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| {{About|the number|the pop music album|The Golden Ratio (album)|calendar dates|Golden number (time)}}
| | I like Auto racing. Seems boring? Not!<br>I try to learn German in my spare time.<br><br>My web page: [http://www.kcomemphis.org/Korean_School_Gallery2/main.php?g2_itemId=1615 FIFA Coin generator] |
| [[File:Golden ratio line.svg|thumb|Line segments in the golden ratio]]
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| [[File:SimilarGoldenRectangles.svg|right|thumb|A [[golden rectangle]] with longer side <span style="color:blue;">'''''a'''''</span> and shorter side <span style="color:red;">'''''b'''''</span>, when placed adjacent to a square with sides of length <span style="color:blue;">'''''a'''''</span>, will produce a [[Similarity (geometry)|similar]] golden rectangle with longer side <span style="color:green;">'''''a + b'''''</span> and shorter side <span style="color:blue;">'''''a'''''</span>. This illustrates the relationship <math> \frac{a+b}{a} = \frac{a}{b} \equiv \varphi</math>.]]
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| In [[mathematics]], two quantities are in the '''golden ratio''' if their [[ratio]] is the same as the ratio of their [[sum]] to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities ''a'' and ''b'' with ''a'' > ''b'',
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| :<math> \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,</math>
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| where the Greek letter [[Phi (letter)|phi]] (<math>\varphi</math>) represents the golden ratio. Its value is:
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| :<math>\varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.</math><!-- PLEASE DO NOT add additional decimal places here. There is long-standing consensus that additional decimal places here do not contribute to understanding. Thank you.
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| --><ref name="quadform">The golden ratio can be derived by the [[quadratic formula]], by starting with the first number as 1, then solving for 2nd number ''x'', where the ratios (''x'' + 1)/''x'' = ''x''/1 or (multiplying by ''x'') yields: ''x'' + 1 = ''x''<sup>2</sup>, or thus a quadratic equation: ''x''<sup>2</sup> − ''x'' − 1 = 0. Then, by the quadratic formula, for positive ''x'' = (−''b'' + √(''b''<sup>2</sup> − 4''ac''))/(2''a'') with ''a'' = 1, ''b'' = −1, ''c'' = −1, the solution for ''x'' is: (−(−1) + √((−1)<sup>2</sup> − 4·1·(−1)))/(2·1) or (1 + √(5))/2.</ref>
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| The golden ratio is also called the '''golden section''' (Latin: ''sectio aurea'') or '''golden mean'''.<ref name="livio">{{Cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|isbn=0-7679-0815-5|url=http://books.google.com/books?id=w9dmPwAACAAJ}}</ref><ref>Piotr Sadowski, ''The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight'', Cranbury NJ: Associated University Presses, 1996</ref><ref name="dunlap">Richard A Dunlap, ''The Golden Ratio and Fibonacci Numbers'', World Scientific Publishing, 1997</ref> Other names include '''extreme and mean ratio''',<ref name="Elements 6.3">Euclid, ''[http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Elements]'', Book 6, Definition 3.</ref> '''medial section''', '''divine proportion''', '''divine section''' (Latin: ''sectio divina''), '''golden proportion''', '''golden cut''',<ref>Summerson John, ''Heavenly Mansions: And Other Essays on Architecture'' (New York: W.W. Norton, 1963) p. 37. "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> and '''golden number'''.<ref>Jay Hambidge, ''Dynamic Symmetry: The Greek Vase'', New Haven CT: Yale University Press, 1920</ref><ref>William Lidwell, Kritina Holden, Jill Butler, ''Universal Principles of Design: A Cross-Disciplinary Reference'', Gloucester MA: Rockport Publishers, 2003</ref><ref name = "Pacioli">Pacioli, Luca. ''[[De divina proportione]]'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.</ref>
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| Some twentieth-century [[artist]]s and [[architect]]s, including [[Le Corbusier]] and [[Dalí]], have proportioned their works to approximate the golden ratio—especially in the form of the [[golden rectangle]], in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be [[aesthetics|aesthetically]] pleasing (see [[Golden ratio#Applications and observations|Applications and observations]] below).
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| [[Mathematician]]s since [[Euclid]] have studied the properties of the golden ratio, including its appearance in the dimensions of a [[regular pentagon]] and in a [[golden rectangle]], which can be cut into a square and a smaller rectangle with the same [[aspect ratio]]. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as [[financial market]]s, in some cases based on dubious fits to data.<ref>{{Cite news|first=Steven|last=Strogatz|authorlink=Steven Strogatz|title=Me, Myself, and Math: Proportion Control|newspaper=[[New York Times]]|date=September 24, 2012|url=http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/}}</ref>
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| At least 1,000,000,000,000 decimal digits are known.{{fact|date=January 2014}}
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| {{TOC limit|3}}
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| ==Calculation==
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| {| style="float:right; margin:10px" class="wikitable"
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| | colspan="2" style="text-align:center;"| {{Irrational numbers}}
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| |[[Binary numeral system|Binary]]
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| | 1.1001111000110111011...
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| | [[Decimal]]
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| | 1.6180339887498948482...
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| | [[Hexadecimal]]
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| | 1.9E3779B97F4A7C15F39...
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| | [[Continued fraction]]
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| | <math>1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}</math>
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| |-
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| | [[Algebraic number|Algebraic form]]
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| | <math>\frac{1 + \sqrt{5}}{2}</math>
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| | [[Infinite series]]
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| | <math>\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}</math><br />
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| |}
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| Two quantities ''a'' and ''b'' are said to be in the ''golden ratio'' ''φ'' if
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| :<math> \frac{a+b}{a} = \frac{a}{b} = \varphi.</math>
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| One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,
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| :<math>\frac{a+b}{a} = 1 + \frac{b}{a} = 1 + \frac{1}{\varphi}.</math>
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| Therefore,
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| :<math> 1 + \frac{1}{\varphi} = \varphi. </math>
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| Multiplying by ''φ'' gives
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| :<math>\varphi + 1 = \varphi^2</math>
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| which can be rearranged to
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| :<math>{\varphi}^2 - \varphi - 1 = 0.</math>
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| Using the [[quadratic formula]], two solutions are obtained:
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| :<math>\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots</math>
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| and
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| :<math>\varphi = \frac{1 - \sqrt{5}}{2} = -0.6180\,339887\dots</math>
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| Because ''φ'' is the ratio between positive quantities ''φ'' is necessarily positive:
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| :<math>\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots</math> .
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| ==History==
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| [[File:Phi uc lc.svg|thumb|200px|Mathematician [[Mark Barr]] proposed using the first letter in the name of Greek sculptor [[Phidias]], ''phi'', to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the [[Reciprocal (mathematics)|reciprocal]] of the golden ratio, 1/φ.<ref name="MathWorld GR Conjugate"/>]]
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| [[File:Michael Maestlin.jpg|thumb|[[Michael Maestlin]], first to publish a decimal approximation of the golden ratio, in 1597]]
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| The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to [[Mario Livio]]:
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| {{quote|Some of the greatest mathematical minds of all ages, from [[Pythagoras]] and [[Euclid]] in [[ancient Greece]], through the medieval Italian mathematician [[Fibonacci|Leonardo of Pisa]] and the Renaissance astronomer [[Johannes Kepler]], to present-day scientific figures such as Oxford physicist [[Roger Penrose]], have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.<ref>Mario Livio,''The Golden Ratio: The Story of Phi, The World's Most Astonishing Number'', p.6</ref>}}
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| [[Ancient Greece|Ancient Greek]] mathematicians first studied what we now call the golden ratio because of its frequent appearance in [[geometry]]. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular [[Pentagram#Geometry|pentagram]]s and [[pentagon]]s. [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' ([[Greek language|Greek]]: {{lang|grc|Στοιχεῖα}}) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been ''cut in extreme and mean ratio'' when, as the whole line is to the greater segment, so is the greater to the lesser."<ref>"῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν" as translated in {{cite book |title=Euclid's Elements of Geometry |author=Richard Fitzpatrick (translator) |year=2007 |isbn=978-0615179841}}, p. 156</ref> Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e., the golden ratio.<ref>Euclid, ''[http:/.aleph0.clarku.edu/~djoyce/java/elements/toc.html Elements]'', Book 6, Proposition 30.</ref> Throughout the ''Elements'', several propositions ([[theorem]]s in modern terminology) and their proofs employ the golden ratio.<ref>Euclid, ''[http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Elements]'', Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.</ref>
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| The golden ratio is explored in [[Luca Pacioli]]'s book ''[[De divina proportione]]'' of 1509.
