21. Golden Ratio Conjugate -- From MathWorld Number Theory Constants Continued Fraction Constants Number Theory Constants GoldenRatio. golden ratio Conjugate. The quantity, (1). where is the golden ratio. http://mathworld.wolfram.com/GoldenRatioConjugate.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Constants Continued Fraction Constants ... Golden Ratio Golden Ratio Conjugate The quantity where is the golden ratio . The golden ratio conjugate is sometimes also called the silver ratio . A quantity similar to the Feigenbaum constant can be found for the n th continued fraction representation Taking the limit of gives Golden Ratio Silver Ratio search Eric W. Weisstein. "Golden Ratio Conjugate." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/GoldenRatioConjugate.html Wolfram Research, Inc.

22. Golden Ratio 11374 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 ConstructionsYou can actually construct the golden ratio with a compass and straight edge. http://www.geocities.com/robinhuiscool/Goldenratio.html

THE GOLDEN R A T I O I'm not going to go into much detail about the golden ratio. The Golden Ratio, or phi, is perhaps the most perfect number in all mathematics. It is equal to the squareroot of 5 plus 1, divided 2. (Sqrt(5)+1)/2 = It's the only number that if squared, is equal to itself plus one. In other words, Phi^2 = Phi+1. And if you took it's reciprical, it's equal to Phi-1. Phi^-1 = Phi-1. Most things in nature follow the golden ratio pattern. Look at your own fingers for example. Measure the length of the longest finger bone. Then measure the shorter one next to it. Divide the longer one by the shorter one. You should get a number close to 1.168. All parts of the human body are poportional to the golden ratio. If you face is in this ratio you are said to be beautiful. The golden ratio is an irrational number, therfore it can't be written as a regular fraction. You could however, get a close estimate. One way, is by dividing Fibonacci numbers. Fibonacci numbers basically follow the pattern 1,1,2,3,5,8,13,21.... each number is the sum of the two before it. 2+3=5,5+3=8,8+5=13 and so on. See a list of the first 1,000 Fibonacci Numbers

23. Art Gallery Of Mark Ivan A virtual art gallery containing the Phyconic art works of the artist Mark Ivan and showing how this art is created using the golden ratio. http://sidewalksalon.bizhosting.com/

Internet Store and Ecommerce Solution Provider - BizHosting.com Choose an ISP NetZero High Speed Internet Dial up $14.95 or NetZero Internet Service $9.95 Home Page Art Gallery of Mark Ivan Welcome Thanks for viewing my gallery, hope you enjoy your visit. Featured Artwork Old Bridge Many centuries ago, there was a bridge spanning the gap from the mainland to the dwelling on the great rock... Original size: 9x12 in./23x31 cm Favorite Links National Gallery of Art - D.C. Gallery displaying one of the finest collections of art from all periods. Albright-Knox Gallery - Buffalo, NY gallery displaying modern American and European art. Copenhagen Art Gallery - Denmark gallery displaying the work of contemporary Scandinavian artists. Finderz Gallery - Gamesroom displaying free games to play on the internet.

24. Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence The golden ratio/Golden Mean, the Golden Rectangle, and the relation between theFibonacci Sequence and the golden ratio. golden ratio, Fibonacci Sequence. http://mathforum.org/dr.math/faq/faq.golden.ratio.html

Ask Dr. Math: FAQ G olden R atio, F ibonacci S equence Dr. Math FAQ Classic Problems Formulas Search Dr. Math ... Dr. Math Home Please tell me about the Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio. The Golden Ratio The golden ratio is a special number approximately equal to 1.6180339887498948482. We use the Greek letter Phi to refer to this ratio. Like Pi, the digits of the Golden Ratio go on forever without repeating. It is often better to use its exact value: The Golden Rectangle A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio. In other words, if one side of a Golden Rectangle is 2 ft. long, the other side will be approximately equal to Now that you know a little about the Golden Ratio and the Golden Rectangle, let's look a little deeper. Take a line segment and label its two endpoints A and C. Now put a point B between A and C so that the ratio of the short part of the segment (AB) to the long part (BC) equals the ratio of the long part (BC) to the entire segment (AC): The ratio of the lengths of the two parts of this segment is the Golden Ratio. In an equation, we have

