1. The Golden Ratio Provides information about the golden ratio and Fibonacci numbers, and how they relate to biology, art, and ancient Egyptian art. http://www.geocities.com/CapeCanaveral/Station/8228/

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

2. Fibonacci Numbers & The Golden Ratio Link Web Page A long list of links to pages about Fibonacci and his numbers, the golden ratio and applications in art and science. http://pw1.netcom.com/~merrills/fibphi.html

The Fib-Phi Link Page The Best Fibonacci Numbers and the Golden Section Far and away the best single source for Fibonacci and Golden Ratio browsing. Truly remarkable breadth and detail. Take the time to peruse the myriad of links; you will be rewarded. Kudos are definitely in order for Ron Knott! Getting Started The Life and Numbers of Fibonacci Brief history and a quick walk through the concepts, this web site addresses the basic and more advanced issues elegantly and concisely. Written by Dr. Ron Knott and D. A. Quinney. Who was Fibonacci? Dr. Ron Knott's excellent resources at our disposal again, describing the man and his contributions to mathemtatics. Also be sure to visit his other pages, specifically his Fibonacci Numbers and the Golden Section page. Relation between the Fibonacci Sequence and the Golden Ratio Dr. Math's discussion of the Golden Ratio, Rectangle and Fibonacci sequence. Simple layout and concise graphics aid the initial learning experience. Ask Dr. Math Another Dr. Math web site, this one containing all the questions gathered pertaining to Fibonacci and Golden Ratio. Rabbit Numerical Series Ed Stephen's page has some cute rabbits and quickly describes the derivation of the Fibonacci Series and Golden Ratio.

3. Golden Ratio Provides a golden ratio GreekFace activity for 5th grade and up, plus a link for extra information. http://www.markwahl.com/golden-ratio.htm

A Golden Ratio Activity and Resource par excellence! From Mark Wahl Scroll below for a neat 2-page Golden Ratio activity (It will take some extra time to finish loading while that happens, read on). Select each page of it at a time and print it for use with students. 2) Or, if you have been searching for any of the following keywords, a click on one will take you to an excellent book resource A Mathematical Mystery Tour ) for weeks of personal, home, or classroom learning about the Golden Ratio and Fibonacci Numbers: Golden Ratio Phi Ratio Divine Proportion Golden Mean ... Fibonacci Numbers 3) Or you can go to this website and see a selection of creative books, links, and information on math learning that goes beyond these topics: Mark Wahl Learning Services and Books Now, the Golden Ratio . It has fascinated layman and mathematician for centuries. It seems like magic that it turns up in such different arenas as pine cones earth-moon and planet relationships, the Cheops Pyramid in Egypt, the

4. Golden Ratio Custombuilt massage tables, massage chairs, spa equipment, and other products. http://www.goldenratio.com/

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5. ThinkQuest : Library : The Golden Ratio Created by Team C005449 { this site can be viewed with anybrowser } Click here to enter. media/splasherific.gif. http://library.thinkquest.org/C005449/

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.

6. The Golden Ratio Extension of the number. http://www.cs.arizona.edu/icon/oddsends/phi.htm

Last updated March 27, 1996 Icon home page

7. Deep Secrets A new theory that uses diagrammatic geometry to reveal a possible connection between the Great Pyramid, the golden ratio and the ancient Egyptian Royal Cubit. http://www.sover.net/~rc/deep_secrets/

Introduction The Hexagon Measuring the Earth The Great Pyramid ... Links Deep Secrets The Great Pyramid, The Golden Ratio and The Royal Cubit This site provides a new, and perhaps for some a controversial, explanation for the rationale behind the design parameters of the Great Pyramid of Giza. Learn here the historical significance of both the "golden ratio" and the equal-sided pentagon (and pentagram); a new theory for the derivation of the ancient Egyptian Royal Cubit; a diagrammatic method by which the square root of any number can be derived; how to diagrammatically derive a trigonometric table; a relatively easy to follow presentation of Euclid's derivation of the 36 angle ; and a newly added theory detailing the architect's diagram for the interior design parameters of the Great Pyramid, a theory which includes an accurate prediction for the location of the "doors" in the Queen's Chamber "air shafts" Introduction As one delves into the exterior design details of the Great Pyramid, two striking numerical correlations emerge from the data, and they compel the serious student either to explain these correlations as being nothing more than coincidence or to deal with their implications. These findings are: 1) that the pyramid's cross-section, as defined by its slant height of 611.5 feet divided by one half the length of a side (= 377

