41. Mr. Narain's Golden Ratio WebSite Jan 3, 2003 This page is meant to be a basic introduction to one of the most amazing discoveries in mathematics the Golden Ratio. http://cuip.uchicago.edu/~dlnarain/golden/

Welcome to Mr. Narain's Golden Ratio Page! This page is meant to be a basic introduction to one of the most amazing discoveries in mathematics: the Golden Ratio. Please use the navigation bar on the left at any time to take you where you want to go. You can always come back to this page by clicking on the "Home" button. Please read the Introduction below and then go to the Activities page. The activities are meant to be done in sequence. After you complete all of them, you should go to the Assessment page to find out how much you have learned. Finally, you may use the Feedback button to communicate with Mr. Narain and tell him what you liked/disliked about this website. If you are a teacher, please visit the Teacher's Page to learn more about instruction for this website. Finally, the Links page will take you to a list of other Golden Ratio websites, many of them far more involving than this one. Introduction What is the Golden Ratio?

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

Thanks for searching for our Golden ratio page and its wonderful ideas and activities. The address has changed! No problem, just click here to access it: Golden Ratio Page Thanks for your patience! Mark Wahl Learning Services and Books. [Below is some of the text version of the page but clicking "Golden Ratio Page" above will get you to the updated, upgraded version of it: A Golden Ratio Activity and Resource par excellence! 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:

43. The "Amen Break" And Golden Proportion A Comment about my Amen Break and the Golden Ratio article. by Michael S. Schneider M.Ed. Mathematics Author of A Beginner's Guide To Constructing The Universe http://www.constructingtheuniverse.com/Amen Break and GR.html

A Comment about my "Amen Break and the Golden Ratio" article by Michael S. Schneider M.Ed. Mathematics Author of A Beginner's Guide To Constructing The Universe Two years ago I posted an article called "The Amen Break and the Golden Ratio" and it was visited by hundreds of thousands of interested people. But the original article is no longer available, although parts of it have been copied onto many websites and is probably cached in various places. I wrote that I'd noticed that the major peaks of the Amen Break seem "Man" by Libby Reid. For those of you who actually want to learn about geometry and its appearances in Nature and Art here are my Constructing The Universe Activity Books allowing you to Create and Explore the Geometric Patterns of Nature and Worldwide Art, Crafts, Architecture and Design Popular with Artists, Craftspeople, Architects, Designers, Home Schools and anyone who wishes to be inspired by the relationships of mathematics with nature and human creativity. 670 pages filled with hands-on geometric activities.

44. Golden Ratio: Information From Answers.com golden ratio The number 1.618~, unique in that its value equals the ratio of its integer part to its fractional part, i.e. Meeting this condition requires, for fractional part http://www.answers.com/topic/golden-ratio

var isReferenceAnswers = true; BodyLoad('s'); On this page Library Animal Life Health History, Politics, Society ... Technology Golden ratio Measures and Units: golden number Home Library Science Measures and Units golden ratio The number 1.618~, unique in that its value equals the ratio of its integer part to its fractional part, i.e. Meeting this condition requires, for fractional part f , that f f , hence that f f = 1, hence that f f - 1 = 0which has the sole real solution f = 0.618 033 988 75~.Thus the golden ratio, usually represented by the Greek letter phi, is ϕ = 1.618 033 988 75~and its reciprocal equals f . Known to Pythagoras and his followers over 2 000 years ago, the ratio is much vaunted as being the perfect proportion for very many facets of art and architecture. It is inherent in the opening definition that f is the reciprocal of ϕ, i.e. ϕ f , and that the ratio 1:0.618~ is identically 1.618~:1. Thus, if a line is cut into two sections having mutual proportions equal to the golden ratio, the ratio of the whole line to the larger section is also the golden ratio. Such a cutting is called a golden section , with the larger section (obviously 61.8~% of the whole) called the

45. Golden Ratio The Golden Ratio truely is a unique number in mathematics! It occurs often in nature, often in rivalry with the greatest irrational number of all time, p! http://hptgn.tripod.com/golden.htm

