41. Golden Ratio As well as constructions to divide a line in the golden ratio, Euclid gives Here is how the golden ratio comes into the construction of a pentagon. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html

The Golden ratio Number theory index History Topics Index Version for printing Euclid , in The Elements , says that the line AB is divided in extreme and mean ratio by C if AB AC AC CB Although Euclid does not use the term, we shall call this the golden ratio . The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron . Here is how the golden ratio comes into the construction of a pentagon. First construct an isosceles triangle whose base angles are double the vertex angle. This is done by taking a line AB and marking C on the line in the golden ratio. Then draw a circle with centre A radius AB . Mark D on the circle so that AC CD BD . The triangle ABD has the property that its base angles are double its vertex angle. Now starting with such a triangle ABD draw a circle through A B and D . Then bisect the angle ADB with the line DE meeting the circle at E . Note that the line passes through C , the point dividing AB in the golden ratio. Similarly construct

42. The Golden Ratio You are @ The golden ratio Golden Rectangle Golden Spiral Biology Populations Botany Aesthetics Pythagorean Greeks Romans Renaissance http://library.thinkquest.org/C005449/home.html

You are @ The Golden Ratio home intro constructions ... contact Welcome to this site! When you think of math, do you think of beauty? Do you think of stuff like pinecones and sunflowers? What does Leonardo da Vinci have to do with anything? What do the Greeks, Romans, and people of the Renaissance have in common? Well, if you don't know, you're in the right place...We're going to explain all that, and then some, in order to help you understand that yes, math can be beautiful too! To start looking through our page, click on the links you see on the top or the links under the header or at the bottom of the page. If at any time you get lost, get help , or better yet, just start with the tour Site Map Home Tour *Recommended for new visitors! Introduction Fibonacci Sequence Equations Golden Mean ... back to top Created by Andi, Mel, and Shuj for Thinkquest Internet Challenge 2000

43. The Golden Rectangle And The Golden Ratio The symbol often used for the golden ratio is ø (phi). The golden ratio (and the golden triangle) shows up (as an exact fit) in mathematics in many http://www.jimloy.com/geometry/golden.htm

Return to my Mathematics pages Go to my home page The Golden Rectangle and the Golden Ratio 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. By the way, a more accurage value is 1.6180339887498948482045868343656 . . .

44. Guardian Unlimited | Technology | 1.618 Is The Magic Number What makes the golden ratio special is the number of mathematical properties The golden ratio is the only number whose square can be produced simply by http://www.guardian.co.uk/online/science/story/0,12450,875198,00.html

@import url(/external/styles/global/0,14250,,00.css); Skip to main content Read today's paper Sign in Register Go to: Guardian Unlimited home UK news World news Newsblog Archive search Arts Books Business EducationGuardian.co.uk Film Football Jobs MediaGuardian.co.uk Money The Observer Politics Science Shopping SocietyGuardian.co.uk Sport Talk Technology Travel Been there Audio Email services Special reports The Guardian The northerner The wrap Advertising guide Crossword Soulmates dating Headline service Syndication services Events / offers Help / contacts Feedback Information GNL press office Living our values Newsroom Reader Offers Style guide Travel offers TV listings Weather Web guides Working at GNL Guardian Weekly Money Observer Public Home Technology blog Ask Jack blog Gamesblog ... Science Search Online articles Ask Jack The golden rule It links art, music and even architecture. Marcus Chown on an enigmatic number Thursday January 16, 2003 The Guardian 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.

45. The Golden Ratio And Aesthetics It was Euclid who first defined the b golden ratio /b , and ever since people have been fascinated by its extraordinary properties. http://plus.maths.org/issue22/features/golden/

about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 22 November 2002 Contents Features More or Less In a spin The golden ratio and aesthetics The best medicine? Career interview Career interview: Medical statistician Regulars Plus puzzle Pluschat Mystery mix Reviews 'The Golden Ratio' 'Euclid's Window' 'Elements of Grace' and 'Copernican Notes' 'Calculus' ... posters! November 2002 Features The golden ratio and aesthetics 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. The origins of the divine proportion 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:

46. 'The Golden Ratio' The golden ratio The story of phi, the Extraordinary Number of Nature, Art and Beauty For Mario Livio, phi, the golden ratio, evokes this holy wonder, http://plus.maths.org/issue22/reviews/book2/

about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 22 November 2002 Contents Features More or Less In a spin The golden ratio and aesthetics The best medicine? Career interview Career interview: Medical statistician Regulars Plus puzzle Pluschat Mystery mix Reviews 'The Golden Ratio' 'Euclid's Window' 'Elements of Grace' and 'Copernican Notes' 'Calculus' ... posters! November 2002 Reviews 'The Golden Ratio' reviewed by Helen Joyce The Golden Ratio: The story of phi, the Extraordinary Number of Nature, Art and Beauty 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

