81. Randomhouse.com | The Golden Ratio By Mario Livio The golden ratio The Story of PHI, the World s Most Astonishing Number. http://www.randomhouse.com/catalog/display.pperl?isbn=0767908163

82. Construction By Golden Ratio which is 51deg50 . With such a base angle, the ratio of the hypotenuseto half the base is phi, the golden ratio. The design was http://alumni.cse.ucsc.edu/~mikel/sriyantra/golden.html

Division in Mean and Extreme Ratio The base angle of the largest trinalges of most representations of Sri Yantra are about 52 degrees, close to the base angle of the Great Pyrmaid of Cheops , which is 51deg50'. With such a base angle, the ratio of the hypotenuse to half the base is phi, the Golden Ratio The design was investigated with the largest two triangles having this base angle. The construction continues similar to the 7x7 grid method. The result is striking, for apart from the fact that the error (at Z and W) is not noticeable, the overall shape is very simlar to examples from the literature . The error is only .3616% of length of the diameter. A small change in base angle results in large error - a 53 degree angle has an error of 9.125%! 1. Draw a rectangle of 2 x 1 units (ABCD) for construction of the large pyramid- angled triangles on base DC. The centre of DC is E. 2. Draw the diaganol DB. It's length is sqr-root 5, by Pythagoras. 3. Draw a circle of radius one centered at point B. Extend the diagonal. 4. Bisect the extended diaganol, creating two line segments of length sqr-root 5 plus 1, divided by 2. This length is phi!

83. PlanetMath: Golden Ratio golden ratio, (Definition). The golden ratio , or , has the value. and so on. Thesegive us. which implies. golden ratio is owned by akrowne. (view preamble). http://planetmath.org/encyclopedia/GoldenRatio.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List golden ratio (Definition) The "Golden Ratio", or , has the value gets its rather illustrious name from the fact that the Greeks thought that a rectangle with ratio of side lengths of about 1.6 was the most pleasing to the eye. Classical Greek architecture is based on this premise. Above: The golden rectangle; has plenty of interesting mathematical properties , however. Its value is exactly The value is often called and are the two roots of the recurrence relation given by the Fibonacci sequence . The following identities hold for and and so on. These give us which implies "golden ratio" is owned by akrowne view preamble View style: HTML with images page images TeX source Other names: golden number Cross-references: identities Fibonacci sequence recurrence relation roots ... rectangle There are 4 references to this object.

84. Golden Ratio golden ratio. You can read more about the number below, and there are links toother golden ratio sites,. and you can hear the programme. ratio. golden ratio. http://www.simonsingh.net/Golden_Ratio.html

Golden Ratio Back to 5 Numbers More about the Golden Ratio GOLDEN RATIO You can read more about the number below, and there are links to other golden ratio sites and you can hear the programme golden ratio Golden Ratio Leonardo Fibonacci was an Italian mathematician with a penchant for decimalization and rabbits! Having introduced the numbers to 9 to Europe (like some medieval Big Bird from Sesame Street), he turned his attention to a different series of numbers: The Fibonacci sequence is generated by adding the previous two numbers in the list together to form the next and so on and so on... Divide any number in the Fibonacci sequence by the one before it, for example 55/34, or 21/13, and the answer is always close to 1.61803. This is known as the Golden Ratio, and hence Fibonacci's Sequence is also called the Golden Sequence. Unlikely though it might seem, this series of numbers is the common factor linking rabbits, cauliflowers and snails. Fibonacci used his sequence of numbers to investigate the population growth of his favourite furry lop-eared friend, the rabbit. He based his model on a maximum-security bunny heaven where rabbits cannot escape or die, and the problem he devised goes like this...

85. BrainPhire?Fractal Golden Ratio Harmonics In BrainWaves As Mechanism Of Euphoria When this key signature reaches 1 / the .618 golden ratio (red line above) is correlatedextensively by thousands of users to the moment of openness/wonder/love http://www.soulinvitation.com/brainphire/

