61. Activity 2 Activity 2 The golden ratio. The golden ratio is a number that occurs in bothmathematics and in nature. This number is known as the golden ratio. http://homepage.mac.com/efithian/Geometry/Activity-02.html

Activity 2 The Golden Ratio The Golden Ratio is a number that occurs in both mathematics and in nature. In this activity you will examine how this ratio occurs aesthetically, geometrically, and mathematically. Your Favorite Rectangle Examine the five rectangles drawn below from several different views. Choose the one rectangle that is most appealing to you and place an X on it. Use a metric ruler to measure both sides of each rectangle to the nearest millimeter. For each rectangle, divide the length of the longest side by the length of the shortest side and write this ratio on the rectangle. Which rectangle was the most liked in the entire class?_ What is its ratio?_ Beauty of the Body The Golden Ratio can also be found in the dimensions of the human body. Measure your height to the nearest centimeter and record it below. Measure the distance of your navel from the floor to the nearest centimeter and record that also. Divide your height by your navel height to find the height/navel ratio rounded to two decimal places and record that below. Height = Navel Height = Ratio = The Pentagram The Golden Ratio also appears in the measurements of many common geometric figures. The common pentagram on the next page has the golden ratio hidden in its structure. This star is composed of a regular pentagon with an isosceles triangle on each side. Use a metric ruler to measure the lengths of the segments AB, AC, AD, and DC to the nearest millimeter. Divide to find the ratios of AB to AC; AC to AD; and AD to DC rounded to two decimal places.

62. BBC - Radio 4 - 5 Numbers - The Golden Ratio Science, 5 NUMBERS PROGRAMME 3 THE golden ratio, MISSED A PROGRAMME? Go tothe Listen Again page, Monday to Friday 11-15 March 2002, The golden ratio, http://www.bbc.co.uk/radio4/science/5numbers3.shtml

CATEGORIES TV RADIO COMMUNICATE ... INDEX SEARCH 18 Mar 2003 Text only BBC Homepage BBC Radio Radio 4 Programme Finder: A-Z Listen Again What's On Presenter Biogs ... Factual Services: Tickets Tour Radio 4 Have Your Say Weather ... Help Like this page? Send it to a friend! 5 NUMBERS - PROGRAMME 3: THE GOLDEN RATIO MISSED A PROGRAMME? Go to the Listen Again page Simon Singh investigates five very important numbers. Monday to Friday 11-15 March 2002 The Golden Ratio Leonardo Fibonacci was an Italian mathematician with a penchant for decimalization and rabbits! Having introduced the numbers to 9 to Europe (like some medieval Big Bird from Sesame Street), he turned his attention to a different series of numbers: The Fibonacci sequence is generated by adding the previous two numbers in the list together to form the next and so on and so on... Divide any number in the Fibonacci sequence by the one before it, for example 55/34, or 21/13, and the answer is always close to 1.61803. This is known as the Golden Ratio, and hence Fibonacci's Sequence is also called the Golden Sequence. Unlikely though it might seem, this series of numbers is the common factor linking rabbits, cauliflowers and snails. Fibonacci used his sequence of numbers to investigate the population growth of his favourite furry lop-eared friend, the rabbit. He based his model on a maximum-security bunny heaven where rabbits cannot escape or die, and the problem he devised goes like this...

63. BBC - Science - Human Face - Beauty Many formulas have been used in this attempt, the most famous of these beingthe golden ratio (Phi), or Golden Proportion. This is a ratio of 1 to 1.618. http://www.bbc.co.uk/science/humanbody/humanface/beauty_golden_mean.shtml

CATEGORIES TV RADIO COMMUNICATE ... INDEX SEARCH TUESDAY 18th March 2003 Text only BBC Homepage Science Human Body ... Help Like this page? Send it to a friend! Beauty Can beauty be measured? Right through history and across different cultures scientists and artists have tried to calculate the perfect dimensions for a beautiful face. Many "formulas" have been used in this attempt, the most famous of these being the Golden Ratio (Phi) , or Golden Proportion. This is a ratio of 1 to 1.618. The ancient Greeks were aware of this and used it in their art and architecture, and it was felt that this numerical association was so perfect and seen in so many places in the natural world, that it came close to the divine, and so these proportions are responsible for the uniformity of faces of thousands of statues and paintings throughout the history of western art. Click here to find out how to use the beauty mask Some cosmetic surgeons have even begun using the Golden Ratio as a guide for their work. Dr. Stephen Marquardt has discovered a 'Universal Beauty Mask' mathematically created from the Golden Ratio. In our programme he demonstrated that the grid-like mask 'fits' beautiful faces of all races and claims that most people (including babies) are naturally and instinctively drawn to faces that conform to the 1:1.618 Golden Ratio, and that we instinctively find them beautiful. Dr Marquardt also thinks that beauty is still in the eye of the beholder to some extent, because through our experience we may 'fine tune' our preferences. The faces of our loved ones and people we admire may become more beautiful to us over time, because of what they mean to us. However, he says, the Golden Ratio mask is the essence of facial beauty.

