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Golden Ratio: more books (52)

A key to golden ratio geometry by Joan Moore, 1998
Golden Ratio: Mathematics, Irrational number, Mathematical constant, Phidias, Algebraic number, Golden rectangle, Plato, Euclid, Fibonacci, Luca Pacioli, Johannes Kepler, Charles Bonnet, Roger Penrose
Approximating the mean waiting time under the golden ratio policy (Research report RC. International Business Machines Corporation. Research Division) by Thomas K Philips, 1988
The Golden Ratio - Story Of Phi, The World's Most Astonishing Number by Mario Livio, 2003-01-01
The Golden Ratio The Story of PHI the Worlds Most Astonishing Number
Discover it!: Fractions, area, perimeter, Pythagoras, golden ratio, limits by Manuel Dominguez, 1986
Beyond the Golden Ratio by Daljit S. Jandu, 2008-02-07
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number by Mario Livio, 2002-01-01
The Golden Section: An Ancient Egyptian and Grecian Proportion by Steven L Griffing, 2007-11-19
Number Theory and the Periodicity of Matter by Jan C. A. Boeyens, Demetrius C. Levendis, 2010-11-30
Nature: An entry from Macmillan Reference USA's <i>Macmillan Reference USA Science Library: Mathematics</i> by Bart Geerts, 2002
GEOMETRY AND PLANNING: An entry from Gale's <i>Arts and Humanities Through the Eras</i>
Geometry in nature and Persian architecture [An article from: Building and Environment] by M. Hejazi,
Will the rest of the world live like America? [An article from: Technology in Society] by J.H. Ausubel,

lists with details

'The Golden Ratio'
November 2002. Reviews. The golden ratio . reviewed by Helen Joyce. The GoldenRatio The story of phi, the Extraordinary Number of Nature, Art and Beauty.
http://plus.maths.org/issue22/reviews/book2/

Extractions: 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' ... poster! November 2002 Reviews 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.
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

Extractions: June 06, 2004 EDT 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.' "
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)
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
Fibonacci Numbers And The Golden Ratio
Fibonacci numbers and the golden ratio. They called any such rectanglea Golden Rectangle, and so became known as the golden ratio.
http://www.math.umass.edu/~dhayes/math597a/ugcd2/node3.html

Extractions: Next: The Euclidean algorithm stops Up: Analyzing the efficiency of Previous: The worse case: Fibonacci The ancient Greek geometers believed that a rectangle with sides in the ratio was the most pleasing to the eye. They called any such rectangle a Golden Rectangle , and so became known as the Golden Ratio . Naturally, the Greeks defined the Golden Ratio geometrically instead of writing it as a number. Definition 792 A Golden Rectangle is a rectangle R having the following property. If a square on one of the shorter sides of R is removed from R , then the remaining rectangle is similar to R In other words, if a square on one side of a Golden Rectangle is removed from the rectangle, the rectangle that remains is also golden! If u and u v are, respectively, the short and the long side of the Golden Rectangle above, then clearly which shows that or . Solving, we find , demonstrating that the numerical and geometric definitions are the same. The default window in which 2-dimensional graphics are drawn in the Mathematica software package is a Golden Rectangle.
Activities
The golden ratio in Everyday Objects. The golden ratio in Architecture.The golden ratio in Art. The golden ratio in Nature. The Perfect Face.
http://cuip.uchicago.edu/~dlnarain/golden/activities.htm

Extractions: Please click on the Activity Number to go there. The Golden Ratio in Everyday Objects The Golden Ratio in Architecture The Golden Ratio in Art Constructing a Golden Rectangle (Method One) Constructing a Golden Rectangle (Method Two) Constructing a Golden Spiral The Golden Ratio in Nature The Perfect Face Home Activities Conclusions Assessment ... Email Last Updated January 3, 2003 dlnarain@yahoo.com
The Golden Ratio In Art
The golden ratio in Art. Now let s go back and try to discover the golden ratioin art. Directions for finding evidence of the golden ratio in each painting
http://cuip.uchicago.edu/~dlnarain/golden/activity3.htm

Extractions: The Golden Ratio in Art Now let's go back and try to discover the Golden Ratio in art. We will concentrate on the works of Leonardo da Vinci, as he was not only a great artist but also a genius when it came to mathematics and invention. Your task is to find at least one of the following da Vinci paintings on the Internet. Make sure that you find the entire painting and not just part of it. The best way to do this is to use a search engine. I suggest either Google or Lycos . Type the name of the painting you wish to find into the search engine and see what you can come up with. Once you find the painting, return to this site for instructions on how to find the Golden Ratio. If you wish, you may borrow the image you find by right-clicking on the image and selecting "Save Image As...". Then save it to the desktop. This way you have the image on the computer you are using. If you have the capability (a color printer) it might be a good idea to print the image out as well, but this is not completely necessary. List of paintings to look for: The Annunciation Madonna with Child and Saints The Mona Lisa St. Jerome
Mr. Narain's Golden Ratio WebSite
Welcome to Mr. Narain s golden ratio Page! This page is meant to moreinvolving than this one. Introduction. What is the golden ratio?
http://cuip.net/~dlnarain/golden/

