Frettlöh
| first1 = D.
| last2 = Harriss
| first2 = E.
| last3 = Gähler
| first3 = F.
| title = Robinson Triangle
| website = Tilings Encyclopedia
| url = https://tilings.math.uni-bielefeld.de/substitution/robinson-triangle/
}}
{{pb}}
{{cite journal
| last = Clason
| first = Robert G
| year = 1994
| title = A family of golden triangle tile patterns.
| journal = The Mathematical Gazette
| volume = 78
| number = 482
| pages = 130–148
| doi = 10.2307/3618569
| jstor = 3618569
| s2cid = 126206189
}}
</ref>

<ref name=moscovich>
{{cite book
| last = Moscovich
| first = Ivan
| author-link = Ivan Moscovich
| year = 2004
| title = The Hinged Square & Other Puzzles
| location = New York
| publisher = Sterling
| url = https://books.google.com/books?id=4n7PvKC1dfwC&pg=PA122
| page = 122
| isbn = 9781402716669
}}
</ref>

<ref name=shellspirals>
{{cite journal
| last = Peterson
| first = Ivars
| author-link = Ivars Peterson
| date = 1 April 2005
| title = Sea shell spirals
| journal = [[Science News]]
| url = http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
| access-date = 10 November 2008
| archive-date = 3 October 2012
| archive-url = https://web.archive.org/web/20121003045834/http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
| url-status = dead
}}
</ref>

<ref name=gutenberg>
{{cite book
| last = Man
| first = John
| year = 2002
| title = Gutenberg: How One Man Remade the World with Word
| pages = 166–167
| publisher = Wiley
| isbn = 9780471218234
| url = https://archive.org/details/gutenberghowonem00john/page/166
| quote = The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.
}}
</ref>

<ref name=Fechner>
{{cite book
| title = Vorschule der Ästhetik
| language = de
| trans-title = Preschool of Aesthetics
| last = Fechner
| first = Gustav
| year = 1876
| publisher = Breitkopf & Härtel
| location = Leipzig
| pages = 190–202
| url = https://archive.org/details/vorschulederaest12fechuoft/page/n203/
}}
</ref>

<ref name=osler>
{{cite journal
| last = Osler
| first = Carol
| title = Support for Resistance: Technical Analysis and Intraday Exchange Rates
| journal = Federal Reserve Bank of New York Economic Policy Review
| year = 2000
| volume = 6
| issue = 2
| pages= 53–68
| quote = 38.2 percent and 61.8 percent retracements of recent rises or declines are common,
| url= http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
| archive-url = https://web.archive.org/web/20070512155447/http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
| archive-date = 2007-05-12
| url-status = live
}}
</ref>

<ref name=magicdow>
{{cite report
| last1 = Batchelor
| first1 = Roy
| author1-link = Roy Batchelor
| last2 = Ramyar
| first2 = Richard
| year = 2005
| title = Magic numbers in the Dow
| publisher = Cass Business School
| url = http://openaccess.city.ac.uk/16276/
| pages = 13, 31
}}
Popular press summaries can be found in:
{{cite news
| last = Stevenson
| first = Tom
| date = 2006-04-10
| title = Not since the 'big is beautiful' days have giants looked better
| newspaper = The Daily Telegraph
| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
}}
{{cite news
| date = 2006-09-23
| title = Technical failure
| magazine = The Economist
| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
}}
</ref>

<ref name=greatpyramid>
{{cite book
| last = Herz-Fischler
| first = Roger
| year = 2000
| isbn = 0-88920-324-5
| publisher = Wilfrid Laurier University Press
| title = The Shape of the Great Pyramid
}} The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
{{pb}}
{{cite book
| last = Rossi
| first = Corinna
| author-link = Corinna Rossi
| title = Architecture and Mathematics in Ancient Egypt
| year = 2004
| publisher = Cambridge University Press
| pages = 67–68
| url = https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/
| quote = there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to {{tmath|\varphi}}, and {{tmath|\varphi}} itself as a number, do not fit with the extant Middle Kingdom mathematical sources
}}; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
{{pb}}
{{cite journal
| last1 = Rossi
| first1 = Corinna
| last2 = Tout
| first2 = Christopher A.
| year = 2002
| title = Were the Fibonacci series and the Golden Section known in ancient Egypt?
| journal = [[Historia Mathematica]]
| volume = 29
| issue = 2
| doi = 10.1006/hmat.2001.2334
| pages = 101–113
| hdl = 11311/997099
| hdl-access = free
}}
{{pb}}
{{cite journal
| last = Markowsky
| first = George
| year = 1992
| title = Misconceptions about the Golden Ratio
| journal = [[The College Mathematics Journal]]
| volume = 23
| issue = 1
| doi = 10.2307/2686193
| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
| publisher = Mathematical Association of America
| pages = 2–19
| jstor = 2686193
| quote = It does not appear that the Egyptians even knew of the existence of {{tmath|\varphi}} much less incorporated it in their buildings
| access-date = 2012-06-29
}}
</ref>