47. The Golden Section And The Golden Rectangle It is incredible but the golden ratio number 1.618 is the most irrational among all What has the golden ratio to do with beauty or anatomy ? http://www.mlahanas.de/Greeks/GoldenSection.htm

The Golden Section and the Golden Rectangle Michael Lahanas Der Goldene Schnitt und das Goldene Rechteck Geometry has two great treasures: one is the Theorem of Pythagoras ; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. Johannes Kepler Euclid in Book 6, Proposition 30 shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". Euclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (ie is the same as the ratio AG/AB). AB/AG = AG/GB. If AB = 1 and AG = x then GB = 1-x and 1/x = x/(1-x) and it follows x = 1-x, i.e. x - x + 1 = The golden section was found by the Pythagoreans who used the Pentagram formed, by the diagonals of a regular Pentagon , as a symbol of their school. It was probably known much earlier in Egypt as some ratio of lengths and heights of Pyramids suggest. Pythagoras may have obtained this knowledge when he visited Egypt. Theano of Thurii Theorem of the golden mean.

48. Pentagram & The Golden Ratio The ratio has become known as the golden ratio or golden section. length of the side of the pentagon (AB) is the golden ratio . http://www.contracosta.cc.ca.us/math/Pentagrm.htm

The Pentagram Golden Ratio Geometry has two great treasures: one the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. Johann Kepler (1571-1630) T he 'ratio' has become known as the golden ratio or golden section This ratio can be found in many places: in art, architecture, and mathematics. Consider the construction of the regular pentagon. If the side AB of a regular pentagon (see figure to the right) has unit length, then any diagonal, such as AC, has length and this is the golden ratio Indeed, in any similar pentagon/pentagram configuration the ratio of the length of the side of the pentagram (AC) to the length of the side of the pentagon (AB) is the golden ratio Notice also the diagonals of the pentagon form another regular pentagon in the center of the figure with, of course, the potential for additional diagonals to be drawn, thus generating the golden ratio again as well as another regular pentagon further inside the figure. Presumably this could continue indefinitely. equal to the reciprocal solving the equation: In this next figure (to the left) is a regular pentagon with an inscribed pentagram.

49. CameraHobby - E-Book On The Golden Ratio, Chapter 16 Ancient Greeks were not the only ones who understood the golden ratio. Egyptians used the golden ratio for their pyramids and the layout of the three great http://www.camerahobby.com/Ebook-GoldenRatio_Chapter16.htm

Home Learning e-Book on Photography Table of Contents Photography e-Book Chapter 16 - The Golden Ratio Loosely related to the rule of thirds is the Golden Ratio also referenced to the Golden Rectangle. This is, as far as I am able to decipher from a layperson's perspective, a mathematical look at human aesthetics. Mathematicians seem to love to apply numbers to what seemingly could or should not have numbers applied to them but what do I know, as there are many geniuses out there seeking a single mathematical formula that would explain the nature of the universe and of life itself. Mathematicians have even come up with a formula for the human decision-making process, better known as Game Theory or the Zero Sum Gain. The Golden Ratio has purportedly been a profound influence since ancient times with Greeks utilizing the Golden Ratio in their buildings such as the Parthenon at the Temple of Athena on the Acropolis. During the Renaissance when European artists rediscovered the styles of the ancient world, the Golden Ratio was utilized for their sculptures and paintings. Leonardo da Vinci being the most prominent Renaissance artist known to have used the Golden Ratio for great works such as the Mona Lisa. The more traditional physical shape of the Golden Ratio is the golden rectangle. This rectangle is comprised of a square and one-half of another square that is the same dimension together, as seen below. It can be seen as another example of the rule of thirds as the rectangle can be comprised of three equally sized smaller rectangles. Technically, the golden rectangle is comprised of two parts that follow the Fibonacci sequence.

50. Fotogenetic - 35mm Film And The Golden Rectangle The Golden Rectangle and golden ratio appear in some very interesting places. the ratio of consecutive numbers increasingly approaches the golden ratio. http://fotogenetic.dearingfilm.com/golden_rectangle.html

home articles gallery downloads ... about the author 35mm Film and the Golden Rectangle Oskar Barnack (1879-1936) "In the first years of the art, photographers were mainly concerned with hauling their heavy plate cameras from one location to the next. Their trials and tribulations stimulated Oskar Barnack into seeking a completely new method of taking photographs. Even as early as 1905, he had the idea of reducing the format of negatives and then enlarging the photographs after they had been exposed. It was ten years later, as development manager at Leica, that he was able to put his theory into practice. He took an instrument for taking exposure samples for cinema film and turned it into the world's first 35 mm camera: the 'Ur-Leica'. At the time, the miniature film format of 24 x 36 mm was created by simply doubling the cinema film format. The first photos - of outstanding quality for the time - were made in 1914. Progress was interrupted by the First World War, so the first LEICA (Leitz Camera) did not go into serial production until 1924, being presented to the public in 1925."

