1. Golden Ratio - Wikipedia, The Free Encyclopedia In mathematics, and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio http://en.wikipedia.org/wiki/Golden_ratio

Golden ratio From Wikipedia, the free encyclopedia Jump to: navigation search For the Ace of Base album, see The Golden Ratio (album) The golden section is a line segment divided according to the golden ratio: The total length a + b is to the longer segment a as a is to the shorter segment b In mathematics and the arts , two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant , approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea ) and golden mean Other terms encountered include extreme and mean ratio medial section divine proportion divine section (Latin: sectio divina golden proportion golden cut golden number , and mean of Phidias The golden ratio is often denoted by the Greek letter phi , usually lower case ( The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically: This equation has one positive solution in the algebraic irrational number At least since the Renaissance , many artists and architects 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 aesthetically pleasing.

2. Golden Ratio Sep 2, 2010 Although not naming it the Golden Ratio, around 300 BCE Euclid of The Golden Section, or Golden Ratio, divides a line at a point such http://freemasonry.bcy.ca/symbolism/golden_ratio/index.html

SYMBOLISM ARCHITECTURE INDEX OF PAPERS Download a 1.1MB PDF essay on the Golden Rato by clicking here View a 520k PowerPoint presentation on the Golden Rato by clicking here The golden section Although not naming it the Golden Ratio, around 300 BCE Euclid of Alexandria defined the proportion: "A straight line is said to have been cut into extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser." The precise value of the Golden Ratio is an irrational number, phi (1.61803399...), and, like the irrational number pi (3.14159265...) or e (2.71828183...), was considered by the Pythagoreans to be so horrific that its knowledge should be kept secret. The Golden Section, or Golden Ratio, divides a line at a point such that the smaller part relates to the greater as the greater relates to the whole: the ratio of the lengths of the two sides is equal to the ratio of the longer side to the sum of the two sides. a/b = b/a+b = a+b/a+2b = a+2b/2a+3b Fra Luca Paciola (1445-1517) published Divina Proportione The theological and philosophical implications in the name Divine Proportion It was first termed the Golden Section by German mathematician, Martin Ohm in a footnote in the 1835 second edition of

3. Golden Ratio - CreationWiki, The Encyclopedia Of Creation Science Sep 26, 2010 The golden ratio, otherwise known as the Divine Proportion or Phi, is a mathematical ratio with special properties and aesthetic http://creationwiki.org/Golden_ratio

SERVER UPGRADE COMPLETE! Please consider supporting this site Golden ratio From CreationWiki, the encyclopedia of creation science Jump to: navigation search The golden ratio , otherwise known as the Divine Proportion or Phi , is a mathematical ratio with special properties and aesthetic significance. An enormous number of things in the universe are engineered around the ratio, ranging from the human body to the ark of the covenant to snail shells to the orbits of the planets. Some argue that the prevalence of the Golden Ratio is positive evidence of a common design plan uniting a wide variety of phenomena which share only their creator in common. Contents Calculating = Examples Architecture Biology ... See Also Calculating = Phi is derived by dividing a line segment into two parts in such a way that the ratio of the smaller segment to the larger segment is the same as the ratio of the large segment to the whole. The number is irrational, meaning it never ends or repeats in a decimal system. To the first ten decimals, it is 1.6180339887 ... A golden rectangle is one in which the ratio of length to height is 1:phi.

4. The Math Forum - Math Library - Golden Ratio/Fibonacci The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/golden_ratio/

Browse and Search the Library Home Math Topics Arithmetic/Early Number Sense/About Numbers : Golden Ratio/Fibonacci Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Fibonacci Numbers and the Golden Section - Ron Knott Information about the Fibonacci series, including a brief biography of Fibonacci, the numerical properties of the series, and the ways it is manifested in nature. Fibonacci numbers are closely related to the golden ratio (also known as the golden mean, golden number, golden section) and golden string. Includes: geometric applications of the golden ratio; Fibonacci puzzles; the Fibonacci rabbit binary sequence; the golden section in art, architecture, and music; using Fibonacci bases to represent integers; Fibonacci Forgeries (or "Fibonacci Fibs"); Lucas Numbers; a list of Fibonacci and Phi Formulae; references; and ways to use Fibonacci numbers to calculate the golden ratio. more>> The Fibonacci Series - Matt Anderson, Jeffrey Frazier, and Kris Popendorf; ThinkQuest 1999 more>> Golden Ratio, Fibonacci Sequence - Math Forum, Ask Dr. Math FAQ About the Golden Ratio (or Golden Section or Golden Mean), the Golden Rectangle, and the Fibonacci Sequence. more>> All Sites - 96 items found, showing 1 to 50

