Geometry.Net - Math_Discover: Golden Ratio

Spa Body Solutions
About golden ratio Bodyworks We are located at the base of Emigrant Peak,on the Yellowstone River in beautiful Montana. As a division
http://www.goldenratiobodyworks.com/info/info.html
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

Extractions: Go to: Guardian Unlimited home UK news World news Archive search Arts Books Business EducationGuardian.co.uk Film Football Jobs 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 Headline service Syndication services Events / offers Help / contacts Information Newsroom Soulmates Style guide Travel offers TV listings Weather Web guides Guardian Weekly Money Observer Online home Ask Jack Web watch Gadgets ... Working IT out 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. What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can be produced simply by adding 1 and whose reciprocal by subtracting 1. If you take a golden rectangle - one whose length-to-breadth is in the golden ratio - and snip out a square, what remains is another, smaller golden rectangle. The golden ratio is also difficult to pin down: it's the most difficult to express as any kind of fraction and its digits - 10 million of which were computed in 1996 - never repeat.
PHI, The Golden Ratio
Provides a definition of Phi, and explains some mathematical background as well as a listing of 1000 Category Science Math Recreations Specific Numbers phiPHI, the golden ratio. (Also known as the golden mean) Written by Paul BourkeMay 1990, Updated January 1995. Definition. 2 dimensional golden ratio.
http://astronomy.swin.edu.au/pbourke/analysis/phi/

Extractions: 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 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
American Phi
Music, story and poetry of the golden ratio.
http://members.aol.com/loosetooth/phi.html
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
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
Golden Ratio References
References for The golden ratio. Books R Fischler, On applications of the goldenratio in the visual arts, Leonardo 14 (1981), 3132; 262-264; 349-351.
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/References/Golden_ratio.html
Blowing The Lid Off Phi
A look at the numerical sequences which underlie the golden ratio and analogous numbers.
http://home.earthlink.net/~conklin76/phi.html

Extractions: Blowing the Lid off Phi F F , is a representation of a number which, when multiplied by itself, is the same as itself plus 1. Graphically: F F is derived from the following equation: and, if you don't know the pattern, it's that each number is the sum of the previous two numbers, so 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, etc. F , here. The first is that if you find the ratios of the successive numbers, the results converge on 1.618..., or F , thus: F F F F F F F F F F F F F F F F F F F F ) and the Fibonacci number series, and while I agree there is significance, I disagree on what it is. F is unique, right? F is 1.6188033989... F represents a number in a series. I will try to illustrate that series. To get it, it's first important to take another look at the formula from which F is derived. F F fits in? At 1. And 2 fits in at 2. What about at 3 and 4? F sequence, the number being multiplied (in red) and the number being added (in blue), added together equal the red number in the next formula. What you also might notice, is that the number in red, MULTIPLIED BY 2, equals the number in blue in the next formula. In the F sequence, you would multiply by 1, instead.
Welcome To The Golden Ratio
This is an informative site on an interesting aspect of Geometry The golden ratio.Category Science Math Geometry
http://members.tripod.com/~ColinCool/mathindex.html
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

Extractions: 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
Welcome To The Golden Ratio
Welcome To Greg and Colin's golden ratio Extravaganza. The GoldenRatio manifests in the whole of creation. Take the ratio of the
http://members.tripod.com/~ColinCool/Pages/Welcome.html
Golden Ratio
Essay and brief introduction by Edwin M. Dickey.Category Science Math Recreations Specific Numbers phiThe golden ratio A Golden Opportunity to Investigate Multiple Representations ofa Problem. Graphic Solution to golden ratio Equation Using Derive ª Software
http://www.ite.sc.edu/dickey/golden/golden.html

Extractions: 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
Golden Ratio
Provides a golden ratio GreekFace activity for 5th grade and up, plus a link for extra information.Category Science Math Education Teaching ResourcesA golden ratio Activity and Resource par excellence! 1) Scroll below MarkWahl. A golden ratio Activity. A GOLDEN GREEK FACE. Toolbox
http://www.markwahl.com/golden-ratio.htm
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 Category Science Math Recreations Specific Numbers Relation between the Fibonacci Sequence and the golden ratio. Dr. Math's discussionof the golden ratio, Rectangle and Fibonacci sequence. The golden ratio.
http://pw1.netcom.com/~merrills/fibphi.html

Extractions: 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.
Golden Ratio
The golden ratio. Contents. Construct the Regular pentagons. Constructthe golden ratio and a golden rectangle. Construct the golden ratio.
http://www.math.csusb.edu/courses/m129/golden/golden_ratio.html