51. Good Stories, Pity They're Not True The great Leonardo Da Vinci is said to have used the golden ratio to proportion the It is also true that the golden ratio is linked to the pentagram http://www.maa.org/devlin/devlin_06_04.html

Search MAA Online MAA Home Devlin's Angle June 2004 Good stories, pity they're not true The enormous success of Dan Brown's novel The Da Vinci Code has introduced the famous Golden Ratio (henceforth GR) to a whole new audience. Regular readers of this column will surely be familiar with the story. The ancient Greeks believed that there is a rectangle that the human eye finds the most pleasing, and that its aspect ratio is the positive root of the quadratic equation x - x - 1 = You are faced with this equation when you try to determine how to divide a line segment into two pieces such that the ratio of the whole line to the longer part is equal to the ratio of the longer part to the shorter. The answer is an irrational number whose decimal expansion begins 1.618. Having found this number, the story continues, the Greeks then made extensive use of the magic number in their architecture, including the famous Parthenon building in Athens. Inspired by the Greeks, future generations of architects likewise based their designs of buildings on this wonderful ratio. Painters did not lag far behind. The great Leonardo Da Vinci is said to have used the Golden Ratio to proportion the human figures in his paintings - which is how the Golden Ratio finds its way into Dan Brown's potboiler. It's a great story that tends to get better every time it's told. Unfortunately, apart from the fact that Euclid did solve the line division problem in his book

52. Mr. Narain's Golden Ratio WebSite Finally, the Links page will take you to a list of other golden ratio websites, The relationship of this sequence to the golden ratio lies not in the http://cuip.net/~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?

53. TLC Human Face Among them is the golden ratio, which is the ratio of 1.618to-1. Using the golden ratio one can also make golden triangles, pentagons and decagons. http://tlc.discovery.com/convergence/humanface/articles/mask.html

54. Math & Art: The Golden Rectangle The special property of the Golden Rectangle is that the ratio of its length NoteFor more information of the golden ratio and Its appearance in nature http://educ.queensu.ca/~fmc/october2001/GoldenArt.htm

The Golden Rectangle is a unique and a very important shape in mathematics. The Golden Rectangle appears in nature, music, and is also often used in art and architecture. The special property of the Golden Rectangle is that the ratio of its length to the width equals to approximately 1.618: The Golden Rectangle is considered to be one of the most pleasing and beautiful shapes to look at, which is why many artists have used it in their work. The two artists, who are perhaps the most famous for their use of the golden ratio, are Leonardo Da Vinci and Piet Mondrian. Leonardo Da Vinci: Leonardo Da Vinci was a great Italian Renaissance painter as well as a scientist and inventor, who lived in 15th century. In his art, Da Vinci carefully examined the proportions of the human body and found many occurrences of the golden ratio and golden rectangles. Art Masterpiece : Mona Lisa You probably heard of "Mona Lisa" before. This is one of the most famous paintings in the world, and is a very good example of Da Vinci's use of the golden ratio in art. If you draw a rectangle around Mona Lisa's face, that rectangle will turn out to be golden. The dimensions of the painting itself also form a golden rectangle. As well, the proportions of Mona Lisa's body exhibit several golden ratios. For example, a golden rectangle can be drawn from her neck to just above the hands.

55. EFTel - High Speed Internet - Australian Broadband ADSL ISP The Golden Rectangle is said to be one of the most visually satisfying of all 3 is about 1;1.6 This is the so called golden ratio, or Golden Section, http://www.q-net.net.au/~lolita/symmetry.htm

EFTel Speak Fast Dial-up Broadband ADSL Satellite Broadband ... Employment Start your phone SAVINGS today with Speak! Just click here to check out the details. Best value ADSL plans EFTel's Freedom 256 and Freedom 512 are listed in the Australian Net Guide's list of Australia's Best Value Broadband Deals! Can I get ADSL? Using: Google MSN Yahoo Australia Yahoo Lycos Web Crawler Excite GoEureka Anzwers White Pages Yellow Pages Auto Business ... Employment

56. Proportion And The Golden Ratio - Mathematics And The Liberal Arts The author shows how the golden ratio occurs in music and art. The author mentions Luca Pacioli s statements on the golden ratio in De Divina http://math.truman.edu/~thammond/history/Proportion.html

Proportion and the Golden Ratio - Mathematics and the Liberal Arts To expand search, see Art . Laterally related topics: Symmetry Perspective Fractals in Art Weaving ... Origami , and Mazes The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Comput. Math. Appl. Part B (1986), no. 1-2, 3962. SC: 92A27 (01A99 52-01), MR: 838 136. Certainly an unorthodox essay. It may be hard to understand the author's terms