25. Math Forum - Ask Dr. Math Archives: High School Fibonacci Sequence/Golden Ratio Browse High School Fibonacci Sequence/golden ratio. golden ratio 01/03/1998 Doyou have any topics that I can use in my term paper about the golden ratio? http://mathforum.org/library/drmath/sets/high_fibonacci-golden.html

Ask Dr. Math High School Archive Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ TOPICS This page: Fibonacci sequence, golden ratio Search Dr. Math See also the Dr. Math FAQ golden ratio, Fibonacci sequence Internet Library golden ratio/ Fibonacci HIGH SCHOOL About Math Analysis Algebra basic algebra ... Trigonometry Browse High School Fibonacci Sequence/Golden Ratio Stars indicate particularly interesting answers or good places to begin browsing. Appearances of the Golden Number Why does the irrational number phi = (1 + sqrt(5))/2 appear in so many biological and non-biological applications? Calculating the Fibonacci Sequence Is there a formula to calculate the nth Fibonacci number? Congruum Problem I have found a reference to Fibonacci and his congruum problem. But something has me stumped... Fibonacci sequence in nature, Golden Mean, Golden Ratio I need examples of where the Fibonacci sequence is found in nature and how it relates to the Golden Mean. Fibonacci Series I was helping an Algebra student with a "bonus" problem recently. It asked something about drawing a spiral using the Fibonacci series. What is this series? Does it draw a spiral? Golden Ratio Do you have any topics that I can use in my term paper about the golden ratio?

26. Golden Ratio Design Local doctor designs medical database applications for the Palm, as well as providing other Web design services. Includes samples of sites and PERL scripts, links, and news on spring peepers. http://www.tonywitte.com/

Web and Palm OS Applications for Health Care LMRP Tool The Golden Ratio or Golden Mean is a number revered since antiquity that appears with suprising frequency in natural designs and great works of art. At Golden Ratio Design we believe well-designed computer applications and web tools are recognizable for their simplicity and elegance. We bring this concept to web-based and Palm OS applications for health care clinical desion making and productivity. The Palm OS is the dominant operating system for handheld computing, and for good reasons: simplicity, utility, dependability and a wide range of available software. As a physician dealing with volumes of critical information on a regular basis, the value of handheld computing has been very apparent to me. I have created a few applications for the Palm OS of use to the medical community, and am at work on other applications placing needed databases literally in the palm of the clinician's hand. LabCode NCD - Medical Necessity coding tool using latest Medicare medical necessity requirements included in the 23 lab test National Coverage Decisions. LabCode LMRP - Medical Necessity coding tool using latest local medical review policy medical necessity requirements.

27. Cynthia Lanius' Lessons: The Golden Ratio golden ratio. 1.61803398874989484820. This ratio, called the golden ratio, notonly appears in art and architecture, but also in natural structures. http://math.rice.edu/~lanius/Geom/golden.html

Cynthia Lanius Thanks to PBS for permission to use the Pyramid photo. Golden Ratio If you need a definition If you were going to design a rectangular TV screen or swimming pool, would one shape be more pleasing to the eye than others? Since the early Greeks, a ratio of length to width of approximately 1.618, has been considered the most visually appealing. This ratio, called the golden ratio, not only appears in art and architecture, but also in natural structures. Estimate the ratio of the length to width in the rectangles below: length width Answers Golden Ratio Table of Contents Introduction Find golden rectangles. Build golden rectangles. Confirm the ratio using algebra. Back to the ... Email any comments to lanius@math.rice.edu URL http://math.rice.edu/~lanius/Geom/golden.html

28. The Golden Ratio Explanation, presence in biology, art, and ancient art. http://www.geocities.com/jyce3

This site is devoted to the Golden Ratio. In this page, you will find information about the golden ratio, Fibonacci numbers, and how they relate to biology, art, and ancient Egyptian art.  The Golden Counter says: people have visited this site Introduction to the Golden Ratio and Fibonacci Numbers Biology Art Ancient Art and Mathematics ... View My Guestbook Email jyce3@yahoo.com Created April 1999 by the Proprietors title graphic courtesy of Alex Lumen My URL: http://zap.to/goldenratio I got it for free at http://come.to