Extractions: 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?
Golden Ratio Homepage
The golden ratio. Introduction to The golden ratio. It is my intention to highlightand explain the key points of this number. A Definition of the golden ratio.
http://students.bath.ac.uk/ma1cam/GRpage.html

Extractions: The Golden Ratio Introduction to The Golden Ratio Since the early Greeks, the ratio of length to width of approximatly 1.618 has been identified as being the most asthetically pleasing. This ratio is used in architecture, art and, along with Fibonaccis numbers, appears frequently in nature. It also, as a number, posseses very interesting qualities. It is my intention to highlight and explain the key points of this number. A Definition of the Golden Ratio As mentioned before the Golden Ratio is approximatly 1.618 (click here for a more accurate approximation). Here is a definition explaining how the ratio was found. Divide a line into two sections such that the ratio of the whole line over the larger section is equal to the larger section over the smaller section. The point on the line that divides the two sections is called the Divine Cut. The Golden Ratio is 1/a=a/b. The Golden Ratio is often refered to by the greek letter Phi ) and sometimes tau ), refering to the cut that is made in the definition. On this website I will take Phi to stand for the Golden Ratio.
The Golden Ratio In Art
THE golden ratio IN THE ARTS. Throughout the centuries, artists andmusicians pursuing aesthetics - have turned to natures own
http://students.bath.ac.uk/ma1caab/art.html

Extractions: Throughout the centuries, artists and musicians - pursuing aesthetics - have turned to natures own tactics and emulated the golden ratio in thier own creations. There is evidence of this from ancient times - for example this aztec decoration. The space between the two heads is Phi times the width of the heads. The renaissance was a period of renewed interest and advancement in the arts. Many artists which have become 'household names' are from this era, and it turns out that some of the most famous artists of all time had a very mathematical approach to thier work. In Michaelangelo's renowned life size sculpture 'David', the naval is at the golden section of the figure. Leonardo da Vinci was a mathematician and a scientist as much as he was an artist: along with other techniques he frequently used the divine proportion in his paintings. The example on the left is a more modern one - 'Post' by Picasso - using a golden mean gauge we can see that the lines are spaced to the golden proportion. Another interesting example of its application is in the De Stjil art movement of the 20 th century: these apparantly random black lines and blocks of primary colour are in fact mostly constructed to the golden section. The illustration on the right is from a painting by Mondrian, and again uses the same gauge to illustrate our point.
Golden Ratio And Architecture
golden ratio and Architecture. The Golden above. Also, the spacesbetween the columns are in proportion to the golden ratio. The
http://educ.queensu.ca/~fmc/october2001/GoldenArc.htm

Extractions: Golden Ratio and Architecture 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: Example 1: Ancient Egyptians, Mesopotamians and Greeks were aware of the beauty of the Golden Rectangle and used it to create many different buildings. One of the most famous and beautiful buildings, built in ancient Greece on the Acropolis, is called the Parthenon. It was created about 2500 years ago. Golden Ratio appears in several places in the design of the Parthenon. The dimensions of the front of the temple form a perfect golden rectangle, as shown on the diagram above. Also, the spaces between the columns are in proportion to the golden ratio. The golden rectangles can be seen in almost all exterior dimensions of this great masterpiece.
The Geometry Junkyard: Pentagonal Geometry And The Golden Ratio
The Geometry Junkyard. Pentagonal Geometry and the golden ratio. Fibonaccispirals, Ned May. The golden ratio in an equilateral triangle.
http://www.ics.uci.edu/~eppstein/junkyard/pent.html

Extractions: Pentagonal Geometry and the Golden Ratio This page includes geometric problems defined on regular pentagons, involving pentagonal angles, or based on the golden ratio (the ratio of diagonal to side length in a regular pentagon). A Brunnian link . Cutting any one of five links allows the remaining four to be disconnected from each other, so this is in some sense a generalization of the Borromean rings. However since each pair of links crosses four times, it can't be drawn with circles. Constructing a regular pentagon inscribed in a circle, by straightedge and compass. Scott Brodie. Also described by M. Gallant Cut-the-knot logo . With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon. Digital Diffraction , B. Hayes, Amer. Scientist 84(3), May-June 1996. What does the Fourier transform of a geometric figure such as a regular pentagon look like? The answer can reveal symmetries of interest to crystallographers. The downstairs half bath . Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles. Equilateral pentagons . Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them. Equilateral pentagons that tile the plane , Livio Zucca.
The Golden Ratio
This site is all about the golden ratio and how it applies to biology,art, and ancient Egyptian art. domain V3. The golden ratio.
http://zap.to/goldenratio
NPR : 'The Golden Ratio'
But in his new book i The golden ratio /i , author Mario Livio examines the mysteriesof pi s lesserknown cousin, phi a number that has both counfounded
http://www.npr.org/display_pages/features/feature_1191723.html