8. The Golden Ratio And The Fibonacci Numbers The golden ratio and The Fibonacci Numbers. The golden ratio All RightsReserved. The golden ratio and The Fibonacci Numbers, Note 1. http://www.friesian.com/golden.htm

The Golden Ratio and The Fibonacci Numbers The Golden Ratio ) is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one: . Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one: . Since that equation can be written as , we can derive the value of the Golden Ratio from the quadratic equation, , with a = 1 b = -1 , and c = -1 . The Golden Ratio is an irrational number, but not a transcendental one (like ), since it is the solution to a polynomial equation. This gives us either or . The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal. The Golden Ratio can also be derived from trigonometic functions: = 2 sin 3 /10 = 2 cos ; and = 2 sin /10 = 2 cos 2 . The angles in the trigonometric equations in degrees rather than radians are o o o , and 72 o , respectively. The Golden Ratio seems to get its name from the Golden Rectangle , a rectangle whose sides are in the proportion of the Golden Ratio. The theory of the Golden Rectangle is an aesthetic one, that the ratio is an aesthetically pleasing one and so can be found spontaneously or deliberately turning up in a great deal of art. Thus, for instance, the front of the Parthenon can be comfortably framed with a Golden Rectangle. How pleasing the Golden Rectangle is, and how often it really does turn up in art, may be largely a matter of interpretation and preference. The construction of a Golden Rectangle, however, is an interesting exercise in the geometry of the Golden Ratio

9. PHI, The Golden Ratio Provides a definition of Phi, and explains some mathematical background as well as a listing of 1000 digits of the number. http://astronomy.swin.edu.au/pbourke/analysis/phi/

PHI, the golden ratio (Also known as the golden mean Written by Paul Bourke May 1990, Updated January 1995 Definition Break a line segment into two such that the ratio of the whole to the longest segment is the same as the ratio of the two segments. From the diagram below. The condition can expressed as a/b = 1/a. This can be rearranged and expressed as a quadratic. There are two solutions, phi-1 and -phi where This is the original Greek definition, often phi-1 is used instead. Solution of a quadratic Normally the quadratic for which phi is the quoted solution is The solutions being phi and phi-1 Relationships phi x+1 = phi x + phi x-1 Continued fractions phi = phi = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + .....)))) Relationship to the Fibonnaci series Consider the first order Fibonnaci series x , x , x ..... x i ..... where x i = x i-1 + x i-2 The ratio This tends to phi as i tends to infinity. That is, the ratio of consecutive terms in such a series approaches phi, this is true independent of the starting points of the series. The zero order series starts with 1 and 1 as below. 1 1 2 3 5 8 13 21 34 55 89 etc the ratio of consecutive pairs are 1 0.5 0.67 0.6 0.625 0.6154 0.619 0.6176 0.6182 etc

10. Golden Ratio Essay and brief introduction by Edwin M. Dickey. http://www.ite.sc.edu/dickey/golden/golden.html

The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem Edwin M. Dickey College of Education University of South Carolina MATHEMATICS TEACHER October 1993 Figure 1 The simple elegance of the algebraic expression stands out in glaring contrast to the mind numbing English language expression of the same idea. Why do we study algebra? Because it provides us with an effective and efficient means of communicating certain ideas. Given the definition of the Golden Ratio in algebraic language, one can now investigate methods of finding the numbers satisfying the statement through other representations. The algebraic analysis takes the form of solving the equation: . This can be done by multiplying the equation by 1 + x and solving the resulting quadratic equation using the quadratic formula. This type of analysis yielding two solutions: is familiar to algebra teachers. The graphical analysis of the original problem can be accomplished by again manipulating the original equation into the form x^2 + x - 1= and graphing the relation y = x^2 + x - 1. To solve the equation one can "zoom in" on the point where the curve crosses the x-axis (where the curve y = x^2 + x - 1 crosses the line y = 0). Figure 2 illustrates how the computer algebra system

11. Steve, Jeanette, And Marc's Final Project Project with art references and object construction lessons. http://www.geom.uiuc.edu/~demo5337/s97b/

Welcome to the home page for OUR FINAL PROJECT T HE G OLDEN R ATIO Presented to you by: Steve Blacker, Jeanette Polanski, and Marc Schwach The purpose of this web page is to provide an introduction to the Golden Ratio and Fibonacci Sequence. Instead of simply supplying definitions and asking the student to engage in mindless practice, our idea is to have the student work through several activities to discover the applications of the Golden Ratio and Fibonacci Sequence. Enjoy! Please work through the following activities in the order given: Fibonacci Sequence Fibonacci in Nature Discover the Golden Ratio Golden Ratio in Art ... Worksheet Suggested Activities for Further Investigation (found on the web) Gain further practice in hand-constructing a golden rectangle golden spiral , and a golden section Here are some pretty descent activities you can work through for further understanding. Check out some of Dr. Math's responses to some excellent questions about the Golden Ratio and Fibonacci Sequence. For a slightly more rigorous look at the golden ratio, take a look at this page Here are some cool puzzles you can play with to pursue your investigations further.