29. Sacred Geometry Jewelry And Healing Tools By Gretchen McPherson Jewelry and power objects based on Platonic solids, Archimedian solids, and the golden ratio. Necklaces, pendants, earrings, altar pieces by artist Gretchen McPherson. http://www.lotuslazuli.com/

lotus lazuli is a collection of sacred geometry jewelry and power objects handcrafted by jewelry artist, gretchen mcpherson The Platonic solids tetrahedron, octahedron, cube, icosahedron and dodecahedron , as well as Archimedian solids, the Merkaba, Flower of Life, fibonacci sequence, phi spiral and golden ratio are the major motifs employed. A 3-dimensional labyrinth is created within each shape, directing the energy flow of carefully chosen semi-precious and precious stones held inside its net. Math, spirituality and beauty meet in these unique necklaces pendants earrings and altar pieces.

30. ThinkQuest : Library : The Golden Ratio Welcome to this site! When you think of math, do you think of beauty?Do you think of stuff like pinecones and sunflowers? What http://library.thinkquest.org/C005449/home.html

Index Math Geometry The Golden Ratio When people think of math, do they think of beauty? Do they think of things like pinecones and sunflowers? What does Leonardo da Vinci have to do with this? What do the Greeks, Romans, and people of the Renaissance have in common? They all a mathematics concept in common: the Golden Ratio. The most irrational number in the world is a basis for many things: math, art, architecture, biology, and this site explains how. Visit Site 2000 ThinkQuest Internet Challenge Students Andrei Grupul Scolar H. Coanda, Rm.Valcea, Romania Shujun Thomas Jefferson High School for Science and Technology, Great Falls, VA, United States Melissa Potomac Falls High School, Potomac Falls, VA, United States Coaches Randy Potomac Falls High School, Alexandria, VA, United States Emilia Potomac Falls High School, Rm. Valcea, Romania Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site.

31. Golden Ratio The golden ratio. Although Euclid does not use the term, we shall callthis the golden ratio. The definition appears in Book VI but http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html

The Golden ratio Number theory index History Topics Index Euclid , in The Elements , says that the line AB is divided in extreme and mean ratio by C if AB AC AC CB 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, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron . Here is how the golden ratio comes into the construction of a pentagon. First construct an isosceles triangle whose base angles are double the vertex angle. This is done by taking a line AB and marking C on the line in the golden ratio. Then draw a circle with centre A radius AB . Mark D on the circle so that AC CD BD . The triangle ABD has the property that its base angles are double its vertex angle. Now starting with such a triangle ABD draw a circle through A B and D . Then bisect the angle ADB with the line DE meeting the circle at E . Note that the line passes through C , the point dividing AB in the golden ratio. Similarly construct

32. The Golden Rectangle And The Golden Ratio The Golden Rectangle and the golden ratio. a/b = (a+b)/a This fraction, (a+b)/a,is called the golden ratio (or golden section or golden mean). http://www.jimloy.com/geometry/golden.htm

Return to my Mathematics pages Go to my home page The Golden Rectangle and the Golden Ratio click here for the alternative Golden Rectangle and Golden Ratio page This diagram shows a golden rectangle (roughly). I have divided the rectangle into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us: a/b = (a+b)/a This fraction, (a+b)/a, is called the golden ratio (or golden section or golden mean). Above I have defined the golden rectangle, and then said what the golden ratio is, in terms of the rectangle. Alternatively, I could have defined the golden ratio, using the above equation. And then a golden rectangle becomes any rectangle that exhibits this ratio. From our equation, we see that the ratio a/b=1/2+sqr(5)/2 -1/2+sqr(5)/2 or 0.61803398875 . . .) is called the golden ratio. Also, other mathematical quantities are called phi. The golden ratio is also called tau. Some people call the bigger one (1.61803398875 . . .) Phi (an uppercase phi) and the smaller one (0.61803398875 . . .) phi.