5. Golden Ratio Although Euclid does not use the term, we shall call this the golden ratio. The definition appears in Book VI but there is a construction given in Book II, http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html

6. Golden Ratio - InfoVis:Wiki DEFINITION. The golden ratio is an irrational mathematical constant with the approximated value of 1.6180339887. It is the relation between two quantities, where the ratio of the sum http://www.infovis-wiki.net/index.php?title=Golden_Ratio

7. Golden Ratio - Encyclopedia Article - Citizendium This is a draft article, under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to a disclaimer. http://en.citizendium.org/wiki/Golden_ratio

Golden ratio From Citizendium, the Citizens' Compendium Jump to: navigation search addthis_pub = 'citizendium'; addthis_logo = ''; addthis_logo_color = ''; addthis_logo_background = ''; addthis_brand = 'Citizendium'; addthis_options = ''; addthis_offset_top = ''; addthis_offset_left = ''; Main Article Talk Related Articles Bibliography External Links This is a draft article , under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to edit intro (PD) Diagram: Joel Holdsworth demonstrating a simple method to construct a Golden Ratio rectangle illustrating geometric proportions in an Attic Greek amphora, including the Golden Proportion: .618 PD Image The old townhall in Leipzig . The tower is positioned between the right (a) and left (b) sections so that equals the golden ratio. The golden ratio is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity. The ratio itself is also known as the mean and extreme ratio , the golden section , the golden mean , and the divine proportion . The earliest existing description is found in the Elements of Euclid (Book 5, definition 3):

8. Category:Golden Ratio - Wikimedia Commons The golden ratio (Latin sectio aurea) \frac 1 \varphi = \varphi 1;\; \varphi = English The golden ratio is an irrational number http://en.wikipedia.org/wiki/Commons:Category:Golden_ratio

Category:Golden ratio From Wikimedia Commons, the free media repository Jump to: navigation search The golden ratio Latin sectio aurea): Deutsch: Der Goldene Schnitt ist eine irrationale Zahl English: The golden ratio is an irrational number Italiano: Sezione aurea è un numero irrazionale Subcategories This category has the following 4 subcategories, out of 4 total. B Body proportions (4 F) G Golden angle (5 F) Golden rectangle (26 F) Golden triangles (20 F) Pages in category "Golden ratio" This category contains only the following page. G Golden ratio Media in category "Golden ratio" The following 169 files are in this category, out of 169 total. Zolotoe setshenie.jpg 45,608 bytes 3,139 bytes 13,467 bytes Academ PlatonicDodec... 2,804 bytes Academ PlatonicDodec... 3,620 bytes Academ scale ratio s... 3,409 bytes Academ six unfolded ... 4,727 bytes AcademViews Platonic... 3,796 bytes Academviews Platonic... 6,028 bytes Aspect ratios-mathem... 12,542 bytes Basel Elisabethenkir... 1,574,772 bytes Calcul-de-phi.jpg 12,232 bytes Ceanpea.jpg

9. Golden Ratio - Simple English Wikipedia, The Free Encyclopedia The golden ratio is an irrational number. If a person tries to write it, it will never stop and never be the same again and again, but it will start this http://simple.wikipedia.org/wiki/Golden_ratio

Golden ratio From Wikipedia, the free encyclopedia Jump to: navigation search If a person has one number a and another smaller number b , he can make the ratio of the two numbers by dividing them. Their ratio is a b . He can make another ratio by adding the two numbers together a b and dividing this by the larger number a . The new ratio is ( a b a . If these two ratios are equal to the same number, then that number is called the golden ratio . The Greek letter phi ) is usually used as the name for the golden ratio. For example, if b = 1 and a b , then . The second ratio ( a b a is then . Because these two ratios are equal, this is true: One way to write this number is is the number which, when multiplied by itself, makes 5: The golden ratio is an irrational number . If a person tries to write it, it will never stop and never be the same again and again, but it will start this way: 1.6180339887... An important thing about this number is that a person can subtract 1 from it or divide 1 by it. Either way, he will find the same number: change Golden rectangle The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:

10. Golden_ratio www.awarmsoftfear.com/ CachedGolden Ratio Spiral the Breath of BrahmaThe Golden Ratio/Mean/Spiral is the mathematical expression of the life breath of a living universe. http://www.awarmsoftfear.com/

11. All Items On Etsy, A Global Handmade And Vintage Marketplace. Shop for unique golden_ratio from our artisan community. Buy and sell handmade or vintage items, art and supplies. Share stories through millions of items http://www.etsy.com/search_results.php?includes[0]=tags&search_query=golden_

12. File:Golden Ratio.svg - Wikimedia Commons =Golden Ratio, ab = 1.61 = (1+sqr......Aug 21, 2009 golden_ratio.svg (SVG file, nominally 397 × 70 pixels, file size 1 KB) ({{ Information http://commons.wikimedia.org/wiki/File:Golden_Ratio.svg

File:Golden Ratio.svg From Wikimedia Commons, the free media repository Jump to: navigation search File File history ... Golden_Ratio.svg (SVG file, nominally 397 × 70 pixels, file size: 1 KB) edit Summary Description Golden Ratio.svg Golden Ratio, a:b = 1.61 Date Source original here , but SVG: own work Author Petar Marjanovic 19:34, 8 January 2008 (UTC) Permission Reusing this file See below. edit Licensing This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. www.creativecommons.org/licenses/by-sa/3.0 CC-BY-SA-3.0 Creative Commons Attribution-Share Alike 3.0 true true File history Click on a date/time to view the file as it appeared at that time.

13. Mathematical Constants — Sage Reference Manual V4.5.3 sage a = pi + e + golden_ratio + log2 + euler_gamma + catalan + khinchin + twinprime + . sage gr = golden_ratio sage RR(gr) 1.61803398874989 sage R http://www.sagemath.org/doc/reference/sage/symbolic/constants.html

Navigation index modules next previous ... Constants Mathematical constants The following standard mathematical constants are defined in Sage, along with support for coercing them into GAP, GP/PARI, KASH, Maxima, Mathematica, Maple, Octave, and Singular: sage: pi pi sage: e # base of the natural logarithm e sage: NaN # Not a number NaN sage: sage: # natural logarithm of the real number 2 sage: sage: catalan # the Catalan constant catalan sage: khinchin khinchin sage: twinprime twinprime sage: mertens mertens sage: brun brun Support for coercion into the various systems means that if, e.g., you want to create sage: maxima pi %pi sage: singular pi pi sage: gap pi pi sage: gp pi 3.141592653589793238462643383 # 32-bit 3.1415926535897932384626433832795028842 # 64-bit sage: pari pi sage: kash pi # optional sage: mathematica pi # optional Pi sage: maple pi # optional Pi sage: octave pi # optional Arithmetic operations with constants also yield constants, which can be coerced into other systems or evaluated. sage: a pi e a pi + 4/5*e sage: maxima a %pi+4*%e/5 sage: RealField a # 15 *bits* of precision sage: gp a 5.316218116357029426750873360 # 32-bit

14. Golden Ratio - Definition The golden ratio is a number, approximately 1.618, that possesses many interesting properties. It was studied by ancient mathematicians due to its frequent http://www.wordiq.com/definition/Golden_ratio

Golden ratio - Definition The golden ratio is a number , approximately 1.618, that possesses many interesting properties. It was studied by ancient mathematicians due to its frequent appearance in geometry . Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean golden section golden number or divine proportion It is usually denoted by the Greek letter (phi) with (tau) being less common. Contents showTocToggle("show","hide") 1 Origin of name 2 Definition of the Golden Ratio 3 Properties 4 Alternate forms ... 11.3 Other Origin of name The name "golden ratio" first seemed to have been used in the form sectio aurea , "golden section", by Leonardo da Vinci sculptor Phidias , who was long believed to have used the golden ratio in his designs. Definition of the Golden Ratio Two quantities are said to be in the golden ratio , if "the whole is to the larger as the larger is to the smaller", i.e. if

15. Golden Ratio: Facts, Discussion Forum, And Encyclopedia Article Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from http://www.absoluteastronomy.com/topics/Golden_ratio

Home Discussion Topics Dictionary ... Login Golden ratio Golden ratio Topic Home Discussion Definition Overview In mathematics Mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.... and the art Art Art is the process or product of deliberately arranging elements in a way to affect the senses or emotions. It encompasses a diverse range of human activities, creations, and modes of expression, including music, literature, film, photography, sculpture, and paintings... s, two quantities are in the golden ratio if the ratio Ratio In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient... of the sum of the quantities to the larger quantity is equal to (=) the ratio Ratio In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient...