If the Source of Becoming Informed, and of Knowing - Lies in How Much CHARGE Information can Be Contained or Embedded in the Body.. And - Peak Awareness is Peak Charge- Then The Goal of EDUCATION Should be that Peak.. Consider - the Physics of Euphoria and Peak Awareness ( even BLISS) as the Ultimate Educator BrainPhire? Study in Russia Indicates Fractal Golden Ratio Harmonics in BrainWaves as Mechanism of Euphoria / Active Visualization / Bliss? May Support Neurofeedback Solutions to ATTENTION DISORDERS Based on IMPLOSION. notes from Dan Winter and James Barrett, Oct 17,2001, Index Study in Russia (Konstantin Korotkov, Gas Discharge Visualization Group) http://www.heartbeat2000.com/korotkov.htm , kindly reprinted by James Barrett (HeartLink Project), heartbeat2000.com , shows that Bliss / Euphoria - active visualization states, measureably correspond to Golden Mean Fractal Ratio between EEG contained harmonics see BLISS TUNER Above: Dr Korotkov- shows by GDV measurement that the altered state / bliss / euphoria which accompanies increased perception - has consistent qualities. The GDV area (aura size) should correlate to increased heart coherence. "I am reminded of the day Dr.Ed Wilson, Research Director Monroe Institute, decided to quit Bob Monroe and travel to my farm to make our film, when he discovered Fibonacci (Phi directed) audio heterodynes in ear headphones measureably induced (Lexicor's) 'definition' of transcendance in Brain Wave Mapping. He was so excited we had found parallels in the Heart EKG for compassion." Dan Winter

86. Investors Hub - Fibonacci Golden Ratio Of 1.6180339887498949 Fibonacci golden ratio of 1.6180339887498949 (FIBO), Level II News - Quote- Chart, The Fibonacci golden ratio is based upon simple numerology. http://www.investorshub.com/boards/board.asp?board_id=1462

87. The Golden Ratio The golden ratio. Curriculum Tie The students will use the GoldenRatio (1.618) to experience how to set up and work with ratios. http://www.uen.org/Lessonplan/preview.cgi?LPid=6685

88. Golden Ratio Spring Harbor Press. The golden ratio by Mario Livio. The golden ratiois 1.61803Â and is symbolized by the Greek letter f, Phi. http://www.springharborpress.com/golden_ratio.htm

Spring Harbor Press The Golden Ratio by Mario Livio. Broadway Books, 294 pages, includes an Appendix of equations, a bibliography of further reading, and an index. $24.95 Books about science are popular these days. The physicist Brian Greene wrote a best seller untangling string theory and last year the cosmologist Hawkings gave us a lavishly illustrated coffee table tome, compressing the universe into a nutshell. Books about numbers are not that popular. But those who like to play with calculations and geometric diagrams will want to look at Mario Livio’s entertaining book about a very special numeral. In The Golden Ratio Livio tells the story of Phi, "the world’s most astonishing number." There are some numbers which turn up on special occasions and which have such curious characteristics that mathematicians eventually name them or, more precisely, symbolize them by a letter, usually from the Greek alphabet. Probably the best known numeral to be honored this way is the one you come up with when you divide the circumference of a circle by its diameter – the endless 3.14159 , the number called Pi, symbolized by the Greek p. The Golden Ratio is 1.61803 and is symbolized by the Greek letter f, Phi. Phi is not so well known as Pi, but it’s considerably more interesting and has been called the Golden Number, the Golden Ratio, the Golden Section, perhaps because of its esthetic value.

89. Phi, The Golden Ratio The golden ratio, aka The Divine Proportion. Logarithmic SpiralGolden Triangle.Logarithmic SpiralGolden Rectangle. Logarithmic SpiralChambered Nautilus. http://www.cord.edu/faculty/andersod/phi.html

The Golden Ratio, aka The Divine Proportion Logarithmic SpiralGolden Triangle Logarithmic SpiralGolden Rectangle Logarithmic SpiralChambered Nautilus Spiral Galaxy M100 Spiral Galaxy M31 Artists Math May Seminar Math 300

90. Golden Mean The golden mean (proportio divina or sectio aurea), also called golden ratio, goldensection, golden number or divine proportion, usually denoted by the Greek http://www.fact-index.com/g/go/golden_mean.html

Main Page See live article Alphabetical index Golden mean The golden mean proportio divina or sectio aurea ), also called golden ratio golden section golden number or divine proportion , usually denoted by the Greek letter phi , is the number the unique positive real number with and the equally interesting property 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 Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference: After simple algebraic manipulations (multiply the first equation with a b or the second equation with ( a b b ), both of these equations are seen to be equivalent to and hence The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio". Uses and Aesthetics The ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, often used it when building monuments (e.g., the Parthenon ). The

91. Golden Ratio A Key Component Of Isotiles Isotiles use the golden ratio to achieve the relationship between the two trianglesthat enables all the geometrical shapes to be made. The golden ratio. http://www.isotiles.com/goldenratio.htm

The Golden Ratio The Golden Ratio was known to the ancients, and is the next most significant irrational number after pi. Its value (to the first 10 decimal places) is When a larger pentagon is made with Isotiles, with its side now g, it is clear that the longer length (diagonally from each base corner to the apex) is g+1. This is the golden ratio: g is such a value that 1/g=g/(g+1) and g/1=(g+1)/g This is the relationship that makes so many symmetrical tessellations possibile with Isotiles, and gives them a unique quality.