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| The first known approximation of the (inverse) golden ratio by a [[decimal fraction]], stated as "about 0.6180340", was written in 1597 by [[Michael Maestlin]] of the [[University of Tübingen]] in a letter to his former student [[Johannes Kepler]].<ref>{{cite web| url = http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html | title = The Golden Ratio | work = The MacTutor History of Mathematics archive | accessdate = 2007-09-18}}</ref>
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| Since the 20th century, the golden ratio has been represented by the [[Greek alphabet|Greek letter]] '''''φ''''' ([[phi]], after [[Phidias]], a sculptor who is said to have employed it) or less commonly by '''''τ''''' ([[tau]], the first letter of the [[ancient Greek]] root τομή—meaning ''cut'').<ref name="livio"/><ref>{{Mathworld|title=Golden Ratio|urlname=GoldenRatio}}</ref>
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| ===Timeline===
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| Timeline according to Priya Hemenway:<ref name=Hemenway,P>{{Cite book
| |
| | last = Hemenway
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| | first = Priya
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| | title = Divine Proportion: Phi In Art, Nature, and Science
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| | year = 2005
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| | publisher = Sterling
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| | location = New York
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| | isbn = 1-4027-3522-7
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| | pages = 20–21
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| }}</ref>
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| * [[Phidias]] (490–430 BC) made the [[Parthenon]] statues that seem to embody the golden ratio.
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| * [[Plato]] (427–347 BC), in his ''[[Timaeus (dialogue)|Timaeus]]'', describes five possible regular solids (the [[Platonic solids]]: the [[tetrahedron]], [[cube]], [[octahedron]], [[dodecahedron]], and [[icosahedron]]), some of which are related to the golden ratio.<ref>{{cite web
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| | last = Plato
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| | authorlink = Plato
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| | year = 360 BC) (Benjamin Jowett trans.
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| | url = http://classics.mit.edu/Plato/timaeus.html
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| | title = Timaeus
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| | publisher = The Internet Classics Archive
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| | accessdate = 2006-05-30
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| }}</ref>
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| * [[Euclid]] (c. 325–c. 265 BC), in his ''[[Euclid's Elements|Elements]]'', gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ἄκρος καὶ μέσος λόγος).<ref name="Elements 6.3"/>
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| * [[Fibonacci]] (1170–1250) mentioned the [[Sequence|numerical series]] now named after him in his ''[[Liber Abaci]]''; the ratio of sequential elements of the [[Fibonacci number|Fibonacci sequence]] approaches the golden ratio asymptotically.
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| * [[Luca Pacioli]] (1445–1517) defines the golden ratio as the "divine proportion" in his ''Divina Proportione''.
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| * [[Michael Maestlin]] (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a [[decimal fraction]].
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| * [[Johannes Kepler]] (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,<ref name=tatt>
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| {{Cite book
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| | title = Elementary number theory in nine chapters | |
| | edition = 2nd
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| | author = James Joseph Tattersall
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| | publisher = [[Cambridge University Press]]
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| | year = 2005
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| | isbn = 978-0-521-85014-8
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| | page = 28
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| | url = http://books.google.com/?id=QGgLbf2oFUYC&pg=PA29&dq=golden-ratio+limit+fibonacci+ratio+kepler&q=golden-ratio%20limit%20fibonacci%20ratio%20kepler
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| }}</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]].
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| * [[Charles Bonnet]] (1720–1793) points out that in the spiral [[phyllotaxis]] of plants going [[clockwise]] and counter-clockwise were frequently two successive Fibonacci series.
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| * [[Martin Ohm]] (1792–1872) is believed to be the first to use the term ''goldener Schnitt'' (golden section) to describe this ratio, in 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>
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| * [[Édouard Lucas]] (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
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| * Mark Barr (20th century) suggests the Greek letter phi ('''φ'''), the initial letter of Greek sculptor Phidias's name, as a [[symbol]] for the golden ratio.<ref>{{Cite book
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| | last = Cook
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| | first = Theodore Andrea
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| | title = The Curves of Life
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| | origyear = 1914
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| | url = http://books.google.com/?id=ea-TStM-07EC&pg=PA420&dq=phi+mark+barr+intitle:The+intitle:Curves+intitle:of+intitle:Life
| |
| | year = 1979
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| | publisher = Dover Publications
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| | location = New York
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| | isbn = 0-486-23701-X
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| }}</ref>
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| * [[Roger Penrose]] (b. 1931) discovered in 1974 the [[Penrose tiling]], a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.<ref>{{citation|title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics|first=Martin|last=Gardner|authorlink=Martin Gardner|publisher=W. W. Norton & Company|year=2001|isbn=9780393020236|page=88|url=http://books.google.com/books?id=orz0SDEakpYC&pg=PA88}}.</ref> This in turn led to new discoveries about [[quasicrystal]]s.<ref>{{citation|title=Introduction to the Mathematics of Quasicrystals|first=Marko V.|last=Jaric|publisher=Elsevier|year=2012|isbn=9780323159470|page=x|url=http://books.google.com/books?id=OToVjZW9CKMC&pg=PR10|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.}}</ref>
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| ==Applications and observations==
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| ===Aesthetics===
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| {{See also|History of aesthetics (pre-20th-century)}}
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| ''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. ''De Divina Proportione'' contains illustrations of regular solids by [[Leonardo da Vinci]], Pacioli's longtime friend and collaborator.
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| ===Architecture===
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| [[File:Acropolis of Athens 01361.JPG|thumb|right|250px|Many of the proportions of the [[Parthenon]] are alleged to exhibit the golden ratio.]]
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| The [[Parthenon]]'s façade as well as elements of its façade 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 example, Midhat J. Gazalé says, "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 [[decagon]]s 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.
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| A 2004 geometrical analysis of earlier research into 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]]. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction.