Build your own FREE website at Tripod.com Share: Facebook Twitter Digg reddit document.write(lycos_ad['leaderboard']); document.write(lycos_ad['leaderboard2']); Golden Ratio The Golden Ratio truely is a unique number in mathematics! It occurs often in nature, often in rivalry with the greatest irrational number of all time, p ! But unlike it's rival p , the Golden Ratio is a constructable number! Officially, the Golden Ratio is denoted by the symbol . By definition: Enter any positive number: Golden Section Draw AB (yellow). Draw BC (magenta), so that BC AB and BC is half of AB Draw a line through points CA (red). Draw a circle at C , crossing B . Circle C crosses CA at D Draw a circle at A , crossing D . Circle A crosses AB at E The ratio of AE to EB Another way of calculating the Golden Ratio, is the use of Fibonacci numbers. A Fibonacci sequence is of this form: f f f n = f n-2 + f n-1 , for n Golden Rectangle Construct square ABCD Extend DC Find the midpoint M of DC Draw a circle at M , crossing B . It intersects DC at E DE is the length of the new rectangle. The ratio of DE to AD Golden Spiral

46. Inter.View To George Cardas An interview with George Cardas, describing his use of the Golden Ratio in high-end 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.

47. 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.

48. Welcome To Golden Ratio Woodworks Online ordering of over 1200 products for massage, therapy, beauty and spa professionals. http://goldenratio.com/

Spare parts and repairs for all Golden Ratio Massage Tables 406-222-5928 to order (8AM to 6PM -mountain time-weekdays) fivesweeneys@hotmail.com Plus huge clearance sale on massage tables made in USA We can repair any Golden Ratio portable table for $75 plus $50 return shipping. A cut down bicycle box or large box can be used to send it along with a check for $125 made out to Timothy Sweeney. Call first then send to: Timothy Sweeney 123 South E Street, Livingston MT 59047 Fully adjustable face rest , heavy duty, weighs 25 ounces. $70 for frame only, $100 with pillow (plus $10 shipping). Fits all Golden Ratio tables 9.5" from center to center hole spacing. Straight frame head rest $25 With pillow $55 Tilting frame repair $35 - Includes return shipping. Send it along with a check for $35 made out to Timothy Sweeney. fivesweeneys@hotmail.com Assorted bolsters $30 9" half round or 6" round Contoured Pillow with foam $30 Armrest/footrest combo $95 QT repair -Includes return shipping. Send it along with a check for $35 made out to Timothy Sweeney. Footrest/table extender $50 Black Lynak transformer $100 4 Rubber feet for legs $6 4 Rubber bumpers $8 Extension leg knobs $5 each Star, smooth salon, Oak or Maple

49. 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

50. Golden Ratio How to generate the number, GSP script for dividing segments, rectangle and other shapes, the rabbit problem and references. 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.

51. Golden Ratio Software Informer: Latest Version Download, News And Info About Thi It is possible that the download link provided above does not refer to the intended version. If you are looking for Golden Ratio, you may try searching for a download link http://golden-ratio.software.informer.com/

52. Constructing The Golden Rectangle A description of the golden rectangle with formulas and drawings. http://jwilson.coe.uga.edu/EMT669/Student.Folders/May.Leanne/Leanne's Page/Golde

Constructing the Golden Rectangle With respect to the Golden Ratio by Leanne May The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The ratio is close to 1.618. Whoever first discovered these intriguing manifestations of geometry in nature must have been very excited about the discovery. A study of the Golden Ratio provides an intereting setting for enrichement activities for older students. Ideas involved are: ratio, similarity, sequences, constructions, and other concepts of algebra and goemetry. Finding the Golden Ratio. Consider a line segment of a length x+1 such that the ratio of the whole line segment x+1 to the longer segment x is the same as the ratio of the line segment, x, to the shorter segment, 1. Thus

53. Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence The Golden Ratio/Golden Mean, the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio. 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