64. Geometry In Art & Architecture Unit 2 The golden ratio Squaring the Circle in the Great Pyramid. golden ratio. Sowhat is this golden ratio that the Great Pyramid is supposed to contain? 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.

65. Golden Ratio Altn Oran(golden ratio) tum dogada rastlasabilecegimiz bir oran.Peki nerden geliyorbu?Bu Pi sayisinin gizeminden gelmekte..Mesela bir altin dikdortgen cizelim http://okanss.tripod.com/okanss/id16.html

Get Four DVDs for $.49 each. Join now. Tell me when this page is updated okanss Home Stand inside your love.. Pi Golden Ratio Golden Ratio Solution Spirals People Are Strange Unknown Soldier ... ArAgOrN Golden Ratio Formula for the Golden Ratio Phi is the golden ratio. It is the irrational number that is equal to (sqrt(5)+1)/2 = 1.618033989... Using your calculator to find Phi : Enter the number 1. Add 1. Take its reciprocal. Add 1. Take its reciprocal. Add 1. Take its reciprocal. Continue this. You should be converging on the Golden Ratio EGER BUNLARDA NE DIYORSANIZ,USTTEKI RESME TIKLAYIN. Altn Oran(golden ratio) tum dogada rastlasabilecegimiz bir oran.Peki nerden geliyor bu?Bu Pi sayisinin gizeminden gelmekte..Mesela bir altin dikdortgen cizelim.Bir kosesini x alip,diger kosesinin uzunlugunu pi(x)alirsak bir altin dikdortgen olusur. Daha sonra burada kucuk kenari yarcap alan bir daire cizersek,elimizde baska bir altin dikdortgen olur Bunu sonsuz kere tekrarlarsak elimizde bir spiral yaratan dikdortgenler zinciri olusur. Spirals

66. Golden Ratio Solution 1.61803398874989484820458683436563811772030917980576286213544862270526046281890 244970720720418939113748475408807538689175212663386222353693179318006076672635 http://okanss.tripod.com/okanss/id18.html

Get Four DVDs for $.49 each. Join now. Tell me when this page is updated okanss Home Stand inside your love.. Pi Golden Ratio Golden Ratio Solution Spirals People Are Strange Unknown Soldier 11-A LILARA ... ArAgOrN Golden Ratio Solution

67. Golden Ratio This WorkBook uses Logo style graphics to explore the relations between the GoldenRatio and the Fibonacci sequence. The Greek view of ratio is illustrated. http://www.mathwright.com/book_pgs/book014.html

Been away for a while? Check out our new building by clicking the picture on the right! This WorkBook requires Mathwright Library Player 2000 to read it. To download the book, press the button on the left. A self-extracting file will be downloaded. Either save it to disk and execute it later, or simply select "Open it" from the popup dialog. This places the book, along with its documentation, on the Start, Programs, Mathwright Library menu, so that you may read it whenever you like. Size: 110 KB Find similar WorkBooks in the Rooms below: Categories: Home Study Visualization Math and Computers Subjects: Fractions Fibonacci Sequence Logo graphics Geometry ... Sequences and series Title: Golden Ratio and the Fibonacci Sequence Book Description: Author: James White Suggested Use: Visualization of an irrational number Topics: golden ratio, fibonacci sequence, Logo graphics

68. Illumin-Fall 2000 mathematics Phi. More commonly called the golden ratio or Golden Section, Phi has a numeric value of approximately 1.618. Although not http://www.usc.edu/dept/engineering/illumin/archives/fall2000/athehouse/phi/