<ref name=Polemic>
{{cite journal
| last = Van Mersbergen
| first = Audrey M.
| year = 1998
| title = Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic
| journal = Communication Quarterly
| volume = 46
| number = 2
| pages = 194–213
| doi = 10.1080/01463379809370095
}}
</ref>

<ref name=mathinstinct>
{{cite book
| last = Devlin
| first = Keith J.
| year = 2005
| title = The Math Instinct
| url = https://archive.org/details/isbn_9781560258391/page/108
| location = New York
| publisher = Thunder's Mouth Press
| page = 108
}}
</ref>

<ref name=gazalé>
{{cite book
| last = Gazalé
| first = Midhat J.
| author-link = Midhat J. Gazalé
| year = 1999
| title = Gnomon: From Pharaohs to Fractals
| publisher = Princeton
| page = 125
| isbn = 9780691005140
| url = https://archive.org/details/gnomonfrompharao0000gaza/page/125/
| url-access = limited
}}
</ref>

<ref name=foutakis>
{{cite journal
| last = Foutakis
| first = Patrice
| year = 2014
| title = Did the Greeks Build According to the Golden Ratio?
| journal = Cambridge Archaeological Journal
| volume = 24
| number = 1
| pages = 71–86
| doi = 10.1017/S0959774314000201
| s2cid = 162767334
}}
</ref>

<ref name=centrepompidou1>
[http://mediation.centrepompidou.fr/education/ressources/ENS-futurisme2008/ENS-futurisme2008-07-section-or.html ''Le Salon de la Section d'Or''], October 1912, Mediation Centre Pompidou
</ref>

<ref name=centrepompidou2>
[http://bibliothequekandinsky.centrepompidou.fr/imagesbk/RP225%5C001/M5050_X0031_PER_P02251912001.pdf ''Jeunes Peintres ne vous frappez pas !'', La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, no. 1, 9 octobre 1912, pp.&nbsp;1–7] {{Webarchive
| url = https://web.archive.org/web/20201030080014/http://bibliothequekandinsky.centrepompidou.fr/imagesbk/RP225%5C001/M5050_X0031_PER_P02251912001.pdf
| date = 2020-10-30
}}, Bibliothèque Kandinsky
</ref>

<ref name=seuratclaims>
{{cite journal
| last = Herz-Fischler
| first = Roger
| year = 1983
| title = An Examination of Claims Concerning Seurat and the Golden Number
| journal = Gazette des Beaux-Arts
| volume = 101
| pages = 109–112
| url = https://people.math.carleton.ca/~rhfischl/PUBLICATIONS/seurat_GN.pdf
}}</ref>

<ref name=herbert>
{{cite book
| last = Herbert
| first = Robert
| year = 1968
| title = Neo-Impressionism
| page = 24
| publisher = Guggenheim Foundation
| url = https://archive.org/details/neoimpressionism0000herb
| url-access = limited
}}
</ref>

<ref name=camfield>
{{cite journal
| last = Camfield
| first = William A.
| date = March 1965
| doi = 10.1080/00043079.1965.10788819
| issue = 1
| journal = The Art Bulletin
| pages = 128–134
| title = Juan Gris and the golden section
| volume = 47
}}
</ref>

<ref name=juangris>
{{cite book
| last = Green
| first = Christopher
| year = 1992
| title = Juan Gris
| publisher = Yale
| pages = 37–38
| isbn = 9780300053746
| url = https://archive.org/details/juangris0000gree/page/37/
| url-access = limited
}}
{{pb}}
{{cite book
| last = Cottington
| first = David
| year = 2004
| title = Cubism and Its Histories
| publisher = Manchester University Press
| pages = 112, 142
| url = https://books.google.com/books?isbn=0719050049
}}
</ref>

<ref name=allard>
{{cite journal
| last = Allard
| first = Roger
| date = June 1911
| title = Sur quelques peintres
| journal = Les Marches du Sud-Ouest
| pages = 57–64
}}
Reprinted in
{{cite book
| editor1-last = Antliff
| editor1-first = Mark
| editor2-last = Leighten
| editor2-first = Patricia
| title = A Cubism Reader, Documents and Criticism, 1906–1914
| year = 2008
| publisher = The University of Chicago Press
| pages = 178–191
| url = https://archive.org/details/cubismreaderdocu0008unse/page/178/
}}
</ref>

<ref name=bouleau>
{{cite book
| last = Bouleau
| first = Charles
| year = 1963
| title = The Painter's Secret Geometry: A Study of Composition in Art
| publisher = Harcourt, Brace & World
| pages = 247–248
| url = https://archive.org/details/painterssecretge0000char/page/247/
}}
</ref>

<!-- END REFLIST --> }}