33. American Phi Music, story and poetry of the golden ratio. http://members.aol.com/loosetooth/phi.html

34. Math Forum - Ask Dr. Math Archives: High School Fibonacci Sequence/Golden Ratio A list of questions gathered pertaining to Fibonacci and golden ratio. http://mathforum.org/dr.math/tocs/golden.high.html

Ask Dr. Math High School Archive Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ TOPICS This page: Fibonacci sequence, golden ratio Search Dr. Math See also the Dr. Math FAQ golden ratio, Fibonacci sequence Internet Library golden ratio/ Fibonacci HIGH SCHOOL About Math Analysis Algebra basic algebra ... Trigonometry Browse High School Fibonacci Sequence/Golden Ratio Stars indicate particularly interesting answers or good places to begin browsing. Appearances of the Golden Number Why does the irrational number phi = (1 + sqrt(5))/2 appear in so many biological and non-biological applications? Calculating the Fibonacci Sequence Is there a formula to calculate the nth Fibonacci number? Congruum Problem I have found a reference to Fibonacci and his congruum problem. But something has me stumped... Fibonacci sequence in nature, Golden Mean, Golden Ratio I need examples of where the Fibonacci sequence is found in nature and how it relates to the Golden Mean. Fibonacci Series I was helping an Algebra student with a "bonus" problem recently. It asked something about drawing a spiral using the Fibonacci series. What is this series? Does it draw a spiral? Golden Ratio Do you have any topics that I can use in my term paper about the golden ratio?

35. Fibonacci Numbers & The Golden Ratio Link Web Page - About This Page About This Page. Welcome to Dawson Merrill s Fibonacci and golden ratio Link webpage. If asked to sum up the golden ratio in a word, I would reply, Growth . http://pw1.netcom.com/~merrills/aboutthispage.html

The Fib-Phi Link Page About This Page Welcome to Dawson Merrill's Fibonacci and Golden Ratio Link web page. Please note that unless otherwise stated each link takes you off my page to an Internet site, the author of which deserves all the credit for his or her efforts. It is my hope that I am maintaining a single point of focus for all mathematicians, scientists, researchers, hobbyists, and explorers everywhere in providing a well organized link to the appropriate Internet resource. I want to take this opportunity to thank Dr. Gerald Alexanderson of Santa Clara University who first planted the seed and watered the thirst for my personal pursuit of the Golden Ratio. It began in a phone conversation when I was a teenager in 1980. Although not comprehending much of what Dr. Alexanderson said while on the phone, I feverishly scribbled down as much and as fast as I could. Then after the call, putting pencil, compass, and straightedge to paper, I began to catch a glimpse of the beauties of which he spoke. Ever since, it has been my life's most inspiring and enthralling quest. I'd also like to thank Dr. Ron Knott

36. Index Mathematical calculations; explanations of Phi, the golden ratio and Golden Rectangles; examples from art, architecture, music and nature. http://www.geocities.com/cyd_conner

Fibonacci cynthia conner Joan McDuff curr 356 february 2001 A big thank you to my dad, who first introduced me to the wonders of Fibonacci, and to Joan McDuff and Lynda Colgan for their support and guidance. Picture Credit: Columbia University Library, D.E. Smith Collection I created this site as my term project for the 2000/2001 Elementary Math curriculum course at Queen's University. I have made every attempt to reference the graphics and text which I have gleaned from various sources. Many of these resources are web-based and I have included links to the sites. However, due to the ever-changing nature of the Web, some of these links may be broken and I apologize in advance for any inconvenience. Most of the links are green , and the button in the upper left corner of each page will always take you back to familar territory! Click here to enter It was very hard to do this... maybe even charge this, but I'm a nice guy and will let you have it free. Either View-Source or copy below. NOTE: Put this after your /html tag.

