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* [[Roger Penrose]] (lahir 1931) menemukan pola simetris yang menggunakan rasio emas dalam bidang [[tegel aperiodik]], yang mengarah kepada temuan baru mengenai [[quasikristal]].
== Penerapan dan pengamatan ==
=== Estetika ===
''De Divina Proportione'', sebuah buku tiga volume karya[[Luca Pacioli]], diterbitkan pada 1509. Pacioli, seorang pendeta [[Fransiskan]], dikenal sebagai ahli matematika, tetapi ia diketahui tertarik dan terlatih dalam bidang seni. ''De Divina Proportione'' mengeksplorasi matematika dari rasio emas. Sering disebutkan bahwa Pacioli adalah penganjur penerapan rasio emas untuk menghasilkan proporsi yang harmonis, indah, dan menyenangkan. Livio menunjukkan bahwa penafsiran ini salah akibat salah penerjemahan yang dirunut terjadi pada 1799, bahwa Pacioli sesungguhnya menganjurkan sistem [[Vitruvius]] untuk proporsi rasional.<ref name="livio"/> Pacioli juga melihat signifikansi iman Katolik dalam rasio ini, hal inilah yang mendorongnya memberi judul karyanya seperti demikian. Terdapat ilustrasi bangun karya [[Leonardo da Vinci]], sahabat dan kolega Pacioli, ''De Divina Proportione'' menjadi pengaruh utama bagi seniman dan arsitek.{{Citation needed|date=February 2011}}
==Applications and observations==
===Aesthetics===
{{See also|History of aesthetics (pre-20th-century)}}
=== Arsitektur ===
''De Divina Proportione'', a three-volume work by [[Luca Pacioli]], was published in 1509. Pacioli, a [[Franciscan]] [[friar]], was known mostly as a mathematician, but he was also trained and keenly interested in art. ''De Divina Proportione'' explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the [[Vitruvius|Vitruvian]] system of rational proportions.<ref name="livio"/> Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by [[Leonardo da Vinci]], Pacioli's longtime friend and collaborator, ''De Divina Proportione'' was a major influence on generations of artists and architects alike.{{Citation needed|date=February 2011}}
[[ FileBerkas:Acropolis of Athens 01361.JPG|thumb|right|250px| ManyProporsi ofbeberapa the proportions ofbagian [[ the Parthenon]] are alleged to exhibitdianggap themenampilkan goldenrasio ratioemas.]] ▼===Architecture===
Fasad dan elemen Parthenon serta bagian lainnya disebut dipengaruhi persegi panjang emas.<ref>Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic", ''Communication Quarterly'', Vol. 46 No. 2, 1998, pp 194-213.</ref> Sementara para ilmuwan lainnya menolak anggapan bahwa Yunani menghubungkan keindahan dengan rasio emas.
▲[[File:Acropolis of Athens 01361.JPG|thumb|right|250px|Many of the proportions of [[the Parthenon]] are alleged to exhibit the golden ratio.]]
The Parthenon's facade as well as elements of its facade and elsewhere are said by some to be circumscribed by golden rectangles.<ref>Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic", ''Communication Quarterly'', Vol. 46 No. 2, 1998, pp 194-213.</ref> Other scholars deny that the Greeks had any aesthetic association with golden ratio. For exampleMisalnya, Midhat J. Gazalé saysmengatakan, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the ''Elements'' (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a [[regular polyhedron]] whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."<ref>Midhat J. Gazalé , ''Gnomon'', Princeton University Press, 1999. ISBN 0-691-00514-1</ref> And [[Keith Devlin]] says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook ''Elements'', written around 300 BC, showed how to calculate its value."<ref>Keith J. Devlin ''The Math Instinct: Why You're A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs)'', [http://books.google.com/books?id=eRD9gYk2r6oC&pg=PA108 p. 108]. New York: Thunder's Mouth Press, 2005, ISBN 1-56025-672-9</ref> Near-contemporary sources like [[Vitruvius]] exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
A geometrical analysis of the [[Mosque of Oqba|Great Mosque of Kairouan]] reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.<ref>Boussora, Kenza and Mazouz, Said, ''The Use of the Golden Section in the Great Mosque of Kairouan'', Nexus Network Journal, vol. 6 no. 1 (Spring 2004), [http://www.emis.de/journals/NNJ/BouMaz.html]</ref> They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the [[minaret]].
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