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| The Swiss [[architect]] [[Le Corbusier]], famous for his contributions to the [[modernism|modern]] [[International style (architecture)|international style]], centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."<ref>Le Corbusier, ''The Modulor'' p. 25, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6</ref>
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| Le Corbusier explicitly used the golden ratio in his [[Modulor]] system for the [[scale (ratio)|scale]] of [[Proportion (architecture)|architectural proportion]]. He saw this system as a continuation of the long tradition of [[Vitruvius]], Leonardo da Vinci's "[[Vitruvian Man]]", the work of [[Leon Battista Alberti]], and others who used the proportions of the human body to improve the appearance and function of [[architecture]]. In addition to the golden ratio, Le Corbusier based the system on [[anthropometry|human measurements]], [[Fibonacci numbers]], and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the [[Modulor]] system. Le Corbusier's 1927 Villa Stein in [[Garches]] exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.<ref>Le Corbusier, ''The Modulor'', p. 35, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".</ref>
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| Another Swiss architect, [[Mario Botta]], bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in [[Origlio]], the golden ratio is the proportion between the central section and the side sections of the house.<ref>Urwin, Simon. ''Analysing Architecture'' (2003) pp. 154-5, ISBN 0-415-30685-X</ref>
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| In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the [[Naqsh-e Jahan Square]] and the adjacent Lotfollah mosque.<ref>
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| {{Cite book
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| | author = Jason Elliot
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| | title = Mirrors of the Unseen: Journeys in Iran
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| | year = 2006
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| | pages = 277, 284
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| | publisher = Macmillan
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| | isbn = 978-0-312-30191-0
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| | url = http://books.google.com/?id=Gcs4IjUx3-4C&pg=PA284&dq=intitle:%22Mirrors+of+the+Unseen%22+golden-ratio+maidan
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| }}</ref>
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| ===Painting===
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| [[File:Pentagram and human body (Agrippa).jpg|right|thumb|The drawing of a man's body in a pentagram suggests relationships to the golden ratio.<ref name="Sadowski"/>]]
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| The 16th-century philosopher [[Heinrich Agrippa]] drew a man over a [[pentagram]] inside a circle, implying a relationship to the golden ratio.<ref name="Sadowski">
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| {{cite book
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| | title = The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight
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| | author = Piotr Sadowski
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| | publisher = University of Delaware Press
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| | year = 1996
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| | isbn = 978-0-87413-580-0
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| | page = 124
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| | url = http://books.google.com/books?id=RNFqRs3Ccp4C&pg=PA124
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| }}</ref>
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| [[Leonardo da Vinci]]'s illustrations of [[polyhedra]] in ''[[De divina proportione]]'' (''On the Divine Proportion'') and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.<ref>''Leonardo da Vinci's Polyhedra'', by [[George W. Hart]][http://www.georgehart.com/virtual-polyhedra/leonardo.html]</ref> But the suggestion that his ''[[Mona Lisa]]'', for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings.<ref>{{cite web|url=http://plus.maths.org/issue22/features/golden/|author=Livio, Mario|accessdate=2008-03-21|title=The golden ratio and aesthetics}}</ref> Similarly, although the ''[[Vitruvian Man]]'' is often<ref>"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." {{cite web|author=Keith Devlin|url=http://www.maa.org/external_archive/devlin/devlin_05_07.html|title=The Myth That Will Not Go Away|accessdate=September 26, 2013|date=May 2007}}</ref> shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.<ref>{{cite web|author=Donald E. Simanek|url=http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm|title=Fibonacci Flim-Flam|accessdate=April 9, 2013}}</ref>
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| | |
| [[Salvador Dalí]], influenced by the works of [[Matila Ghyka]],<ref>{{cite video |people=Salvador Dalí |date=2008 |title=The Dali Dimension: Decoding the Mind of a Genius |url= |format=DVD |language=English |publisher=Media 3.14-TVC-FGSD-IRL-AVRO|url=http://www.dalidimension.com/eng/index.html}}</ref> explicitly used the golden ratio in his masterpiece, ''[[The Sacrament of the Last Supper]]''. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.<ref name=livio/><ref>Hunt, Carla Herndon and Gilkey, Susan Nicodemus. ''Teaching Mathematics in the Block'' pp. 44, 47, ISBN 1-883001-51-X</ref>
| |
| | |
| [[Piet Mondrian|Mondrian]] has been said to have used the golden section extensively in his geometrical paintings,<ref>Bouleau, Charles, ''The Painter's Secret Geometry: A Study of Composition in Art'' (1963) pp.247-8, Harcourt, Brace & World, ISBN 0-87817-259-9</ref> though other experts (including critic [[Yve-Alain Bois]]) have disputed this claim.<ref name=livio/>
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| | |
| A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).<ref>Olariu, Agata, ''Golden Section and the Art of Painting'' [http://arxiv.org/abs/physics/9908036/ Available online]</ref> On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.<ref>Tosto, Pablo, ''La composición áurea en las artes plásticas – El número de oro'', Librería Hachette, 1969, p. 134–144</ref>
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| {{-}}
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| | |
| ===Book design===
| |
| [[File:Medieval manuscript framework.svg|thumb|Depiction of the proportions in a medieval manuscript. According to [[Jan Tschichold]]: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."<ref>[[Jan Tschichold]]. ''The Form of the Book'', pp.43 Fig 4. "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>]]
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| {{Main|Canons of page construction}}
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| | |
| According to [[Jan Tschichold]],<ref>[[Jan Tschichold]], ''The Form of the Book'', Hartley & Marks (1991), ISBN 0-88179-116-4.</ref>
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| | |
| <blockquote>There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.</blockquote>
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| | |
| ===Design===
| |
| Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing cards, posters, wide-screen televisions, photographs, light switch plates and cars.<ref>
| |
| {{Cite journal
| |
| |title=The golden section: A most remarkable measure
| |
| |first=Ronald|last=Jones
| |
| |journal=The Structurist
| |
| |volume=11|year=1971
| |
| |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?
| |
| }}</ref><ref>
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| {{cite book
| |
| | title = Famous problems and their mathematicians
| |
| | author = Art Johnson
| |
| | publisher = Libraries Unlimited
| |
| | year = 1999
| |
| | isbn = 978-1-56308-446-1
| |
| | page = 45
| |
| | url = http://books.google.com/?id=STKX4qadFTkC&pg=PA45&dq=switch+%22golden+ratio%22#v=onepage&q=switch%20%22golden%20ratio%22&f=false
| |
| | quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
| |
| }}</ref><ref>
| |
| {{cite book
| |
| | title = The mathematics of harmony: from Euclid to contemporary mathematics and computer science
| |
| | edition =
| |
| | author = Alexey Stakhov, Scott Olsen, Scott Anthony Olsen
| |
| | publisher = World Scientific
| |
| | year = 2009
| |
| | isbn =978-981-277-582-5
| |
| | page = 21
| |
| | url = http://books.google.com/?id=K6fac9RxXREC&pg=PA21&dq=%22credit+card%22+%22golden+ratio%22+rectangle#v=onepage&q=%22credit%20card%22%20%22golden%20ratio%22%20rectangle&f=false
| |
| | quote = A credit card has a form of the golden rectangle.
| |
| }}</ref><ref>
| |
| {{cite book
| |
| | title = Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel
| |
| | author = Simon Cox
| |
| | publisher = Barnes & Noble Books
| |
| | year = 2004
| |
| | isbn = 978-0-7607-5931-8
| |
| | url = http://books.google.com/?id=TbjwhwLCEeAC&q=%22golden+ratio%22+postcard&dq=%22golden+ratio%22+postcard
| |
| | 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>
| |
| {{cite web
| |
| | title = THE NEW RAPIDE S : Design
| |
| | url = http://www.astonmartin.com/cars/rapide-s/rapide-s-design
| |
| | quote = The ‘Golden Ratio’ sits at the heart of every Aston Martin.