HOME WELCOME GLOSSARY ARCHIVES ... SCHOOL OF ENGINEERING around Phi: A Magic Number Explained the house Jeremy Zechar It is human fascination with the infinite that also leads us to perhaps the most "underrated" irrational number in mathematics: Phi. More commonly called the "Golden Ratio" or "Golden Section," Phi has a numeric value of approximately 1.618. Although not as well known as pi or e, this number can be found in architecture, art, and nature. Herein, we discover the magic of Phi. You are here : home athehouse phi Mathematics in the Public Eye Film director Robert Zemeckis recently quipped that one of the fundamental commandments of Hollywood filmmaking is to eliminate numbers in a movie for fear that the audience will be confused. A handful of filmmakers, however, have disregarded this rule and actually made critically successful movies about math. Good Will Hunting , for example, tells the story of a blue collar Boston janitor who also happens to be a mathematical genius. Throughout the film, we see Matt Damon proving complex geometrical problems. In the independent film arena, Pi has been successful; here the protagonist is convinced that stock market patterns can be found in the Talmud and deep within the digits of the number pi.

69. The Golden Ratio the golden ratio. are generally considered as especially harmonious. It's not surprisingthat the golden ratio is widely used painting and in architecture. http://www.ac-noumea.nc/maths/amc/polyhedr/gold_.htm

the golden ratio A simple knot made with a strip of paper, and then carefully flatted is a " golden knot " ; just fold over one of the strip's ends and you get a complete pentagram (convex regular pentagon with its five diagonals which are the sides of a regular star pentagon). The ratio of the sides of this two regular pentagons is the golden ratio j. A rectangle whose length/width ratio is the golden ratio is a golden rectangle . Its construction is simple (with AB=2AU, ABCD and ABC'D' are two golden rectangles, and we get the first by adding a square to the second). The pentagram shows several golden sections and several examples of the two types of golden triangles (isosceles triangles with ratio of the sides equal to j An ellipse inscribed in a golden rectangle is a golden ellipse (ratio of the axis equal to j j Its area is pj j With sequences of embedded golden rectangles or triangles, we get easily nice approximate drawings of the golden spiral As the other logarithmic spirals - also known as equiangle spirals or Bernouilli's spirals - they are invariant by similitude ; Miss Nature uses them to provide harmonious growths (shells, horns...). Golden rectangle, golden ellipse, pentagram, golden spiral... are generally considered as especially harmonious. It's not surprising that the golden ratio is widely used painting and in architecture.

70. Golden Ratio Translate this page LinkageViewer applet http://iaks-www.ira.uka.de/home/egner/linkages/golden.html

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71. Golden Ratio golden ratio. You can read more about the number below, and there are links toother golden ratio sites,. and you can hear the programme. ratio. golden ratio. http://www.simonsingh.net/owtasite/Golden_Ratio.html

Golden Ratio Back to 5 Numbers More about the Golden Ratio GOLDEN RATIO You can read more about the number below, and there are links to other golden ratio sites and you can hear the programme golden ratio Golden Ratio Leonardo Fibonacci was an Italian mathematician with a penchant for decimalization and rabbits! Having introduced the numbers to 9 to Europe (like some medieval Big Bird from Sesame Street), he turned his attention to a different series of numbers: The Fibonacci sequence is generated by adding the previous two numbers in the list together to form the next and so on and so on... Divide any number in the Fibonacci sequence by the one before it, for example 55/34, or 21/13, and the answer is always close to 1.61803. This is known as the Golden Ratio, and hence Fibonacci's Sequence is also called the Golden Sequence. Unlikely though it might seem, this series of numbers is the common factor linking rabbits, cauliflowers and snails. Fibonacci used his sequence of numbers to investigate the population growth of his favourite furry lop-eared friend, the rabbit. He based his model on a maximum-security bunny heaven where rabbits cannot escape or die, and the problem he devised goes like this...

72. Golden Ratio Woodworks Massage Equipment, Portable Massage Tables, Quicklite Mas golden ratio Woodworks massage equipment, portable massage tables, quicklite massagechairs, portable massage chairs, professional massage chairs, stationary http://contractorstools.com/golden_ratio.html

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73. Golden Ratio -- From MathWorld Similar pages The golden ratioThe golden ratio/Fibonnacci Sequence Measure or Number or Fibonnacci Member.Value. First =, Second =. or. golden ratio (1)=. golden ratio (2)=. http://www.astro.virginia.edu/~eww6n/math/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

74. Los Angeles Times: The Golden Ratio FIRST CHAPTER The golden ratio The Story of Phi, the World's Most AstonishingNumber. For bookstores and other literary venues Antiquarian http://www.calendarlive.com/books/cl-first-number,0,6314207.htmlstory?coll=cl-bo