37. Guardian Unlimited | Online | 1.618 Is The Magic Number It s the golden ratio and, arguably, it crops up in more places inart, music and so on than any number except pi. The golden ratio! http://www.guardian.co.uk/online/science/story/0,12450,875198,00.html

@import url(/external/styles/global/0,14250,,00.css); Sign in Register Go to: Guardian Unlimited home UK news World news Archive search Arts Books Business EducationGuardian.co.uk Film Football Jobs Life MediaGuardian.co.uk Money The Observer Online Politics Shopping SocietyGuardian.co.uk Sport Talk Travel Audio Email services Special reports The Guardian The weblog The informer The northerner The wrap Advertising guide Crossword Dating Headline service Syndication services Events / offers Help / contacts Information Living our values Newsroom Reader Offers Style guide Travel offers TV listings Weather Web guides Working at GNL Guardian Weekly Money Observer Home Web watch Ask Jack Feedback ... Life Search Online articles Ask Jack The golden rule It links art, music and even architecture. Marcus Chown on an enigmatic number Thursday January 16, 2003 The Guardian Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887... What's the significance of this number? It's the "golden ratio" and, arguably, it crops up in more places in art, music and so on than any number except pi. Claude Debussy used it explicitly in his music and Le Corbusier in his architecture. There are claims the number was used by Leonardo da Vinci in the painting of the Mona Lisa, by the Greeks in building the Parthenon and by ancient Egyptians in the construction of the Great Pyramid of Khufu.

38. The Golden Ratio The golden ratio. Throughout history, the ratio for length to the eye.This ratio was named the golden ratio by the Greeks. In the world http://www.geom.uiuc.edu/~demo5337/s97b/art.htm

The Golden Ratio Throughout history, the ratio for length to width of rectangles of 1.61803 39887 49894 84820 has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the collumns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. You can take a slide show visit to the Parthenon which is pictured above. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle. How many examples of golden rectangles can you find in the below floorplan of the Parthenon? You may want to print the diagram and measure the distances using a ruler. Following are more examples of art and architecture which have employed the golden rectangle. This first example of the Great Pyramid of Giza is believed to be 4,600 years old, which was long before the Greeks. Its dimensions are also based on the Golden Ratio. The website about the pyramid gives very extensive details on this.

39. Pentagram & The Golden Ratio The Pentagram The golden ratio. Johann Kepler (1571-1630). The ratio has become known as the golden ratio or golden section. http://www.contracosta.cc.ca.us/math/pentagrm.htm

The Pentagram Golden Ratio Geometry has two great treasures: one 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 gold; the second we may name a precious jewel. Johann Kepler (1571-1630) The 'ratio' has become known as the golden ratio or golden section This ratio can be found in many places: in art, architecture, and mathematics. Consider the construction of the regular pentagon. If the side AB of a regular pentagon (see figure to the right) has unit length, then any diagonal, such as AC, has length and this is the golden ratio Notice also the diagonals of the pentagon form another regular pentagon in the center of the figure with, of course, the potential for additional diagonals to be drawn, thus generating the golden ratio again as well as another regular pentagon further inside the figure. Presumably this could continue indefinitely. The golden ratio also appears in comparing consecutive elements of certain kinds of sequences, most notably, the

40. The Golden Ratio And Aesthetics It was Euclid who first defined the b golden ratio /b , and ever sincepeople have been fascinated by its extraordinary properties. http://plus.maths.org/issue22/features/golden/

@import url(../../../newinclude/plus_copy.css); @import url(../../../newinclude/print.css); @import url(../../../newinclude/plus.css); about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 22 November 2002 Contents Features More or Less In a spin The golden ratio and aesthetics The best medicine? Career interview Career interview: Medical statistician Regulars Plus puzzle Pluschat Mystery mix Reviews 'The Golden Ratio' 'Euclid's Window' 'Elements of Grace' and 'Copernican Notes' 'Calculus' ... poster! November 2002 Features The golden ratio and aesthetics by Mario Livio Mario Livio is a scientist and self-proclaimed "art fanatic" who owns many hundreds of art books. Recently, he combined his passions for science and art in two popular books, The Accelerating Universe , which appeared in 2000, and The Golden Ratio reviewed in this issue of Plus . The former book discusses "beauty" as an essential ingredient in fundamental theories of the universe. The latter discusses the amazing appearances of the peculiar number 1.618... in nature, the arts, and psychology. Here he gives us a taster.

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