| |
| }}</ref>
| |
| | |
| ===Music===
| |
| [[Ernő Lendvaï]] analyzes [[Béla Bartók]]'s works as being based on two opposing systems, that of the golden ratio and the [[acoustic scale]],<ref>Lendvai, Ernő (1971). ''Béla Bartók: An Analysis of His Music''. London: Kahn and Averill.</ref> though other music scholars reject that analysis.<ref name="livio"/> French composer [[Erik Satie]] used the golden ratio in several of his pieces, including ''Sonneries de la Rose+Croix''. The golden ratio is also apparent in the organization of the sections in the music of [[Debussy]]'s ''[[Reflets dans l'eau]] (Reflections in Water)'', from ''Images'' (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."<ref name=Smith>Smith, Peter F. ''[http://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83&dq=bartok+intitle:The+intitle:Dynamics+intitle:of+intitle:Delight+intitle:Architecture+intitle:and+intitle:Aesthetics&as_brr=0&ei=WkkSR5L6OI--ogLpmoyzBg&sig=Ijw4YifrLhkcdQSMVAjSL5g4zVk The Dynamics of Delight: Architecture and Aesthetics]'' (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X</ref>
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| | |
| The musicologist [[Roy Howat]] has observed that the formal boundaries of ''[[La Mer (Debussy)|La Mer]]'' correspond exactly to the golden section.<ref>{{Cite book| title = Debussy in Proportion: A Musical Analysis | author = Roy Howat | url = http://books.google.com/?id=4bwKykNp24wC&pg=PA169&dq=intitle:Debussy+intitle:in+intitle:Proportion+golden+la-mer | publisher = Cambridge University Press | year = 1983 | isbn = 0-521-31145-4 }}</ref> Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.<ref>{{Cite book| title = Debussy: La Mer | author = Simon Trezise | publisher = Cambridge University Press | year = 1994 | isbn = 0-521-44656-2 | page = 53 | url = http://books.google.com/?id=THD1nge_UzcC&pg=PA53&dq=inauthor:Trezise+golden+evidence }}</ref>
| |
| | |
| [[Pearl Drums]] positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a [[patent]] on this innovation.<ref>
| |
| {{cite web
| |
| | url = http://www.pearldrum.com/premium-birch.asp
| |
| | title = Pearl Masters Premium
| |
| | accessdate =December 2, 2007
| |
| | publisher = Pearl Corporation
| |
| }}</ref>
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| | |
| Though [[Heinz Bohlen]] proposed the non-octave-repeating [[833 cents scale]] based on [[combination tone]]s, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents ({{audio|Golden ratio on C.mid|Play}}).<ref>"[http://www.huygens-fokker.org/bpsite/833cent.html An 833 Cents Scale: An experiment on harmony]", ''Huygens-Fokker.org''. Accessed December 1, 2012.</ref>
| |
| | |
| ===Nature===
| |
| [[File:Aeonium tabuliforme.jpg|thumb|A detail of an ''[[Aeonium tabuliforme]]'' in [[Trädgårdsföreningen]], [[Göteborg]]]]
| |
| | |
| [[Adolf Zeising]], whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the [[Plant stem|stem]]s of plants and of [[Leaf#Veins|veins]] in leaves. He extended his research to the [[skeleton]]s of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of [[crystal]]s, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.<ref>{{Cite book
| |
| | title = Proportion
| |
| | author = Richard Padovan
| |
| | publisher = Taylor & Francis
| |
| | year = 1999
| |
| | isbn = 978-0-419-22780-9
| |
| | pages = 305–306
| |
| | url = http://books.google.com/?id=Vk_CQULdAssC&pg=PA306&dq=%22contained+the+ground-principle+of+all+formative+striving%22
| |
| }}</ref><ref>{{cite journal|journal=Nexus Network Journal|first=Richard|last=Padovan|title=Proportion: Science, Philosophy, Architecture|volume=4|pages=113–122|doi=10.1007/s00004-001-0008-7|year=2002|issue=1}}</ref> In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all [[structure]]s, [[Visual appearance|form]]s and [[Proportionality (mathematics)|proportions]], whether cosmic or individual, [[Organic chemistry|organic]] or [[inorganic]], [[Acoustics|acoustic]] or [[optical]]; which finds its fullest realization, however, in the human form."<ref>{{cite book|first=Adolf|last=Zeising|title=Neue Lehre van den Proportionen des meschlischen Körpers|year=1854|page=preface|nopp=true}}</ref>
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| In 2010, the journal ''Science'' reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.<ref>{{cite web|url=http://www.eurekalert.org/pub_releases/2010-01/haog-grd010510.php |title=Golden ratio discovered in a quantum world |publisher=Eurekalert.org |date=2010-01-07 |accessdate=2011-10-31}}</ref>
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| | |
| Since 1991, several researchers have proposed connections between the golden ratio and [[human genome]] [[DNA]].<ref>
| |
| J.C. Perez (1991), [http://golden-ratio-in-dna.blogspot.com/2008/01/1991-first-publication-related-to.html "Chaos DNA and Neuro-computers: A Golden Link"], in ''Speculations in Science and Technology'' vol. 14 no. 4, {{ISSN|0155-7785}}.
| |
| </ref><ref>
| |
| Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), [http://www.springerlink.com/content/p140352473151957/?p=d5b18a2dfee949858e2062449e9ccfad&pi=0 "Nucleotide Frequencies in Human Genome and Fibonacci Numbers"], in ''Bulletin of Mathematical Biology,'' {{ISSN|0092-8240}} (print), {{ISSN|1522-9602}} (online). [http://www.springerlink.com/content/p140352473151957/fulltext.pdf PDF full text]
| |
| </ref><ref>{{cite journal |author= Perez, J.-C. |title= Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618 |journal= Interdisciplinary Sciences: Computational Life Science |date=September 2010 |volume= 2 |issue= 3 |pages= 228–240 |pmid= 20658335 |doi= 10.1007/s12539-010-0022-0 }} [http://fr.scribd.com/doc/95641538/Codon-Populations-in-Single-stranded-Whole-Human-Genome-DNA-Are-Fractal-and-Fine-tuned-by-the-Golden-Ratio-1-618 PDF full text]
| |
| </ref><ref>{{cite journal |author= Perez, J.-C. |title= The “3 Genomic Numbers” Discovery: How Our Genome Single-Stranded DNA Sequence Is “Self-Designed” as a Numerical Whole |journal= Applied Mathematics, Biomathematics issue |date=October 2013 |volume= 4 |issue= 10B |pages= 37–53 |doi= 10.4236/am.2013.410A2004 }} [http://file.scirp.org/Html/4-7401586_37457.htm J.C. Perez, The “3 Genomic Numbers” Discovery: How Our Genome Single-Stranded DNA Sequence Is “Self-Designed” as a Numerical Whole, Applied Mathematics (Biomathematics issue October 2013)]</ref>
| |
| | |
| However, some have argued that many of the apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are in fact fictitious.<ref>Pommersheim, James E., Tim K. Marks, and [[Erica Flapan|Erica L. Flapan]], eds. 2010. Number Theory: A Lively Introduction with Proofs, Applications, and Stories. John Wiley and Sons: 82.</ref>
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| | |
| ===Optimization===
| |
| The golden ratio is key to the [[golden section search]].
| |
| | |
| ===Perceptual studies===
| |
| Studies by psychologists, starting with [[Gustav Fechner|Fechner]], have been devised to test the idea that the golden ratio plays a role in human perception of [[beauty]]. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.<ref name="livio" /><ref>[http://plus.maths.org/issue22/features/golden/ The golden ratio and aesthetics], by Mario Livio.</ref>
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| | |
| ==Mathematics==
| |
| | |
| ===Golden ratio conjugate===
| |
| The negative root of the quadratic equation for '''φ''' (the "conjugate root") is
| |
| | |
| :<math>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt{5}}{2} = -0.61803\,39887\dots.</math>
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| | |
| The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ''b/a''), and is sometimes referred to as the ''golden ratio conjugate''.<ref name="MathWorld GR Conjugate">{{MathWorld|title=Golden Ratio Conjugate|urlname=GoldenRatioConjugate}}</ref> It is denoted here by the capital Phi ('''Φ'''):
| |
| | |
| :<math>\Phi = {1 \over \varphi} = \varphi^{-1} = 0.61803\,39887\ldots.</math>
| |
| | |
| Alternatively, '''Φ''' can be expressed as
| |
| | |
| :<math>\Phi = \varphi -1 = 1.61803\,39887\ldots -1 = 0.61803\,39887\ldots.</math>
| |
| | |
| This illustrates the unique property of the golden ratio among positive numbers, that
| |
| | |
| :<math>{1 \over \varphi} = \varphi - 1,</math>
| |
| | |
| or its inverse:
| |
| | |
| :<math>{1 \over \Phi} = \Phi + 1.</math>
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| | |
| This means 0.61803...:1 = 1:1.61803....
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| | |
| ===Short proofs of irrationality===
| |
| | |
| ====Contradiction from an expression in lowest terms====
| |
| [[File:Whirling squares.svg|thumb|right|If φ were [[Rational number|rational]], then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so φ cannot be rational.]]
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| | |
| Recall that:
| |
| | |
| : the whole is the longer part plus the shorter part;
| |
| : the whole is to the longer part as the longer part is to the shorter part.