Extractions: Mark a segment CD of length 1 along L, so that AD has length 2. With compass point at B, mark off distance BD along line M in the direciton opposite A. Call the intersection point E. The ratio of lengths AE/AD is called the golden ratio. This number is usually denoted by the greek letter tau, but I will use g, which is easier to type. What is the length of BD? Hint: use the Pythagorean theorem. What is the length of AE? What is the golden ratio? Express it in two ways: with radicals, and as a decimal approximation. Show that the golden ratio (in radical form) satisfies the polynomial equation x^2 - x - 1 = 0. (Note: the symbol ^ means "raised to the power"; so x^2 means x raised to the 2nd power, or x squared.) Use algebra to rearrange the equation for g in the previous question to show that g^2 = g + 1 (that is, g^2 can be replaced by g+1 whenever convenient)
The Golden Rectangle And The Golden Ratio
The Golden Rectangle and the golden ratio. This fraction, (a+b)/a,is called the golden ratio (or golden section or golden mean).
http://www.jimloy.com/geometry/golden.htm

Extractions: Go to my home page 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.
The Golden Rectangle And The Golden Ratio
The Golden Rectangle and the golden ratio. Return to the primary GoldenRectangle and golden ratio page. © Copyright 1997, Jim Loy.
http://www.jimloy.com/geometry/goldenz.htm

Extractions: Go to my home page jimloy@jimloy.com Return to the primary 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. Supposedly, Pythagoras discovered this ratio. And the ancient Greeks incorporated it into their art and architecture. Apparently, many ancient buildings (including the Parthenon) use golden rectangles. It was thought to be the most pleasing of all rectangles. It was not too thick, not too thin, but just right (Baby Bear rectangles). Because of this, sheets of paper and blank canvases are often somewhat close to being golden rectangles. 8.5x11 is not particularly close to a golden rectangle, by the way.
Deep Secrets
A new theory that uses diagrammatic geometry to reveal a possible connection between the Great Pyramid, Category Science Social Sciences Alternative Egypt PyramidsA new theory that uses diagrammatic geometry to reveal a possible connection betweenthe Great Pyramid, the golden ratio and the ancient Egyptian Royal Cubit.
http://www.sover.net/~rc/deep_secrets/

Extractions: This site provides a new, and perhaps for some a controversial, explanation for the rationale behind the exterior 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 derivation diagram for the interior design parameters of the Great Pyramid. 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 9 feet), embodies a numerical finding equal to the enigmatic ratio known as the "golden ratio" (for a schematic of this situation refer to triangle ABC in
The Golden Ratio And Aesthetics
The golden ratio and aesthetics. by Mario Livio. Mario Livio is ascientist the arts. The golden ratio in the arts. Many books claim
http://plus.maths.org/issue22/features/golden/

Extractions: Issue 22: Jan 03 Issue 21: Sep 02 Issue 20: May 02 Issue 19: Mar 02 Issue 18: Jan 02 Issue 17: Nov 01 Issue 16: Sep 01 Issue 15: Jun 01 Issue 14: Mar 01 Issue 13: Jan 01 Issue 12: Sep 00 Issue 11: Jun 00 Issue 10: Jan 00 Issue 9: Sep 99 Issue 8: May 99 Issue 7: Jan 99 Issue 6: Sep 98 Issue 5: May 98 Issue 4: Jan 98 Issue 3: Sep 97 Issue 2: May 97 Issue 1: Jan 97 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. In the Elements , the most influential mathematics textbook ever written, Euclid of Alexandria (ca. 300 BC) defines a proportion derived from a division of a line into what he calls its "extreme and mean ratio." Euclid's definition reads:
"The Golden Ratio"
The golden ratio . reviewed by Helen Joyce. The golden ratio The storyof phi, the Extraordinary Number of Nature, Art and Beauty.
http://plus.maths.org/issue22/reviews/book2/

Extractions: Issue 22: Jan 03 Issue 21: Sep 02 Issue 20: May 02 Issue 19: Mar 02 Issue 18: Jan 02 Issue 17: Nov 01 Issue 16: Sep 01 Issue 15: Jun 01 Issue 14: Mar 01 Issue 13: Jan 01 Issue 12: Sep 00 Issue 11: Jun 00 Issue 10: Jan 00 Issue 9: Sep 99 Issue 8: May 99 Issue 7: Jan 99 Issue 6: Sep 98 Issue 5: May 98 Issue 4: Jan 98 Issue 3: Sep 97 Issue 2: May 97 Issue 1: Jan 97 reviewed by Helen Joyce Euclid defined what later became known as the Golden Ratio thus: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. In the introduction to this book, the author quotes Einstein as saying The fairest thing we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and science. He who knows it not and can no longer wonder, is as good as dead, a snuffed-out candle. For Mario Livio, phi, the Golden Ratio, evokes this holy wonder, and he sets out to make us feel it too. Telling us that "the Golden Ratio has inspired thinkers of

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