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

Search our catalog or try our: Advanced Search Portable Tables Massage Chairs Spa Tables ... Useful Links Want a free catalog? Portable Massage Tables More all-around leg and knee room than any other portable massage table! 7 Reasons Why It Makes Sense To Buy A Golden Ratio Table Luxurious lotions Soothing Touch Massage videos Pregnancy Aids Spa Tables Beauty and Functionality. These tables answer all your needs Electric Hydraulic Gas Spring 7 Reasons Why It Makes Sense To Buy A Golden Ratio Table Spa Supplies The Ultimate in Client Care Safa Aestheticians' Tools Hydrocollators Moor Spa Massage Chairs The Easiest Setup around! Stable on uneven surfaces Still the lightest at 14 pounds! Only chairs with patented QuickTouch Head Support The wonders of heat and water Steamy Wonder Amerec Steam Embrace Reid Systems Constructed like fine furniture Beautifully upholstered with attention to every detail Flat, Two-Section or Four-Section Salon Top

2. 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.

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

Last updated March 27, 1996 Icon home page

4. 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

5. 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

This page has moved. If you aren't redirected in 5 seconds please click here Please update your links. Thank you! This page accessed times. Changes last made on: Saturday, April 9, 2005, 16:09:11, PST

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

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7. The Golden Ratio Examines the uses/occurances of phi in biology, art, and ancient times. http://library.thinkquest.org/C005449

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8. The Golden Ratio Gives and intro to the golden ratio and its 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

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. The Golden Ratio And The Fibonacci Numbers A presentation of the relationship between the golden ratio and the Fibonacci Numbers from the proceedings of the Friesian School. 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

12. 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.

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 (800) 3451129 sales@goldenratio.com 2005 golden ratio, All Rights Reserved http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Fibonacci Numbers, The Golden Section And The Golden String The Golden section and Geometry The golden section is also called the golden ratio, the golden mean and the divine proportion. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Golden Ratio -- From MathWorld The golden ratio, also known as the divine proportion, golden mean, Livio, M. The golden ratio The Story of Phi, the World s Most Astonishing Number. http://mathworld.wolfram.com/GoldenRatio.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Constants Golden Ratio ... 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"). The term "golden section" ( goldene Schnitt ) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6). has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers . It is also a so-called Pisot Number Given a rectangle having sides in the ratio is defined such that partitioning the original

17. 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

18. The Golden Section - The Number And Its Geometry What is the golden section (or Phi)? We will call the golden ratio (or Golden number) after a greek letter Phi ( http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

19. 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

20. The Golden Ratio Gives and intro to the golden ratio and its presence in biology, art, and ancient art http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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