92. Golden Ratio Lesson Plan Lesson Plan golden ratio of the Human Body Indeed, the ratio of the height ofthe body to the height of the navel is consistent with the golden ratio. http://www.saintjoe.edu/~tsp/lessonplan

Lesson Plan: Golden Ratio of the Human Body Scholar: Todd Huff Teacher: Scott Micklo Contact thf3296@saintjoe.edu Todd Huff's Home Page Lesson Overview Projected Time Schedule for Project ... References Lesson Overview The purpose this lesson is to provide students with a hands-on activity to accompany a unit involving ratios. In this activity, students calculate the ratio of the height of the body to the height of the navel. Students will discover that the ratios are all approximately equal. Indeed, the ratio of the height of the body to the height of the navel is consistent with the Golden Ratio. The lesson, then, introduces the concept of the Golden Ratio and provides students with knowledge to explore the presence of Golden Ratios in their everyday lives. The lesson incorporates several of the standards found in Principles and Standards for School Mathematics as encouraged by the National Council of Teachers of Mathematics. For example, the lesson enables students to connect mathematics to realms outside of mathematics. Specifically, the Goldren Ratio is present in nature, including the human body, and architecture. In addition, the lesson incorporates measurement by having the students use a meter stick to obtain data involving measurable attributes of concrete materials.

93. WNYC - Reading Room: The Golden Ratio The golden ratio The Story of Phi, the World s Most Astonishing Number By MarioLivio Random House Copyright © 2002 Mario Livio ISBN 07679-0815-5. http://www.wnyc.org/books/11057

Listen Online Now Windows Overnight Music Windows BBC World Service ... Schedules Search the Site and Archives Support WNYC with an E-Pledge Thank You Gifts Email Updates Sign Up Now to receive WNYCmail The Golden Ratio: The Story of Phi, the World's Most Astonishing Number By Mario Livio Random House ISBN: 0-7679-0815-5 Available for purchase at Amazon.com Chapter One PRELUDE TO A NUMBER Numberless are the world's wonders.-Sophocles (495-405 b.c.) Less known than pi is another number, phi (f), which is in many respects even more fascinating. Suppose I ask you, for example: What do the delightful petal arrangement in a red rose, Salvador Dali's famous painting "Sacrament of the Last Supper," the magnificent spiral shells of mollusks, and the breeding of rabbits all have in common? Hard to believe, but these very disparate examples do have in common a certain number or geometrical proportion known since antiquity, a number that in the nineteenth century was given the honorifics "Golden Number," "Golden Ratio," and "Golden Section." A book published in Italy at the beginning of the sixteenth century went so far as to call this ratio the "Divine Proportion." 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. As we shall see, the Golden Ratio provides us with an intriguing mingling of the two definitions in that, while defined mathematically, it is claimed to have pleasingly harmonious qualities.

94. The Golden Rectangle This is also referred to as the golden ratio. x. And this is a golden rectangle.x / y = 1.618033989 And, coincidentally, y / x = 0.618033989 http://www.johnkyrk.com/golden.html

T H E G O L D E N R A T I O and the Fibonacci series of numbers The Fibonacci series of numbers is Each number after the first two is the sum of the previous two. Sets of adjacent numbers in the Fibonacci series can be applied to the sides of rectangles to compare their proportions. As the numbers get larger, the ratio between adjacent numbers approaches the value called phi ( f ) which is equal to 1.618033989... This is also referred to as the Golden Ratio. x And this is a golden rectangle. x / y = And, coincidentally, y / x = 0.618033989... It's easy to construct from a square. y

95. The Golden Section / The Golden Ratio In Music, The Arts, And Natural Sciences. Livio, Mario The golden ratio The Story of Phi, the World s Most AstonishingNumber Broadway Books, 2002 ISBN 0767908163. Music and the golden ratio. http://www.personal.uni-jena.de/~x8moma/goldensection.htm

This is a bibliography of sources related to the Golden Section , esp. in Music , the Fine Arts, Architecture, Aesthetics, Gestalt Theory, Theory of Proportion etc. As a perfect introduction, you surely enough will have found Dr. Knott 's pages. Feel free to e-mail me comments and further reading suggestions. UK visitors : Please note that sometimes, it is several quid cheaper to order the books via the American page Amazon.com - God knows why. M.Mus. Martin Morgenstern Royal Holloway College, London Der Goldene Schnitt in Musik, Kunst und Naturwissenschaft Essay empfehlen. HIER Introduction Music and the Golden Ratio Psychomusicology etc. ... Music Aesthetics Some of the books are listed at Amazon: Introduction Livio, Mario The Golden Ratio: The Story of Phi, the World's Most Astonishing Number Broadway Books, 2002 ISBN: With a guide to the history of ideas as impassioned as Livio, even the math-phobic can experience the shock and pleasure of scientific discovery. This thoroughly enjoyable work [indeed] vividly demonstrates to the general reader that, as Galileo put it, the universe is, indeed, written in the language of mathematics. Schneider, Michael S.