57. Golden Ratio (1985) constant known as the golden ratio, also known as the Golden Mean or Golden Section. In mathematical texts, the golden ratio is often represented by the http://www.frogsonice.com/quilts/golden-ratio/

Golden Ratio (1985) approximate size 86 x 91" This quilt is based on the mathematical constant known as the Golden Ratio , also known as the Golden Mean or Golden Section . In mathematical texts, the Golden Ratio is often represented by the Greek letter phi , and it has the value phi == (sqrt (5) + 1) / 2 The Golden Ratio has all sorts of neat mathematical properties. For instance, phi phi and phi phi It's also related to the Fibonacci numbers and shows up in nature in the geometry of sunflowers and nautilus shells, among other things. Click here to find more about phi In terms of the design of this quilt, the sizes of the stars are related to each other by the constant phi . Here's a sketch that shows some of the places where phi shows up in the geometry of the 5-pointed star figure: Detail. It's not a coincidence that I made this quilt in shades of gold! More detail. Still more detail. Back to the main quilting page.

58. The Golden Ratio - The Story Of Phi - The World's Most Astonishing Number - The The golden ratio The Story of Phi The World s Most Astonishing Number Mario Livio Broadway Books, 320 pp, $24.95, reviewed by Christopher Grobe http://www.yalereviewofbooks.com/archive/winter03/review03.shtml.htm

Constant Attention A biography of the number phi. The Golden Ratio: The Story of Phi: The World's Most Astonishing Number Mario Livio Broadway Books, 320 pp, $24.95 reviewed by Christopher Grobe Monograph: Poetry for Dummies? by Deborah Friedell Nonfiction Erasmus and the Age of Reformation book by Johan Huizinga reviewed by Susannah Rutherglen The Golden Ratio: The Story of Phi: ... reviewed by Michael Obernauer Fiction The Cave book by Jose Saramago reviewed by Paige Austin Three Junes ... reviewed by Su Mon Han Features Letter to an Undergraduate from Annabel Patterson Profs' Picks can't judge a cover by its book ... by Elizabeth Archibald Save $4.99 Christopher Grobe is a sophomore in Calhoun College. back to top navigate

59. 10000 Decimal Golden Ratio This is the first publication of the golden ratio to 10000 digits. I got the golden ratio by adding 0.5 to sqr(5) divided by 2. http://www.ac.wwu.edu/~stephan/webstuff/ratio.digits.html

First 10,000 Digits of the Golden Ratio This is the first publication of the Golden Ratio to 10,000 digits. If you know of an earlier one, please let me know . How was this done? Here's how. There's more We're now listed as a Useless Page (search for "gold")! The Digit Warehouse gives the first million digits of the square root of five. I got the Golden Ratio by adding 0.5 to sqr(5) divided by 2. Most computers carry division out to a limited maximum number of decimal places. To divide the first 10,000 digits of sqr(5) by 2, I wrote the following Hypercard script - "long division" by 2. on mouseUp the first 10000 decimal digits of sqr(5) = 2.236067.... are in cd fld 1 when the program's done, add 0.5 to the result put empty into cd fld 2 repeat with i = 1 to 10000 put char i of cd fld 1 after holder if holder mod 2 = then put holder/2 after cd fld 2 put empty into holder else put trunc(holder/2) after cd fld 2 put 1 into holder end if end repeat end mouseUp

60. Golden Ratio Linked To Beauty And Order In Nature - The Daily Cardinal - Science golden ratio linked to beauty and order in nature, Ratio of 1.618 appears in plants, art, human body, The Daily Cardinal, a newspaper of University of http://www.dailycardinal.com/news/2005/03/08/Science/Golden.Ratio.Linked.To.Beau

document.write(''+''); The Daily Cardinal Extras: Student Resources Scholarships Movies Travel ... GradZone What is sexier, a pocket protector, chin pubes, or an asthma inhaler? A pocket Chin pubes An asthma inhaler Comics This listing requires Javascript. We apologize for the inconvenience. Submit a letter Classifieds Back Issues Publishing Policy ... Double Coverage var story_id = 888195; Home Science Golden ratio linked to beauty and order in nature Ratio of 1.618... appears in plants, art, human body By Dinesh Ramde Published: Tuesday, March 8, 2005 In "The Da Vinci Code," author Dan Brown described the number phi, which he claimed occurs in countless occasions in nature. Because of its ubiquity, Brown wrote, phi was dubbed the Divine Proportion by ancient scholars who believed the number was "God's building block for the world." But is the number really all around us? And is it as magical as Brown would have us believe? The number itself, 1.618..., doesn't seem remarkable, but mathematically it has elegant simplicity. For example, 1 divided by phi is the same as phi minus 1: 1 ÷ phi = phi - 1.

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