12. Welcome To The Golden Ratio This is an informative site on an interesting aspect of Geometry The golden ratio. http://members.tripod.com/~ColinCool/mathindex.html

13. Inter.View To George Cardas An interview with George Cardas, describing his use of the golden ratio in highend audio equipment cables. http://www.tnt-audio.com/intervis/cardase.html

TNT Who we are Inter.View to George Cardas - Cardas Cables by Lucio Cadeddu A brief introduction to Golden Ratio freely taken from "Golden sections and sequences in an unstable problem" by Lucio Cadeddu Golden Ratio is an easy concept of elementary geometry which has had, and still has, great relevance both in human designs and in Nature. Recently it has had wide application in HiFi Audio too. Let me write down a brief survey on Golden Ratio and its amazing history. Let us take a segment a of lenght 1. Another segment b is said to be the Golden Section of a if it solves the following equation: b + b - 1 = that is to say the two segments respect the following proportion: a : b = b : (a-b) . In simpler words, given the fact that a has lenght 1, b must be 0.618 approx. Historically the Golden Ratio was well known to the Egyptians who used it for building their pyramids but it achieved wider popularity thanks to the Greek geometers. We have to wait till 1496 in order to have that ratio called "Golden Ratio". Actually the mathematician (Friar) Pacioli wrote a paper called "De Divina Proportione" where he referred to that ratio as a God-given number one can find everywhere in Nature.

14. Golden-ratio.co.uk News, tour dates, biography, discography, lyrics, and photographs. http://www.golden-ratio.co.uk

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15. Fibonacci Numbers, The Golden Section And The Golden String The third of Simon Singh s BBC Radio 4 Five Numbers programmes was all aboutthe golden ratio. It is an excellent introduction to the golden section. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci Numbers and the Golden Section This is the Home page for Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, the Golden section and the Golden string. The Fibonacci numbers are add the last two to get the next The golden section numbers are The golden string is a sequence of 0s and 1s which is closely related to the Fibonacci numbers and the golden section. There is a large amount of information at this site (more than 200 pages if it was printed), so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature. The rest of this page is a brief introduction to all the web pages at this site on Fibonacci Numbers the Golden Section and the Golden String together with their many applications What's New? - the FIBLOG 25 May 2004 Fibonacci Numbers and Golden sections in Nature Fibonacci Numbers and Nature Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements. All involve the Fibonacci numbers - and here's how and why. The Golden section in Nature Continuing the theme of the first page but with specific reference to

16. The Golden Section - The Number And Its Geometry The Golden section ratio Phi. It includes the solution to some problems aboutpyramids but it does not mention anything about the golden ratio Phi. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html

The Golden section ratio: Phi Contents of this Page The icon means there is a Things to do investigation at the end of the section. The icon means there is an interactive calculator in this section. What is the Golden Ratio (or Phi)? A simple definition of Phi A bit of history... Links on Euclid and his "Elements" ... More What is the golden section (or Phi)? We will call the Golden Ratio (or Golden number) after a greek letter, Phi ) here, although some writers and mathematicians use another Greek letter, tau ). Also, we shall use phi (note the lower case p) for a closely related value. A simple definition of Phi There are just two numbers that remain the same when they are squared namely and . Other numbers get bigger and some get smaller when we square them: Squares that are bigger Squares that are smaller is 4 is 9 is 100 One definition of Phi (the golden section number) is that to square it you just add 1 or, in mathematics:

17. Golden Ratio the segment A as the length of segment A is to the length of segment B. If we calculatethese ratios, we see that we get an approximation of the golden ratio. http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenra

Phi: That Golden Number by Mark Freitag Most people are familiar with the number Pi, since it is one of the most ubiquitous irrational numbers known to man. But, there is another irrational number that has the same propensity for popping up and is not as well known as Pi. This wonderful number is Phi, and it has a tendency to turn up in a great number of places, a few of which will be discussed in this essay. One way to find Phi is to consider the solutions to the equation When solving this equation we find that the roots are x = ~ 1.618... or x= We consider the first root to be Phi. We can also express Phi by the following two series. Phi = or Phi = We can use a spreadsheet to see that these two series do approximate the value of Phi. Or, we can show that the limit of the infinite series equals Phi in a more concrete way. For example, let x be equal to the infinte series of square roots. x Squaring both sides we have But this leads to the equation which in turn leads to and this has Phi as one of its roots. Similarly, it can be shown that the limit of the series with fractions is Phi as well. When finding the limit of the fractional series, we can take a side trip and see that Phi is the only number that when one is subtracted from it results in the reciprocal of the number. Phi can also be found in many geometrical shapes, but instead of representing it as an irrational number, we can express it in the following way. Given a line segment, we can divide it into two segments A and B, in such a way that the length of the entire segment is to the length of the segment A as the length of segment A is to the length of segment B. If we calculate these ratios, we see that we get an approximation of the Golden Ratio.

18. Sacred Geometry For Fun And Personal Growth A summary of sacred geometry, covering Pythagoras, the golden ratio, cymatics and creation through sound, plus a practical geometry workshop, geometric graphics, and a meditation for higher consciousness. http://geometry.wholesomebalance.com

Sacred Geometry Home Sacred Geometry Page 2 Geometry Workshop Geometry Quotes Geometric Graphics ... Links Sacred Geometry Sacred Geometry for Fun and Personal Growth. Sacred geometry is a vast and exciting subject. This website will give you a good summary, with some practical exercises and tools to get started, plus further resources to guide you onward. To start with, mathematics is a wonderful clean subject. It is not subject to opinion, nor does it rely on heated debates as to what the correct answer is. And geometry specifically is pure beauty! It is a great synthesizer, merging the linear, rational aspect of math through the left-side of the brain with the graphical, artistic aspect of pattern and beauty through the right brain. It unites the mind and the heart (called the "intelligence of the heart" by the ancient Egyptians), spirit and matter, science and spirituality. These are all apparently separate halves of the Whole. Plato said in the Republic (VII, 527 d, e) that it is through geometry that one purifies the eye of the soul, "since it is by it alone that we contemplate the truth." Mother nature uses geometry everywhere you look, from the spirals of the nautilus shell, sunflower centre and spiral galaxies to the hexagon symmetry of snowflakes, flower petals and honeycomb. Geometry in nature arranges the shapes of the molecules and crystals that make up our bodies and the physical cosmos. It is the key to the creation of the universe. A good primer on this subject is Michael Schneiders interesting book

19. Golden Ratio Some Explorations with the golden ratio. by. Sue Meredith. (Click here forThe Fibonacci Numbers and Nature.). The golden ratio on a Spreadsheet. http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Meredith/Meredith/Golde

Some Explorations with the Golden Ratio by Sue Meredith An exploration with the Golden Ratio offers opportunities to connect an understanding of ratio and proportion to geometry as well as to introduce historic and aesthetic elements to a mathematical concept. The explorations with a spreadsheet demonstrate Fibonacci numbers and the ratio between each pair. The GSP exploration demonstrates how to construct the ratio geometrically. The use of calculators to prove the definition and properties of this ratio are also appropriate. from Web Page by Ned May(nedmay@Moonstar.com History The name Golden Ratio or Golden Number , named "phi" by the Greeks for the Greek scultor Phidias. However, this ratio can be found in art and architecture long before the Greeks. The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the use of this ratio. The length of each side of the base is 756 feet, and the height when built was 481 feet. 756/481=1.571725572. from http://ce.eng.usf.edu/pharos/wonders/Gallery/pyrami.jpg The Parthenon in Athens Greece, built in 440BC, examplifies the use of the golden rectangle in many of the dimensions. The spaces between the columns, height to width, are in proportion to the golden ratio as are most of the exterior dimensions.

20. Golden Ratio -- From MathWorld golden ratio. The legs of a golden triangle are in a golden ratio to its baseand, in fact, this was the method used by Pythagoras to construct . http://mathworld.wolfram.com/GoldenRatio.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 ... Lambrou Golden Ratio The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagram decagon and dodecagon . It is denoted or sometimes (which is an abbreviation of the Greek "tome," meaning "to cut"). is a Pisot-Vijayaraghavan constant . It also has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers Given a rectangle having sides in the ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio Such a rectangle is called a golden rectangle , and successive points dividing a golden rectangle into squares lie on a logarithmic spiral . This figure is known as a whirling square . The legs of a golden triangle are in a golden ratio to its base and, in fact, this was the method used by Pythagoras to construct

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