1. The Golden Section - The Number And Its Geometry We will call the golden ratio (or Golden number) after a greek letter Phi ( ) here, although some writers and http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html

The Golden section ratio: Phi Contents of this Page The line means there is a Things to do investigation at the end of the 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: Phi = Phi + 1 In fact, there are

2. The Golden Ratio Gives and intro to the golden ratio and its presence in biology, art, and ancient art This site is devoted to the golden ratio. In this page, you will find information about the golden ratio, Fibonacci http://www.geom.umn.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.

3. Golden Ratio -- From MathWorld Order portable, salon, spa and stationary tables. Includes a description and a photo of each. http://mathworld.wolfram.com/GoldenRatio.html

Number Theory Constants Continued Fraction Constants Number Theory ... Lambrou Golden Ratio 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 also known as the divine proportion, golden mean, and golden section and is a Pisot-Vijayaraghavan constant . It 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 From the definition of the golden mean combined with the above figure

4. The Golden Ratio And The Fibonacci Numbers A presentation of the relationship between the golden ratio and the Fibonacci Numbers from the proceeding Category Science Math Recreations Specific Numbers phi......The golden ratio and The Fibonacci Numbers. Copyright (c) 1999 Kelley L. Ross, Ph.D.All Rights Reserved. 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

5. Fibonacci Numbers, The Golden Section And The Golden String Fibonacci numbers and the golden section in nature, art, geometry, architecture, music, geometry Category Science Math Specific Numbers Fibonacci Numbers...... Numbers and Nature Fibonacci and the original problem about rabbits where the seriesfirst appears, the family trees of cows and bees, the golden ratio and the 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 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? 10 February 2003 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.

6. Fibonacci Numbers And The Golden Section In Art, Architecture And Music Mathematics lessons for elementary, middle, and high school including geometry. considered the most visually appealing. This ratio, called the golden ratio, not only appears in art and architecture, http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html

Fibonacci Numbers and The Golden Section in Art, Architecture and Music This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio in architecture, art and music. Contents of this page The golden section in architecture The Parthenon and Greek Architecture Modern Architecture Architecture links ... Links to Art sources including Contemporary Artists Fibonacci in Films Fibonacci and Poetry Fibonacci and Music ... More The Golden section in architecture The Parthenon and Greek Architecture Even from the time of the Greeks, a rectangle whose sides are in the golden proportion (1 : 1.618 which is the same as 0.618 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen) and a construction for the golden section point is found in Euclid's Elements in this connection. This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens , Greece but there is no original documentary evidence that this was how the building was designed. (There is a replica of the original building (accurate to one-eighth of an inch!) at

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

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, The Golden Ratio , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to The Golden Ratio click here Back to the Previous Page The Site you have Requested ... The Golden Ratio click here to view this site A ThinkQuest Internet Challenge 2000 Entry Click image for the Site Site Desciption 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. Students Andrei Grupul Scolar H. Coanda Romania Shujun Thomas Jefferson High School for Science and Technology VA, United States

8. Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence Please tell me about the golden ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci http://forum.swarthmore.edu/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

9. ThinkQuest Library Of Entries 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

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, The Golden Ratio , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to The Golden Ratio click here Back to the Previous Page The Site you have Requested ... The Golden Ratio click here to view this site A ThinkQuest Internet Challenge 2000 Entry Click image for the Site Site Desciption 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. Students Andrei Grupul Scolar H. Coanda Romania Shujun Thomas Jefferson High School for Science and Technology VA, United States

10. Golden Ratio How to generate the number, GSP script for dividing segments, rectangle and other shapes, the rabbit Category Science Math Recreations Specific Numbers phi...... 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.

11. Welcome To Golden Ratio Custombuilt massage tables, massage chairs, spa equipment, and other products.Category Business Healthcare Products and Services Massage...... http://www.goldenratio.com/

12. What's New On The Golden Ratio Site Now open—golden ratio's Wellspring Institute—advancing the arts of wellnessand beauty. golden ratio Woodworks 1 800345-1129 PO Box 297 2896 Hwy. http://www.goldenratio.com/whatsnew.html

13. The Golden Ratio Gives and intro to the golden ratio and its presence in biology, art, and ancientart. Introduction to the golden ratio and Fibonacci Numbers. Biology. 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

14. Steve, Jeanette, And Marc's Final Project Project with art references and object construction lessons.Category Science Math Recreations Specific Numbers phi......OUR FINAL PROJECT. THE golden ratio The purpose of this web page is toprovide an introduction to the golden ratio and Fibonacci Sequence. http://www.geom.umn.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.

15. 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 Answer Answer Answer Next: Find the objects around you that are closest to the golden ratio. Build a rectangle approximating the golden ratio. Confirm the ratio with algebra. Back to the Geometry Online Index Email any comments to lanius@math.rice.edu URL http://math.rice.edu/~lanius/Geom/golden.html

16. Cynthia Lanius' Lessons: Algebra Of The Golden Ratio golden ratio Algebra. golden ratio 1.61803, We used this property when we built therectangle, and now we will use it to confirm the value of the golden ratio. http://math.rice.edu/~lanius/Geom/algebra.html

Cynthia Lanius Table of Contents Introduction Find golden rectangles. Build golden rectangles. Confirm the ratio using algebra. Golden Ratio Algebra Golden Ratio 1.61803 One property of golden rectangles is that their length (length + width) = width length We used this property when we built the rectangle , and now we will use it to confirm the value of the golden ratio. When we cross multiply the above proportion, we get l = lw + w or l - lw - w Solving the equation gives us l = w(1 + sqrt5) 2 Divide both sides by w: l = (1 + sqrt5) - - w 2 Enter this into a calculator, and you'll see the approximation of the golden ratio - Geometry Online Index Email any comments to lanius@math.rice.edu URL http://math.rice.edu/~lanius/Geom/algebra.html

17. 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

18. 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?

19. Golden Ratio Conjugate -- From MathWorld golden ratio Conjugate, The quantity, (1). where is the golden ratio.The golden ratio conjugate is sometimes also called the silver ratio. http://mathworld.wolfram.com/GoldenRatioConjugate.html

Number Theory Constants Continued Fraction Constants Number Theory ... 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 Author: Eric W. Weisstein Wolfram Research, Inc.

20. The Golden Ratio Extension of the number.Category Science Math Recreations Specific Numbers phi......Last updated March 27, 1996 1.61803398874989484820458683436563811772030917980576286213544862270526046281890 http://www.cs.arizona.edu/icon/oddsends/phi.htm

Last updated March 27, 1996 Icon home page

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