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| | |
| If we call the whole ''n'' and the longer part ''m'', then the second statement above becomes
| |
| | |
| : ''n'' is to ''m'' as ''m'' is to ''n'' − ''m'',
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| or, algebraically
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| | |
| : <math> \frac nm = \frac{m}{n-m}.\qquad (*) </math>
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| | |
| To say that ''φ'' is rational means that ''φ'' is a fraction ''n''/''m'' where ''n'' and ''m'' are integers. We may take ''n''/''m'' to be in lowest terms and ''n'' and ''m'' to be positive. But if ''n''/''m'' is in lowest terms, then the identity labeled (*) above says ''m''/(''n'' − ''m'') is in still lower terms. That is a contradiction that follows from the assumption that ''φ'' is rational.
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| | |
| ====Derivation from irrationality of √5====
| |
| Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the [[closure (mathematics)|closure]] of rational numbers under addition and multiplication. If <math>\textstyle\frac{1 + \sqrt{5}}{2}</math> is rational, then <math>\textstyle2\left(\frac{1 + \sqrt{5}}{2} - \frac{1}{2}\right) = \sqrt{5}</math> is also rational, which is a contradiction if it is already known that the square root of a non-[[square number|square]] [[natural number]] is irrational.
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| | |
| ===Alternative forms===
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| [[File:Golden mean.png|right|thumb|Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers]]
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| The formula ''φ'' = 1 + 1/''φ'' can be expanded recursively to obtain a [[continued fraction]] for the golden ratio:<ref>{{Cite book| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
| |
| | author = Max. Hailperin, Barbara K. Kaiser, and Karl W. Knight | publisher = Brooks/Cole Pub. Co | year = 1998 | isbn = 0-534-95211-9 | url = http://books.google.com/?id=yYyVRueWlZ8C&pg=PA63&dq=continued-fraction+substitute+golden-ratio }}</ref>
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| | |
| :<math>\varphi = [1; 1, 1, 1, \dots] = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>
| |
| | |
| and its reciprocal:
| |
| | |
| :<math>\varphi^{-1} = [0; 1, 1, 1, \dots] = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>
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| | |
| The [[Convergent (continued fraction)|convergent]]s of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive [[Fibonacci numbers]].
| |
| | |
| The equation ''φ''<sup>2</sup> = 1 + ''φ'' likewise produces the continued [[square root]], or infinite surd, form:
| |
| | |
| :<math>\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}.</math>
| |
| | |
| An infinite series can be derived to express phi:<ref>Brian Roselle, [http://sites.google.com/site/goldenmeanseries/ "Golden Mean Series"]</ref><br />
| |
| :<math>\varphi=\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}.</math>
| |
| | |
| Also:
| |
| | |
| :<math>\varphi = 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ</math>
| |
| :<math>\varphi = {1 \over 2}\csc(\pi/10) = {1 \over 2}\csc 18^\circ</math>
| |
| :<math>\varphi = 2\cos(\pi/5)=2\cos 36^\circ</math>
| |
| :<math> \varphi = 2\sin(3\pi/10)=2\sin 54^\circ. </math>
| |
| | |
| These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a [[pentagram]].
| |
| | |
| ===Geometry===
| |
| [[File:FakeRealLogSpiral.svg|thumb|350px|right|Approximate and true [[golden spiral]]s. The <span style="color:green;">green</span> spiral is made from quarter-circles tangent to the interior of each square, while the <span style="color:maroon;">red</span> spiral is a Golden Spiral, a special type of [[logarithmic spiral]]. Overlapping portions appear <span style="color:olive;">yellow</span>. The length of the side of one square divided by that of the next smaller square is the golden ratio.]]
| |
| The number φ turns up frequently in [[geometry]], particularly in figures with pentagonal [[symmetry]].
| |
| The length of a regular [[pentagon]]'s [[diagonal]] is φ times its side.
| |
| The vertices of a regular [[icosahedron]] are those of [[three]] mutually [[orthogonal]] [[golden rectangle]]s.
| |
| | |
| There is no known general [[algorithm]] to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, ''[[Thomson problem]]''). However, a useful approximation results from dividing the sphere into parallel bands of equal [[surface area]] and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ {{Unicode|≅}} 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory [[artificial satellite|satellite]] [[STARSHINE|Starshine-3]].<ref>{{cite web|url=http://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/ |title=A Disco Ball in Space |publisher=NASA |date=2001-10-09 |accessdate=2007-04-16}}</ref>
| |
| {{-}}
| |
| | |
| ====Dividing a line segment====
| |
| [[File:Goldener Schnitt Konstr beliebt.svg|right|thumb|250px|Dividing a line segment according to the golden ratio]]
| |
| The following [[algorithm]] produces a [[geometric construction]] that divides a [[line segment]] into two line segments where the ratio of the longer to the shorter line segment is the golden ratio:
| |
| # Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the [[hypotenuse]] AC.
| |
| # Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.
| |
| # Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original segment AB into line segments AS and SB with lengths in the golden ratio.
| |
| {{-}}
| |
| | |
| ====Golden triangle, pentagon and pentagram====
| |
| [[File:Golden triangle (math).svg|right|thumb|[[Golden triangle (mathematics)|Golden triangle]]]]
| |
| | |
| =====Golden triangle=====
| |
| The [[Golden triangle (mathematics)|golden triangle]] can be characterized as an [[isosceles triangle]] ABC with the property that [[bisection|bisecting]] the angle C produces a new [[triangle]] CXB which is a [[similar triangle]] to the original.
| |
| | |
| If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.
| |
| | |
| Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ + 1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ<sup>2</sup>. Thus φ<sup>2</sup> = φ + 1, confirming that φ is indeed the golden ratio.
| |
| | |
| Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the [[inverse (mathematics)|inverse]] ratio is φ − 1.
| |
| | |
| =====Pentagon=====
| |
| In a regular pentagon the ratio between a side and a diagonal is <math>\Phi</math> (i.e. 1/φ), while intersecting diagonals section each other in the golden ratio.<ref name = "Pacioli"/>
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| =====Odom's construction=====
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| [[File:Odom.svg|thumb|218 px|Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C.<br/><center><math>\tfrac{|AB|}{|BC|}=\tfrac{|AC|}{|AB|}=\phi</math></center>]]
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| [[George Phillips Odom, Jr|George Odom]] has given a remarkably simple construction for ''φ'' involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the [[Power of a point|intersecting chords theorem]] and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] who published it in Odom's name as a diagram in the ''[[American Mathematical Monthly]]'' accompanied by the single word "Behold!" <ref>{{cite web|title=Quandaries and Queries|url=http://mathcentral.uregina.ca/qq/database/QQ.09.02/mary1.html|publisher=Math Central|accessdate=23 October 2011|author=Chris and Penny}}</ref>
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| | |
| =====Pentagram=====
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| [[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four
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| lengths are in golden ratio to one another.]]
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| The golden ratio plays an important role in the geometry of [[pentagram]]s. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.
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| | |
| The pentagram includes ten [[isosceles triangle]]s: five [[acute triangle|acute]] and five [[obtuse triangle|obtuse]] isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are [[Golden triangle (mathematics)|golden triangle]]s. The obtuse isosceles triangles are [[Golden triangle (mathematics)|golden gnomons]].
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| =====Ptolemy's theorem=====
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| [[File:Ptolemy Pentagon.svg|thumb|The golden ratio in a regular pentagon can be computed using [[Ptolemy's theorem]].]]