96. Golden Ratio Translate this page LinkageViewer applet http://iaks-www.ira.uka.de/home/egner/linkages/golden.html

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97. Geometry In The Natural World THE NATURAL WORLD. The Golden Mean The Golden Mean, or golden ratio as it isknown, is an irrational number just like other important numbers such as Pi. http://www.infinitetechnologies.co.za/articles/geometry1.html

GEOMETRY IN THE NATURAL WORLD The Golden Mean: The Golden Mean, or Golden Ratio as it is known, is an irrational number just like other important numbers such as Pi. This means that it cannot be completely represented by our currently used number system, except as a formula ( Sqr (5)-1)/2. Just like Pi (approx. 3.1416) - Phi, or the Golden Ratio, has an endless number of digits after its decimal point and with no repetition of the digits sequences. Therefore, like other "Transcendental" numbers, its value can only be approximated (using our number system). What is the Golden Ratio, and why is it important? The Golden Ratio is approximately Besides for possessing some remarkable and unique characteristics, the Golden Mean is found in ALL living creatures on Earth. Along with the Fibonacci Sequence (which is a whole-number system approximating the Golden Ratio, discovered by Leonardo Pisano Fibonacci), this ratio is found in plants and animal life wherever one looks. For example, this ratio can be found in fingers one's hand, amongst many other places, and it is prevalent in the skeletal structure of all creatures. The Fibonacci Sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

98. Golden Ratio golden ratio. golden ratio and Fibonacci numbers. Phi Page Golden Section Ratio. Constructionby golden ratio. Egypt Cheops pyramid. Alta Vista golden ratio. http://web.hep.uiuc.edu/home/karliner/golden.html

Golden Ratio Golden Ratio and Fibonacci numbers Phi Page Golden Section Ratio Number Games Construction by Golden ratio ... Lycos Golden Ratio Professor Sever Tipei sent this in response to a student who asked about the Golden Ratio in music: See also Erno Lendvai Bela Bartok : Analysis of His Music Dr. Sever Tipei, Professor of Music Manager, Computer Music Project of the University of Illinois Experimental Music Studios Urbana, Illinois 61801, USA send me mail

99. Golden Palace Of Traffic Free Internet Advertising For Business And Business Opp Manual Surf. 11 ratio, 4 referral levels. http://www.goldenpalaceoftraffic.com

Join Login Buy Credits Join Pro ... Chat Welcome Thank you for choosing Golden Palace of Traffic. Join our traffic exchange free. Our Services Advertise free your web business. May surf for credits or purchase. Our exchange ratio is 1:1. Earn credits by referring new members too earn 10% of level 1 and 2 credits earned and 5% of levels 3 and 4. . You can choose the type of sites you wish to visit! And you can target your visitors to receive the best possible results from your advertising! Pro Membership receive company generated referrals 25% off all purchases only $5/mth Other Features We are offering low rates on credit purchases too! also offer a forum, chat, free classified ads as well as new credit protection features!. Dan's Web Templates FG-A.com

100. The 'Phi-Nest': Source To The Golden Section, Golden Mean, Divine Proportion, Fi Dedicated to providing you with the phinest™ information on The GoldenSection, ratio or Mean. The Divine Proportion. The Fibonacci Series. http://goldennumber.net/

The Golden Number from T he Phi Nest™ Home Phi for Neo'phi'tes Fibonacci Series Golden Sectio ... s You can help support this site by visiting these affiliates: Save on books and other purchases at Amazon Great prices and service on hosting that pays you back Investors: Apply Phi and Fibonacci principles to the stock market Get the benefits of nature's most nutrient-packed foods into your daily diet. Phi: The Golden Number Dedicated to providing you with the p hi n est™ information on: The Golden Section , Ratio or Mean The Divine Proportion The Fibonacci Series and . . . Phi ( The Phi Nest Phi: The number 1.6180339887... Nest: A place providing secure lodging and development Phi Nest ™: A place providing secure lodging and development for insights into the pervasive appearance of Phi in life and the universe. The Da Vinci Code It's not often that you find a intriguing and thought-provoking mystery novel that's a creative and clever blend of mathematics, the creations of Leonardo Da Vinci, sacred geometry, history and religion. The Da Vinci Code by Dan Brown , is creating much awareness and inquiry into the Fibonacci series, the golden section and phi. Yes, these are quite real in mathematics and this site has been devoted to exploring them for years before this book was published.

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