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Sejak abad ke-20, rasio meas diwakili dengan [[huruf Yunani]] '''''Φ''''' atau '''''φ''''' ([[phi]], berdasarkan nama [[Phidias]], (pematung yang disebut-sebut menggunakan rasio ini) atau secara tidak lazim juga dilambangkan dengan '''''τ''''' ([[tau]], huruf pertama untuk kata dalam Yunani kuno τομή yang berarti ''memotong''.<ref name="livio"/><ref>{{Mathworld|title=Golden Ratio|urlname=GoldenRatio}}</ref>
=== Liniwaktu ===
TimelineLiniwaktu according tomenurut Priya Hemenway.<ref name=Hemenway,P>{{Cite book ▼===Timeline===
▲Timeline according to Priya Hemenway.<ref name=Hemenway,P>{{Cite book
| last = Hemenway
| first = Priya
| pages = 20–21
}}</ref>
* [[Phidias]] (490–430 BCSM) mademembuat thepatung-patung [[Parthenon]] statuesyang thatdinilai seemmengandung torasio embody the golden ratioemas.
* [[Plato]] (427–347 BCSM), indalam hiskaryanya ''[[Timaeus (dialogue)|Timaeus]]'', describesmenyebutkan fivelima possiblebangun regularbentuk solidsyang umum (the [[PlatonicBangun solidsPlato]]: the [[tetrahedron]], [[cubekubus]], [[octahedronoktahedron]], [[dodecahedrondodekahedron]], anddan [[icosahedronikosahedron]]), some of which are relatedbeberapa todiantaranya theterkait goldenrasio ratioemas.<ref>{{cite web
| last = Plato
| authorlink = Plato
| accessdate = May 30, 2006
}}</ref>
* [[Euclid]] (c.sekitar 325–c.325–265 265 BCSM), indalam hiskaryanya ''[[Euclid's Elements|ElementsElemen]]'', gavememberikan thecatatan firstpertama recordeddefinisi definitionrasio of the golden ratioemas, whichyang hedisebutnya called, as translated into English,gave "extremerasio andekstrem meandan ratiorata-rata" (Greek: ἄκρος καὶ μέσος λόγος).<ref name="Elements 6.3"/>
* [[Fibonacci]] (1170–1250) mentionedmenyebutkan thederet [[Sequence|numericalbilangan series]]yang nowkini nameddinamai aftersesuai himnamanya indalam hiskaryanya ''[[Liber Abaci]]''; therasio ratioderet of sequential elements of theelemen [[Fibonaccibilangan number|Fibonacci sequence]] approaches the goldenmendekati ratiorasio asymptoticallyemas.
* [[Luca Pacioli]] (1445–1517) definesmendefinisikan therasio goldenemas ratio as thesebagai "divinerasio proportionilahiah" indalam hiskaryanya ''Divina Proportione''.
* [[Michael Maestlin]] (1550–1631) publishesmenerbitkan theperkiraan firstpertama knownyang approximationdiketahui of themengenai (inverseinversi) golden ratiorasio asemas asebagai [[decimalpecahan fractiondesimal]].
* [[Johannes Kepler]] (1571–1630) provesmembuktikan thatbahwa therasio goldenemas ratio is theadalah limit of the ratio ofrasio consecutivekeberlanjutan Fibonacci numbers,<ref name=tatt>
{{Cite book
| title = Elementary number theory in nine chapters
| page = 28
| url = http://books.google.com/?id=QGgLbf2oFUYC&pg=PA29&dq=golden-ratio+limit+fibonacci+ratio+kepler&q=golden-ratio%20limit%20fibonacci%20ratio%20kepler
}}</ref> dan menggambarkan rasio emas sebagai "permata berharga": "Geometri memiliki dua khazanah: yang pertama adalah [[Teorema Pythagoras]], dan yang lainnya adalah pembagian garis menjadi rasio ekstrem dan rata-rata; yang pertama kita dapat mengandaikannya sebagai emas, yang kedua dapat kita namakan sebagai permata berharga." Kedua khazanah ini berpadu dalam [[segitiga Kepler]].
}}</ref> and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the [[Pythagorean theorem|Theorem of Pythagoras]], and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the [[Kepler triangle]].
* [[Charles Bonnet]] (1720–1793) pointsmenyoroti outbahwa thatspiral in the spiraltanaman [[phyllotaxis]] ofberputar plantssearah goingjarum [[clockwise]]jam anddan counter-clockwiseberlawanan werejarum frequentlyjam twobiasanya successivesesuai Fibonaccideret bilangan seriesFibonacci.
* [[Martin Ohm]] (1792–1872) isdipercaya believedsebagai toorang beyang thepertama firstmenggunakan to use the termistilah ''goldener Schnitt'' (goldenbagian sectionemas) tountuk menggambarkan describerasio thisini ratio,pada intahun 1835.<ref>{{Cite book| title = Die Macht der Zahl: Was die Numerologie uns weismachen will | author = Underwood Dudley | publisher = Springer | year = 1999 | isbn = 3-7643-5978-1 | page = 245 | url = http://books.google.com/?id=r6WpMO_hREYC&pg=PA245&dq=%22goldener+Schnitt%22+ohm }}</ref>
* [[Édouard Lucas]] (1842–1891) givesmemberikan theurutan numericalangka sequenceyang nowkini knowndisebut asderet thebilangan Fibonacci sequence its present name.
* Mark Barr (20thabad centuryke-20 M) suggestsmengusulkan thehuruf Greek letterYunani phi ('''φ'''), thesebagai initialinisial letterpematung ofternama Greek sculptorYunani, Phidias's name, as a [[symbol]] forsebagai thelambang goldenrasio ratioemas.<ref>{{Cite book
| last = Cook
| first = Theodore Andrea
| isbn = 0-486-23701-X
}}</ref>
* [[Roger Penrose]] (b.lahir 1931) discovered a symmetrical pattern thatmenemukan usespola thesimetris goldenyang ratiomenggunakan inrasio theemas fielddalam ofbidang [[aperiodictegel tilingaperiodik]]s, whichyang ledmengarah tokepada newtemuan discoveriesbaru aboutmengenai [[quasicrystalsquasikristal]].
==Applications and observations==
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