16. THE GOLDEN RATIO The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C by the founder of geometry as a formalized deductive http://jwilson.coe.uga.edu/EMAT6680/Parveen/golden_ratio.htm

T HE G OLDEN R ATIO by Nikhat Parveen, UGA Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of G old; the second we may name a P recious jewel. Kepler (1571- 1630) In everyday life, we use the word “proportion” either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term “proportion” is used to describe an equality of the type: nine is to three as six is to two. The Golden Ratio provides us with an intriguing mingling, it is claimed to have pleasingly harmonious qualities. The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C by the founder of geometry as a formalized deductive system , Euclid of Alexandria. In Euclid’s words: 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. If the ratio of the length AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio.

17. Golden Ratio: K-12 Lesson Plans, Class Activities & Background Information Golden ratio, K12 lesson plans, class activities, science fair projects and background information. http://www.juliantrubin.com/encyclopedia/mathematics/golden_ratio.html

@import url(http://www.juliantrubin.com/imagesgen/template1.css); @import url(http://www.juliantrubin.com/imagesgen/template2.css); @import url(http://www.juliantrubin.com/imagesgen/template3.css); Home Experiments Math Science Fair Projects Mathematics Jokes and Archimedes ... Warning! Golden Ratio Enter your search terms Submit search form Web www.juliantrubin.com Experiments Home Mathematics Golden Ratio Mathematics Science Fair Projects Home Applied Mathematics Number Theory Miscellany ... The Golden Rectangle (Grades 9-12) Golden Ratio The golden section is a line segment sectioned into two according to the golden ratio . The total length a+b is to the longer segment a as a is to the shorter segment b In mathematics and the arts , two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887. At least since the Renaissance , many artists and architects 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 aesthetically pleasing.

18. Golden Ratio The golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b. http://schools-wikipedia.org/wp/g/Golden_ratio.htm

Golden ratio 2008/9 Schools Wikipedia Selection . Related subjects: Mathematics The golden section is a line segment sectioned into two according to the golden ratio . The total length a+b is to the longer segment a as a is to the shorter segment b In mathematics and the arts , two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887 (from the quadratic formula At least since the Renaissance , many artists and aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically: This equation has as its unique positive solution the algebraic irrational number Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea golden mean golden number , and the Greek letter phi ). Other terms encountered include

19. Golden Ratio - RationalWiki Mar 18, 2010 The golden ratio (also referred to as the golden number or golden section) or has been known for millennia. It is defined as the ratio http://rationalwiki.org/wiki/Golden_ratio

Golden ratio From RationalWiki Jump to: navigation search This article is only a brief description of the subject, and is not intended to give a full explanation. Check out the "see also" or "references" sections, or Wikipedia's article for more detail. The golden ratio (also referred to as the golden number or golden section) or has been known for millennia. It is defined as "the ratio between two numbers such that the lesser is to the greater as the greater is to the sum" The golden section: The golden ratio applied to the division of a line A:B = B:(A+B) B:A = φ = 1.61803398874989484820458683436563811772030917980576 (approx) Its exact value is It has generally been thought to be pleasing and harmonious to human perception and is the basis of much classical architecture. The usage of the Greek letter phi to represent the golden ratio was suggested by mathematician Mark Barr from the first letter of Phidias (ancient Greek, Φειδίας), the sculptor who was alleged to have used it in creating statues for the Parthenon. The golden number (or an approximation) appears often in nature and is the convergent point of the ratio of successive terms of the Fibonacci sequence A rectangle with the ratio of adjacent sides equal to the golden ratio. It is supposedly particularly pleasant, visually.

20. Golden Ratio Evolver.net Jun 24, 2010 A golden thread of synchronicities. A golden section of time. On June 3rd, John, Danielle, Rachel, and I embarked on a series of threads in http://www.evolver.net/group/collective_unconscious_project/discussion/golden_ra

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