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| | |
| The golden ratio properties of a regular pentagon can be confirmed by applying [[Ptolemy's theorem]] to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are ''b'', and short edges are ''a'', then Ptolemy's theorem gives ''b''<sup>2</sup> = ''a''<sup>2</sup> + ''ab'' which yields
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| :<math>{b \over a}={{1+\sqrt{5}}\over 2}.</math>
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| ====Scalenity of triangles====
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| Consider a [[triangle]] with sides of lengths ''a'', ''b'', and ''c'' in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios ''a''/''b'' and ''b''/''c''. The scalenity is always less than φ and can be made as close as desired to φ.<ref>''[[American Mathematical Monthly]]'', pp. 49-50, 1954.</ref>
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| ====Triangle whose sides form a geometric progression====
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| If the side lengths of a triangle form a [[geometric progression]] and are in the ratio 1 : ''r'' : ''r''<sup>2</sup>, where ''r'' is the common ratio, then ''r'' must lie in the range φ−1 < ''r'' < φ, which is a consequence of the [[triangle inequality]] (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If ''r'' = φ then the shorter two sides are 1 and φ but their sum is φ<sup>2</sup>, thus ''r'' < φ. A similar calculation shows that ''r'' > φ−1. A triangle whose sides are in the ratio 1 : √φ : φ is a right triangle (because 1 + φ = φ<sup>2</sup>) known as a [[Kepler triangle]].<ref name=herz/>
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| ====Golden triangle, rhombus, and rhombic triacontahedron====
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| [[File:GoldenRhombus.svg|thumb|One of the rhombic triacontahedron's rhombi]]
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| | |
| [[File:Rhombictriacontahedron.svg|thumb|All of the faces of the rhombic triacontahedron are golden rhombi]]
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| | |
| A [[golden rhombus]] is a [[rhombus]] whose diagonals are in the golden ratio. The [[rhombic triacontahedron]] is a [[convex polytope]] that has a very special property: all of its faces are golden rhombi. In the [[rhombic triacontahedron]] the [[dihedral angle]] between any two adjacent rhombi is 144°, which is twice the isosceles angle of a [[Golden triangle (mathematics)|golden triangle]] and four times its most acute angle.<ref>{{citation|title=Catalan solids derived from three-dimensional-root systems and quaternions|first1=Mehmet|last1=Koca|first2=Nazife Ozdes|last2=Koca|first3=Ramazan|last3=Koç|journal= Journal of Mathematical Physics|volume=51|year=2010|page=043501|doi=10.1063/1.3356985|arxiv=0908.3272}}.</ref>
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| | |
| ===Relationship to Fibonacci sequence===
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| The mathematics of the golden ratio and of the [[Fibonacci number|Fibonacci sequence]] are intimately interconnected. The Fibonacci sequence is:
| |
| | |
| :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....
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| The [[closed-form expression]] (known as [[Jacques Philippe Marie Binet|Binet]]'s formula, even though it was already known by [[Abraham de Moivre]]) for the Fibonacci sequence involves the golden ratio:
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| :<math>F\left(n\right)
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| = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}
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| = {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}.</math>
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| [[File:Fibonacci spiral 34.svg|thumb|250px|right|A [[Fibonacci sequence|Fibonacci spiral]] which approximates the [[golden spiral]], using Fibonacci sequence square sizes up to 34.]]
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| The golden ratio is the [[limit of a sequence|limit]] of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by [[Kepler]]:<ref name="tatt"/>
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| :<math>\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi.</math>
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| Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:
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| :<math>\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)|
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| = \varphi.</math>
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| More generally:
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| :<math>\lim_{n\to\infty}\frac{F(n+a)}{F(n)}={\varphi}^a,</math>
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| | |
| where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when <math>a = 1</math>.
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| Furthermore, the successive powers of φ obey the Fibonacci [[recurrence relation|recurrence]]:
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| :<math>\varphi^{n+1}
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| = \varphi^n + \varphi^{n-1}.</math>
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| This identity allows any polynomial in φ to be reduced to a linear expression. For example:
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| : <math>
| |
| \begin{align}
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| 3\varphi^3 - 5\varphi^2 + 4 & = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\
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| & = 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 \\
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| & = \varphi + 2 \approx 3.618.
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| \end{align}
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| </math>
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| However, this is no special property of φ, because polynomials in any solution ''x'' to a [[quadratic equation]] can be reduced in an analogous manner, by applying:
| |
| :<math>x^2=ax+b</math>
| |
| for given coefficients ''a'', ''b'' such that ''x'' satisfies the equation. Even more generally, any [[rational function]] (with rational coefficients) of the root of an irreducible ''n''th-degree polynomial over the rationals can be reduced to a polynomial of degree ''n'' ‒ 1. Phrased in terms of [[field theory (mathematics)|field theory]], if α is a root of an irreducible ''n''th-degree polynomial, then <math>\Q(\alpha)</math> has degree ''n'' over <math>\Q</math>, with basis <math>\{1, \alpha, \dots, \alpha^{n-1}\}</math>.
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| ===Symmetries===
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| The golden ratio and inverse golden ratio <math>\varphi_\pm = (1\pm \sqrt{5})/2</math> have a set of symmetries that preserve and interrelate them. They are both preserved by the [[fractional linear transformation]]s <math>x, 1/(1-x), (x-1)/x,</math> – this fact corresponds to the identity and the definition quadratic equation.
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| Further, they are interchanged by the three maps <math>1/x, 1-x, x/(x-1)</math> – they are reciprocals, symmetric about <math>1/2</math>, and (projectively) symmetric about 2.
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| More deeply, these maps form a subgroup of the [[modular group]] <math>\operatorname{PSL}(2,\mathbf{Z})</math> isomorphic to the [[symmetric group]] on 3 letters, <math>S_3,</math> corresponding to the [[stabilizer subgroup|stabilizer]] of the set <math>\{0,1,\infty\}</math> of 3 standard points on the [[projective line]], and the symmetries correspond to the quotient map <math>S_3 \to S_2</math> – the subgroup <math>C_3 < S_3</math> consisting of the 3-cycles and the identity <math>() (0 1 \infty) (0 \infty 1)</math> fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.
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| ===Other properties===
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| The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see ''Alternate forms'' above). It is, for that reason, one of the [[continued fraction#A property of the golden ratio φ|worst cases]] of [[Lagrange's approximation theorem]] and it is an extremal case of the [[Hurwitz's theorem (number theory)|Hurwitz inequality]] for [[Diophantine approximation]]s. This may be why angles close to the golden ratio often show up in [[phyllotaxis]] (the growth of plants).<ref>[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Fibonacci Numbers and Nature - Part 2 : Why is the Golden section the "best" arrangement?], from [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/ Dr. Ron Knott's] [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/ Fibonacci Numbers and the Golden Section], retrieved 2012-11-29.</ref>
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| The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:
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| :<math>\varphi^2 = \varphi + 1 = 2.618\dots</math>
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| :<math>{1 \over \varphi} = \varphi - 1 = 0.618\dots.</math>
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| The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally,
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| any power of φ is equal to the sum of the two immediately preceding powers:
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| | |
| : <math>\varphi^n = \varphi^{n-1} + \varphi^{n-2} = \varphi \cdot \operatorname{F}_n + \operatorname{F}_{n-1}.</math>
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| As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ:
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| If <math> \lfloor n/2 - 1 \rfloor = m </math>, then:
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| :<math> \!\ \varphi^n = \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} </math>
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| | |
| :<math> \!\ \varphi^n - \varphi^{n-1} = \varphi^{n-2} . </math>
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| When the golden ratio is used as the base of a [[numeral system]] (see [[Golden ratio base]], sometimes dubbed ''phinary'' or ''φ-nary''), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation.
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| The golden ratio is a [[Fundamental unit (number theory)|fundamental unit]] of the [[algebraic number field]] <math>\mathbb{Q}(\sqrt{5})</math> and is a [[Pisot–Vijayaraghavan number]].<ref>{{MathWorld|urlname=PisotNumber |title=Pisot Number}}</ref> In the field <math>\mathbb{Q}(\sqrt{5})</math> we have <math>\varphi^n = {{L_n + F_n \sqrt{5}} \over 2}</math>, where <math>L_n</math> is the <math>n</math>-th [[Lucas number]].
