Zlatni rez

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Zlatni isječak je linijski segment prepolovljen na dva dijela prema pravilčima zlatnog reza. Ukupna dužina a + b je većem segmentu a isto što je a kraćem segmentu b.

U matematici i umjetnosti, dvije veličine su u zlatnom rezu ako je omjer između sume te dvije veličine i veće od njih jednak sa odnosom veće veličine sa manjom veličinom. Zlatni rez je matematička konstanta, koja približno iznosi 1,6180339887.^[1]

Najkasnije od Renesanse, mnogi umjetnici i arhitekte su nastojali svoje radove praviti prema pravilima zlatnog reza, posebno u obliku zlatnog pravougaonika, u kojem je omjer duže stranice naspram dužine kraće stranice zlatni rez, a vjerovalo se da je ova proporcija estetski zadovoljavajuća. Matematičari su proučavali zlatni rez zbog njegovih jedinstvenih i interesantnih osobina.

Zlatni rez se često označava sa grčkim slovom ϕ (fi). Izgled zlatnog isječka ilustrira geometrijsku vezu koja definiše ovu konstantu. Izraženo algebarski:

$\frac{a+b}{a} = \frac{a}{b} = \varphi\,.$

Ova jednačina ima, kao jedinstveno, pozitivno rješenje, algebarsko iracionalan broj

$\varphi = \frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\ldots\,$ ^[1]

Ostali nazivi, koji se koriste za ili za zlatnom rezu srodne pojmove, su zlatni isječak (latinski: sectio aurea), zlatna sredina, zlatni broj i grčko slovo fi (ϕ).^[2]^[3]^[4] Ostali termini, koji se susreću, jesu ekstremni i srednji omjer, medijalni isječak, božanska proporcija, požanski isječak (latinski: sectio divina), zlatna proporcija, zlatni omjer,^[5], te Fidiasova sredina.^[6]^[7]^[8]

Konstrukcija zlatnog pravougaonika:
1. Konstruišite jedinični kvadrat (crveno).
2. Povucite liniju sa sredine jedne stranice u suprotan ugao.
3. Iskoristite tu liniju kao radijus kako bi nacrtali luk koji definiše dužu dimenziju pravougaonika.

[uredi] Izračunavanje

Spisak brojeva γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
Binarni	1.1001111000110111011...
Decimalni	1,6180339887498948482...
Heksadecimalni	1.9E3779B97F4A7C15F39...
Neprekidni razlomak	$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}}}$
Algebarski oblik	$\frac{1 + \sqrt{5}}{2}$

Za dvije veličine (pozitivni brojevi) a i b se kaže da su u zlatnom rezu ϕ ako vrijedi

$\frac{a+b}{a} = \frac{a}{b} = \varphi\,.$

Ova jednačina jednoznačno definiše ϕ.

Desna jednačina pokazuje da je a = bϕ, što se može zamijeniti u lijevi dio, dajući

$\frac{b\varphi+b}{b\varphi}=\frac{b\varphi}{b}\,.$

Poništavanjem b na obe strane, dobijamo

$\frac{\varphi+1}{\varphi}=\varphi.$

Množenjem obe strane sa ϕ i premještanjem članova vodi do:

${\varphi}^2 - \varphi - 1 = 0.$

Jedino pozitivno rješenje ove kvadratne jednačine je

$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,$

[uredi] Također pogledajte

[uredi] Reference i fusnote

↑ ^1,0 ^1,1 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 = x², or thus a quadratic equation: x² − x − 1 = 0. Then, by the quadratic formula, for positive x = (−b + √(b² − 4ac))/(2a) with a = 1, b = −1, c = −1, the solution for x is: (−(−1) + √((−1)² − 4·1·(−1)))/(2·1) or (1 + √(5))/2.
↑ Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, New York: Broadway Books.
↑ Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: Associated University Presses, 1996
↑ Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
↑ Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) pp.37 . "And the same applies in architecture, to the rectangles 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."
↑ Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920
↑ William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003
↑ Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.

[uredi] Dalje čitanje

Doczi, György [1981] (2005). The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture, Boston: Shambhala Publications.
Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Proportion, New York: Dover Publications.
Joseph, George G. [1991] (2000). The Crest of the Peacock: The Non-European Roots of Mathematics, New Ed., Princeton, NJ: Princeton University Press.
Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design, 3rd Rev. Ed., Charleston, SC: BookSurge.
Schneider, Michael S. (1994). A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science, New York: HarperCollins.
Walser, Hans [Der Goldene Schnitt 1993] (2001). The Golden Section, Peter Hilton trans., Washington, DC: The Mathematical Association of America.

[uredi] Vanjski linkovi

Commons: Zlatni rez

The Pentagram & The Golden Ratio. Geometry instruction with problems to solve.
Khan Amore’s Commentary on the Divine Proportion (HTML; PDF available).
The Golden section ratio: Phi. Information and activities by a mathematics professor.
Eric W. Weisstein, Zlatni rez na MathWorld-u.
The golden number to 17 000 000 000 digits.
"Golden Section" by Michael Schreiber, Wolfram Demonstrations Project, 2007.

[quadform-0] 1,0 ^1,1 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 = x², or thus a quadratic equation: x² − x − 1 = 0. Then, by the quadratic formula, for positive x = (−b + √(b² − 4ac))/(2a) with a = 1, b = −1, c = −1, the solution for x is: (−(−1) + √((−1)² − 4·1·(−1)))/(2·1) or (1 + √(5))/2.

[livio-1] Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, New York: Broadway Books.

[2] Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: Associated University Presses, 1996

[dunlap-3] Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997

[4] Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) pp.37 . "And the same applies in architecture, to the rectangles 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."

[5] Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920

[6] William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003

[7] Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.

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