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Hunt For Golden Ratio
Hunt for golden ratio an Internet Treasure Hunt on golden ratio createdby elaine_bynum Waubonsie Valley HS. What is the golden ratio?
http://www.kn.pacbell.com/wired/fil/pages/huntgoldenrel.html

Extractions: Waubonsie Valley HS Introduction The Questions Internet Resources Grab your brain and head for the further reaches of cyberspace. There's a lot to learn in this information age. Using the Web allows you to discover tons more than you may have ever known possible. Below is a list of questions about the topic of this page. Surf the Internet links on this page to find answers to the questions. Name at least 2 artists that use/used the golden rectangle. Need help with the math? This site has a Fibonnacci Calculator From you hunt you should know how to build a golden rectangle using a compass and ruler, find a golden rectangle in the 'real world'- art or nature or history, and define a golden ratio algebraically. Can you draw a room with golden dimensions? Why would you want to build a room of this size?
CameraHobby - E-Book On The Golden Ratio, Chapter 16
Photography eBook Chapter 16 - The golden ratio. Loosely related to the ruleof thirds is the golden ratio also referenced to the Golden Rectangle.
http://www.camerahobby.com/Ebook-GoldenRatio_Chapter16.htm

Extractions: 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.
Golden Ratio - Wikipedia, The Free Encyclopedia
golden ratio. Two quantities are said to be in the golden ratio, if thewhole is to the larger as the larger is to the smaller , ie if.
http://en.wikipedia.org/wiki/Golden_mean

Extractions: (Redirected from Golden mean The golden ratio proportio divina or sectio aurea ), also called the golden mean golden section golden number or divine proportion , usually denoted by the Greek letter phi , is the number Table of contents 1 Properties 2 Mathematical uses 3 Aesthetic uses 4 See also ... edit It is the unique positive real number with and the equally interesting property Two quantities are said to be in the golden ratio , if "the whole is to the larger as the larger is to the smaller", i.e. if Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference: After simple algebraic manipulations (multiply the first equation with a b or the second equation with ( a b b ), both of these equations are seen to be equivalent to and hence The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio". edit 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
Proportion And The Golden Ratio - Mathematics And The Liberal Arts
Proportion and the golden ratio Mathematics and the Liberal Arts. Theauthor shows how the golden ratio occurs in music and art.
http://math.truman.edu/~thammond/history/Proportion.html

Extractions: 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
The Golden Mean
The ratio of 1 2 was least liked, while the golden ratio was favored by a verylarge margin, which seemed to point to the actual dimensions as generating a
http://www.middlebury.edu/~harris/Humanities/TheGoldenMean.html

Extractions: THE GOLDEN MEAN WITH A New Solution for the Parthenon's GM The Golden Mean is a ratio which has fascinated generation after generation, and culture after culture. It can be expressed succinctly in the ratio of the number "1" to the irrational "l.618034... ", but it has meant so many things to so many people, that a basic investigation of what might is the "Golden Mean Phenomenon" seems in order. So much has been written over the centuries on the Mean, both fanciful imaginings and recondite mathematicizations, that a review of the literature on the subject would be oversize, and probably lose the focus of the problem. This purpose of this paper is to state in the simplest form problems which relate to the Golden Mean, and pursue a variety of directions which aim to explain the origin of this remarkable ratio and its ultimate meaning in the world of mind and matter. The French architect LeCorbusier noted that the human body when measured from foot to navel and then again from navel to top of head, showed average numbers very near to the Golden Ratio. He extended this to height compared with arm-span, and designed doorways consonant with these numbers. But of course much of this was based in averages rather than exact numbers, and so falls into the general area of esthetic design, rather than mathematical proportion. However studies have shown that the patterns of tree- branching adhere to the GM proportion, although again not exactly, while the dendritic cracking in certain metallic alloys which occurs as very low temperatures is basically GM based. In an entirely different area, Duckworth at Princeton found in the early l940's a GM relationship in the length of paragraphs in Vergil's Aeneid, with the figures becoming ever more accurate as larger samples were taken. Lendvai has demonstrated that Bartok used the GM ratio extensively in composing music, the question remaining whether an artist as an educated person uses the GM ratio consciously as a framework for his work, or unconsciously because of its ubiquitous appearance in the world around us, something we sense by living in a GM proportioned world.
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

Extractions: 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.

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