75. Spa Body Solutions pillows. A must for every therapist to support, relax pamper yourclients. golden ratio Essential Massage Stone Sets. $124.95, Set http://www.goldenratiospaworks.com/

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76. The Golden Ratio Essay. The golden ratio. It seems to have gotten its name from the Golden Rectangle,a rectangle whose sides are in the proportion of the golden ratio. http://www.oblivion.net/~ommony/writings/phi.htm

Didrik Wold MAT100C – Gruppe 20 Essay The Golden Ratio The Golden Ratio ( and related names and properties: phi ratio, sacred cut, golden mean, divine proportion, golden number, golden section) If we let 1 be the shorter part of the segment and x be the longer part and the whole segment be the sum (1+x) then we have the following equation: 1/x = x/(x+1) (Side note) This has some interesting results if we expand on it a bit. From this we can see also that: The solution of this equation is sqrt(5))/2 It is important to include Fibonacci numbers and the Fibonacci series in any discussion about the Golden Ratio. The original problem Fibonacci was working with was trying to find out how fast rabbits could breed in ideal circumstances. But that’s not what we’re interested in. We’re interested in the resulting series of numbers. This series is made up of numbers in such a way that each number is the sum of the two numbers before it. This is closely related to the Golden Ratio because if you take the ratio of two successive numbers in Fibonacci's series divide each by the number before it, we will find the following series of numbers: It becomes fairly clear what is happening once we plot the ratios on a graph: The Fibonacci numbers can also be used to create what are known as a Fibonacci rectangle and spiral. These are similar to the Golden Rectangle and Spiral (which are discussed next) but different in some important ways.

77. The Golden Ratio The golden ratio. The golden ratio describes the special relationshipfound in nature between two parts of a whole. It can be described http://www.sacredgeometry.com/golden ratio i.htm

78. Phi - The Golden Section: Did you know that you can find the golden ratio almost anywhere? Comealong with us and discover the magic of the golden ratio. http://www.mm.ocps.net/phi.htm

A Mathematical Phenomenon Powerpoint projects created by 8 th grade students at Maitland Middle School, Maitland, Florida Teachers: Mrs. Jackie Helms and Mrs. Cathy Stephens Did you know that you can find the Golden Ratio almost anywhere? In your body proportions, in architecture, in nature, in art, and even in music? Come along with us and discover the magic of the Golden Ratio. Fibonacci Numbers Golden Ratio and Music G.R. and Music Golden Ratio: Architecture ... MMS

79. Golden Ratio golden ratio This website has been relocated to http//hs.houstonisd.org/debakeyhs/Lessons/goldenratio.htmPlease update your links. pwinkler@houstonisd.org http://outreach.rice.edu/~winkler/goldenratio.html

Golden Ratio This website has been relocated to http://hs.houstonisd.org/debakeyhs/Lessons/goldenratio.htm Please update your links. pwinkler@houstonisd.org

80. Human Face -- The Learning Channel (TLC) -- Human Face Among them is the golden ratio, which is the ratio of 1.618to-1. This ratio canbe used to build so-called golden shapes; for instance, a golden rectangle http://tlc.discovery.com/convergence/humanface/articles/mask.html

Doctor May Have Beauty's Number By Rafe Jones Ever think a number could be beautiful? From the bad luck of 13 to the holiness of 3, people have long ascribed mystical and abstract properties to numbers. One number in particular has been associated with beauty for nearly 2,000 years, and it's not a number you can count to: the so-called "golden section" is an irrational number approximately equal to 0.618. Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have. Beauty Theory According to Dr. Marquardt, beauty is a mechanism to ensure humans recognize and are attracted to other humans. "Other animals recognize members of their own species and have a tremendous reaction when they do," he says, noting dogs' common reaction to other dogs. He adds that humans are highly visual creatures, so we use sight as our primary means of recognition. The most beautiful faces, he claims, are the ones that are the most easily recognizable as human. "Beauty is really just humanness," he says. Dr. Marquardt theorizes that we discern whether a face is obviously human by unconsciously comparing it to an ideal face that lurks in the unreachable recesses of the psyche. Since we have no direct access to this ideal "most human" face, we can't say precisely what it is; however, Dr. Marquardt claims he has captured the ideal face in his beauty mask. He says, "The mask radiates, it advertises and screams: 'human, human, human.' "

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