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| The golden ratio also appears in [[hyperbolic geometry]], as the maximum distance from a point on one side of an [[ideal triangle]] to the closer of the other two sides: this distance, the side length of the [[equilateral triangle]] formed by the points of tangency of a circle inscribed within the ideal triangle, is 4 ln φ.<ref>[http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html Horocycles exinscrits : une propriété hyperbolique remarquable], cabri.net, retrieved 2009-07-21.</ref>
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| ===Decimal expansion===
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| The golden ratio's decimal expansion can be calculated directly from the expression
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| :<math>\varphi = {1+\sqrt{5} \over 2}</math>
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| with √5 ≈ 2.2360679774997896964. The [[square root of 5]] can be calculated with the [[Babylonian method]], starting with an initial estimate such as ''x''φ = 2 and [[iterative method|iterating]]
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| | |
| :<math>x_{n+1} = \frac{(x_n + 5/x_n)}{2}</math>
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| for ''n'' = 1, 2, 3, ..., until the difference between ''x''<sub>''n''</sub> and ''x''<sub>''n''−1</sub> becomes zero, to the desired number of digits.
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| The Babylonian algorithm for √5 is equivalent to [[Newton's method]] for solving the equation ''x''<sup>2</sup> − 5 = 0. In its more general form, Newton's method can be applied directly to any [[algebraic equation]], including the equation ''x''<sup>2</sup> − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,
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| :<math>x_{n+1} = \frac{x_n^2 + 1}{2x_n - 1},</math>
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| for an appropriate initial estimate ''x''φ such as ''x''φ = 1. A slightly faster method is to rewrite the equation as ''x'' − 1 − 1/''x'' = 0, in which case the Newton iteration becomes
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| | |
| :<math>x_{n+1} = \frac{x_n^2 + 2x_n}{x_n^2 + 1}.</math>
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| These iterations all [[quadratic convergence|converge quadratically]]; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with [[arbitrary-precision arithmetic|arbitrary precision]]. The time needed to compute ''n'' digits of the golden ratio is proportional to the time needed to divide two ''n''-digit numbers. This is considerably faster than known algorithms for the [[transcendental number]]s '''[[pi|π]]''' and [[e (mathematical constant)|'''''e''''']].
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| An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers ''F'' <sub>25001</sub> and ''F'' <sub>25000</sub>, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.
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| The golden ratio ''φ'' has been calculated to an accuracy of several millions of decimal digits {{OEIS|id=A001622}}. Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.<ref>{{Cite book| title = The golden number to 17 000 000 000 digits | url = http://www.matematicas.unal.edu.co/airlande/phi.html.en | publisher = Universidad Nacional de Colombia | year = 2008 }}{{dead link|date=June 2011}}</ref>
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| ==Pyramids==
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| [[File:Mathematical Pyramid.svg|right|thumb|250px|A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of <math>1:\sqrt{\varphi}:\varphi</math> and <math>3:4:5\ </math> and <math>1:4/\pi:1.61899\ </math> are of particular interest in relation to Egyptian pyramids.]]
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| Both Egyptian pyramids and those mathematical regular [[square pyramid]]s that resemble them can be analyzed with respect to the golden ratio and other ratios.
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| ===Mathematical pyramids and triangles===
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| A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a ''golden pyramid''. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is <math>\sqrt{\varphi}</math> times the semi-base (that is, the slope of the face is <math>\sqrt{\varphi}</math>); the square of the height is equal to the area of a face, φ times the square of the semi-base.
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| The medial [[right triangle]] of this "golden" pyramid (see diagram), with sides <math>1:\sqrt{\varphi}:\varphi</math> is interesting in its own right, demonstrating via the [[Pythagorean theorem]] the relationship <math>\sqrt{\varphi} = \sqrt{\varphi^2 - 1}</math> or <math>\varphi = \sqrt{1 + \varphi}</math>. This "[[Kepler triangle]]"<ref>{{Cite book| title = The Best of Astraea: 17 Articles on Science, History and Philosophy | url = http://books.google.com/?id=LDTPvbXLxgQC&pg=PA93&dq=kepler-triangle | publisher = Astrea Web Radio | isbn = 1-4259-7040-0 | year = 2006 | author1 = Radio, Astraea Web }}</ref>
| |
| is the only right triangle proportion with edge lengths in [[geometric progression]],<ref name=herz>{{Cite book| title = The Shape of the Great Pyramid | author = Roger Herz-Fischler | publisher = Wilfrid Laurier University Press | year = 2000 | isbn = 0-88920-324-5 | url = http://books.google.com/?id=066T3YLuhA0C&pg=PA81&dq=kepler-triangle+geometric }}</ref> just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in [[arithmetic progression]]. The angle with tangent <math>\sqrt{\varphi}</math> corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").<ref>Midhat Gazale, ''Gnomon: From Pharaohs to Fractals'', Princeton Univ. Press, 1999</ref>
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| A nearly similar pyramid shape, but with rational proportions, is described in the [[Rhind Mathematical Papyrus]] (the source of a large part of modern knowledge of ancient [[Egyptian mathematics]]), based on the 3:4:5 triangle;<ref name = "maor"/> the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).<ref name=Herkommer>{{cite web|url=http://www.petrospec-technologies.com/Herkommer/pyramid/pyramid.htm|title=The Great Pyramid, The Great Discovery, and The Great Coincidence|accessdate=2007-11-25}}</ref> The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,<ref>Lancelot Hogben, ''Mathematics for the Million'', London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, ''Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya'', New York: Simon & Schuster, 2003, p.56</ref> and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.<ref name = "maor"/>
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| Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the [[Kepler triangle]]. This pyramid relationship corresponds to the [[mathematical coincidences|coincidental relationship]] <math>\sqrt{\varphi} \approx 4/\pi</math>.
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| Egyptian pyramids very close in proportion to these mathematical pyramids are known.<ref name=Herkommer/>
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| ===Egyptian pyramids===
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| In the mid-nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the [[Kepler triangle]]; many other mathematical theories of the shape of the pyramids have also been explored.<ref name=herz/>
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| One Egyptian pyramid is remarkably close to a "golden pyramid"—the [[Great Pyramid of Giza]] (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47')<ref name="maor">[[Eli Maor]], ''Trigonometric Delights'', Princeton Univ. Press, 2000</ref> are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.<ref>{{cite book
| |
| |title=The history of mathematics: an introduction
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| |edition=4
| |
| |first1=David M.
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| |last1=Burton
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| |publisher=WCB McGraw-Hill
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| |year=1999
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| |isbn=0-07-009468-3
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| |page=56
| |
| |url=http://books.google.com/books?id=GKtFAAAAYAAJ}}</ref> Several other Egyptian pyramids are very close to the rational 3:4:5 shape.<ref name=Herkommer/>
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| | |
| Adding fuel to controversy over the architectural authorship of the Great Pyramid, [[Eric Temple Bell]], mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.{{Disputed-inline|Egyptian Pyramids|date=November 2013}}<ref>{{Cite book |last=Bell |first=Eric Temple |title=The Development of Mathematics |location=New York |publisher=Dover |year=1940 |page=40 |url=http://books.google.com/books?id=_5KAnw3QMC8C&pg=PA40#v=onepage&q&f=false}}</ref>
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| Michael Rice<ref>Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' pp. 24 Routledge, 2003, ISBN 0-415-26876-1</ref> asserts that principal authorities on the history of [[Egyptian architecture]] have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957).<ref>S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' pp.24 Routledge, 2003</ref> Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.<ref>{{Cite journal
| |
| | last = Markowsky
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| | first = George
| |
| | date =
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| | year = 1992
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| | month = January
| |
| | title = Misconceptions about the Golden Ratio
| |
| | journal = College Mathematics Journal
| |
| | volume = 23
| |
| | issue = 1
| |
| | doi = 10.2307/2686193
| |
| | url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
| |
| | format = PDF
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| | accessdate =
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| | jstor = 2686193
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| | publisher = Mathematical Association of America
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| | pages = 2–19
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| }}</ref>
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| In 1859, the [[Pyramidology|pyramidologist]] [[John Taylor (1781–1864)|John Taylor]] claimed that, in the [[Great Pyramid of Giza]], the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle [[θ]] to the ground, to half the length of the side of the square base, equivalent to the [[Cosecant|secant]] of the angle θ.<ref>Taylor, ''The Great Pyramid: Why Was It Built and Who Built It?'', 1859</ref> The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, [[Richard William Howard Vyse|Howard Vyse]], according to Matila Ghyka,<ref>Matila Ghyka ''The Geometry of Art and Life'', New York: Dover, 1977</ref> reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.