54. Fibonacci And The Golden Ratio Discover how this amazing ratio, revealed in countless proportions throughout nature, applies to the financial markets. http://www.investopedia.com/articles/technical/04/033104.asp

55. Welcome To Golden Ratio Woodworks Custom-built massage tables, massage chairs, spa equipment, and other products. http://www.goldenratio.com/

Spare parts and repairs for all Golden Ratio Massage Tables 406-222-5928 to order (8AM to 6PM -mountain time-weekdays) fivesweeneys@hotmail.com Plus huge clearance sale on massage tables made in USA We can repair any Golden Ratio portable table for $75 plus $50 return shipping. A cut down bicycle box or large box can be used to send it along with a check for $125 made out to Timothy Sweeney. Call first then send to: Timothy Sweeney 123 South E Street, Livingston MT 59047 Fully adjustable face rest , heavy duty, weighs 25 ounces. $70 for frame only, $100 with pillow (plus $10 shipping). Fits all Golden Ratio tables 9.5" from center to center hole spacing. Straight frame head rest $25 With pillow $55 Tilting frame repair $35 - Includes return shipping. Send it along with a check for $35 made out to Timothy Sweeney. fivesweeneys@hotmail.com Assorted bolsters $30 9" half round or 6" round Contoured Pillow with foam $30 Armrest/footrest combo $95 QT repair -Includes return shipping. Send it along with a check for $35 made out to Timothy Sweeney. Footrest/table extender $50 Black Lynak transformer $100 4 Rubber feet for legs $6 4 Rubber bumpers $8 Extension leg knobs $5 each Star, smooth salon, Oak or Maple

56. Golden Ratio Now announcing Phi Day - October 31, 2008 Read the white paper below White Paper Golden Ratio Day - Phi Day http://goldenratio.org/

Now announcing - Phi Day - October 31, 2008 Read the white paper below... White Paper Golden Ratio Day - Phi Day Please click here for the Fib/Phi Link Page Instructions for Downloading 10,000,000 Digits of the Golden Ratio Here are 10 files, each containing 1,000,000 digits, plus a bonus file containing an additional 818600. Sorry I can not host all 15 million at this time. They are shown in order below. Some browsers require you to hold the shift key down while pressing the mouse button and it will prompt you for a place to save it and a filename. phi10000000a.txt phi10000000b.txt phi10000000c.txt phi10000000d.txt ... phi10000000k.txt Email me if you have any questions or problems. This page accessed times.

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

Last updated March 27, 1996 Icon home page

58. Golden Ratio Discovered In Quantum World: Hidden Symmetry Observed For The First Researchers have for the first time observed a nanoscale symmetry hidden in solid state matter. They have measured the signatures of a symmetry showing the same attributes as the http://www.sciencedaily.com/releases/2010/01/100107143909.htm

59. Geometry In Art & Architecture Unit 2 The Golden Ratio Squaring the Circle in the Great Pyramid Twenty years were spent in erecting the pyramid itself of this, which is square, each face is eight plethra, and the http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html

Description and Requirements The Book Bibliography Syllabus ... Introduction The Great Pyramid Music of the Spheres Number Symbolism Polygons and Tilings The Platonic Solids ... Early Twentieth Century Art The Geometric Art of M.C. Escher Later Twentieth Century Geometry Art Art and the Computer Squaring the Circle in the Great Pyramid "Twenty years were spent in erecting the pyramid itself: of this, which is square, each face is eight plethra, and the height is the same; it is composed of polished stones, and jointed with the greatest exactness; none of the stones are less than thirty feet." -Heroditus, Chap. II, para. 124. Slide 2-1: The Giza Pyramids and Sphinx as depicted in 1610, showing European travelers Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 22 Outline: The Great Pyramid The Golden Ratio Egyptian Triangle Squaring the Circle ... Reading The Great Pyramid Slide 2-2: The Great Pyramid of Cheops Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 205 We start our task of showing the connections between geometry, art, and architecture with what appears to be an obvious example; the pyramids, works of architecture that are also basic geometric figures.

60. 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

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