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| ==Disputed observations==
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| Examples of disputed observations of the golden ratio include the following:
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| *Historian John Man states that the pages of the [[Gutenberg Bible]] were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.<ref>Man, John, ''Gutenberg: How One Man Remade the World with Word'' (2002) pp. 166–167, Wiley, ISBN 0-471-21823-5. "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>
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| *Some specific proportions in the bodies of many animals (including humans<ref name=pheasant>{{Cite book |first=Stephen |last=Pheasant |title=Bodyspace |location=London |publisher=Taylor & Francis |year=1998 |isbn=0-7484-0067-2 }}</ref><ref name=vanLaack>{{Cite book |first=Walter |last=van Laack |title=A Better History Of Our World: Volume 1 The Universe |location=Aachen |publisher=van Laach GmbH |year=2001 }}</ref>) and parts of the shells of mollusks<ref name="dunlap"/> are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.<ref name=pheasant/> The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.<ref name=vanLaack/> The [[nautilus]] shell, the construction of which proceeds in a [[logarithmic spiral]], is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;<ref name=moscovich>[[Ivan Moscovich]], ''Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles,'' New York: Sterling, 2004</ref> however, measurements of nautilus shells do not support this claim.<ref>{{Cite journal|title=Sea shell spirals|last=Peterson|first=Ivars|journal=Science News|url=http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals}}</ref>
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| *In investing, some practitioners of [[technical analysis]] use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref>For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in {{Cite journal|author=Osler, 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| url=http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf|format=PDF}}</ref> The use of the golden ratio in investing is also related to more complicated patterns described by [[Fibonacci numbers]] (e.g. [[Elliott wave principle]] and [[Fibonacci retracement]]). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref>[[Roy Batchelor]] and Richard Ramyar, "[http://www.webcitation.org/5reh6NujR Magic numbers in the Dow]," 25th International Symposium on Forecasting, 2005, p. 13, 31. "[http://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html Not since the 'big is beautiful' days have giants looked better]", Tom Stevenson, [[The Daily Telegraph]], Apr. 10, 2006, and "Technical failure", [[The Economist]], Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.</ref>
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| | |
| ==See also==
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| {{div col|colwidth=20em}}
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| * [[Golden angle]]
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| * [[List of works designed with the golden ratio]]
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| * [[Penrose tiling]]
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| * [[Plastic number]]
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| * [[Sacred geometry]]
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| * [[Silver ratio]]
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| {{div col end}}
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| | |
| ==References and footnotes==
| |
| {{reflist|30em}}
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| | |
| ==Further reading==
| |
| {{refbegin|30em}}
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| *{{Cite book
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| | last = Doczi
| |
| | first = György
| |
| | title = The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
| |
| | origyear = 1981
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| | year = 2005
| |
| | publisher = Shambhala Publications
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| | location = Boston
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| | isbn = 1-59030-259-1
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| }}
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| *{{Cite book
| |
| | last = Huntley
| |
| | first = H. E.
| |
| | title = The Divine Proportion: A Study in Mathematical Beauty
| |
| | url =
| |
| | year = 1970
| |
| | publisher = Dover Publications
| |
| | location = New York
| |
| | isbn = 0-486-22254-3
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| }}
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| *{{Cite book
| |
| | last = Livio
| |
| | first = Mario
| |
| | title = The Golden Ratio: The Story of PHI, the World's Most Astonishing Number
| |
| | origyear = 2002
| |
| | url =
| |
| | edition = Hardback
| |
| | year = 2002
| |
| | publisher = Broadway (Random House)
| |
| | location = NYC
| |
| | isbn = 0-7679-0815-5
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| }}
| |
| *{{Cite book
| |
| | last = Joseph
| |
| | first = George G.
| |
| | title = The Crest of the Peacock: The Non-European Roots of Mathematics
| |
| | origyear = 1991
| |
| | url =
| |
| | edition = New
| |
| | year = 2000
| |
| | publisher = Princeton University Press
| |
| | location = Princeton, NJ
| |
| | isbn = 0-691-00659-8
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| }}
| |
| *{{Cite book
| |
| | last = Sahlqvist
| |
| | first = Leif
| |
| | title = Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design
| |
| | edition = 3rd Rev.
| |
| | year = 2008
| |
| | publisher = BookSurge
| |
| | location = Charleston, SC
| |
| | isbn = 1-4196-2157-2
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| }}
| |
| *{{Cite book
| |
| | last = Schneider
| |
| | first = Michael S.
| |
| | title = A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science
| |
| | url =
| |
| | year = 1994
| |
| | publisher = HarperCollins
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| | location = New York
| |
| | isbn = 0-06-016939-7
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| }}
| |
| *{{Cite book
| |
| | last = Stakhov
| |
| | first = A. P.
| |
| | title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science
| |
| | url =
| |
| | year = 2009
| |
| | publisher = World Scientific Publishing
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| | location = Singapore
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| | isbn = 978-981-277-582-5
| |
| }}
| |
| *{{Cite book
| |
| | last = Walser
| |
| | first = Hans
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| | others = Peter Hilton trans.
| |
| | title = The Golden Section
| |
| | origyear = ''Der Goldene Schnitt'' 1993
| |
| | url =
| |
| | year = 2001
| |
| | publisher = The Mathematical Association of America
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| | location = Washington, DC
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| | isbn = 0-88385-534-8
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| }}
| |
| *{{Cite book
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| | last = Scimone
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| | first = Aldo
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| | title = La Sezione Aurea. Storia culturale di un leitmotiv della Matematica
| |
| | year = 1997
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| | publisher = Sigma Edizioni
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| | location = Palermo
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| | isbn = 978-88-7231-025-0}}
| |
| {{refend}}
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| | |
| ==External links==
| |
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| ==={{No more links}}=========-->
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| {{Commons category|Golden ratio}}
| |
| *{{springer|title=Golden ratio|id=p/g044570}}
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| * [http://demonstrations.wolfram.com/GoldenSection/ "Golden Section"] by Michael Schreiber, [[Wolfram Demonstrations Project]], 2007.
| |
| * [http://photoinf.com/Golden_Mean/Eugene_Ilchenko/GoldenSection.html Golden Section in Photography: Golden Ratio, Golden Triangles, Golden Spiral]
| |
| *{{MathWorld|title=Golden Ratio|urlname=GoldenRatio}}
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| *{{cite web|url=http://www.physorg.com/news180531747.html |title=Researcher explains mystery of golden ratio |work=[[PhysOrg]] |date=December 21, 2009 |postscript=<!--None--> }}.
| |
| *{{cite web
| |
| | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html
| |
| | title = The Golden section ratio: Phi
| |
| | last = Knott
| |
| | first = Ron
| |
| | date =
| |
| }} Information and activities by a mathematics professor.
| |
| *[http://web.archive.org/web/20071105084747/http://www.contracosta.cc.ca.us/math/pentagrm.htm The Pentagram & The Golden Ratio]. Green, Thomas M. Updated June 2005. Archived November 2007. Geometry instruction with problems to solve.
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| *{{cite arxiv
| |
| |first1=Robert P.
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| |last1=Schneider
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| |eprint=1109.3216
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| |title=A Golden Pair of Identities in the Theory of Numbers
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| |class=math.HO
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| |year=2011}} Proves formulas that involve the golden mean and the [[Euler totient]] and [[Möbius function]]s.
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| | |
| {{DEFAULTSORT:Golden Ratio}}
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| [[Category:Golden ratio| ]]
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| [[Category:Euclidean plane geometry]]
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| [[Category:Greek inventions]]
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| [[Category:Irrational numbers]]
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| [[Category:Mathematical constants]]
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| [[Category:History of geometry]]
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| [[Category:Visual arts theory]]
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| [[Category:Composition in visual art]]
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| {{Link GA|fr}}
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| {{Link FA|bar}}
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