Geometry.Net - Math_Discover: Golden Ratio

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. Fibonacci spirals , Ned May.
Pentagram & The Golden Ratio
The Pentagram The golden ratio. Johann Kepler (1571-1630). The'ratio' has become known as the golden ratio or golden section.
http://www.contracosta.cc.ca.us/math/pentagrm.htm

Extractions: and this 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. The golden ratio also appears in comparing consecutive elements of certain kinds of sequences, most notably, the
Inter.View To George Cardas
An interview with George Cardas, describing his use of the golden ratio in highend audio equipment cables.Category Science Math Recreations Specific Numbers phi by Lucio Cadeddu. A brief introduction to golden ratio. Let me writedown a brief survey on golden ratio and its amazing history.
http://www.tnt-audio.com/intervis/cardase.html

Extractions: Let us take a segment a of lenght 1. Another segment b is said to be the Golden Section of a if it solves the following equation: b + b - 1 = that is to say the two segments respect the following proportion: a : b = b : (a-b) . In simpler words, given the fact that a has lenght 1, b must be 0.618 approx. Historically the Golden Ratio was well known to the Egyptians who used it for building their pyramids but it achieved wider popularity thanks to the Greek geometers.
Spa And Salon Tables And Equipment By Golden Ratio Beauty
golden ratio Beauty A division of golden ratio Woodworks 1 800345-1129 PO Box 2972896 Hwy. 89 South Emigrant, Montana 59027 406-333-4578 ph. golden ratio.
http://www.goldenratiobeauty.com/

Extractions: Golden Ratio Sports. A line of top quality, tough massage tables built to perform in the sports massage arena. Models include The Master Trainer (heavy-duty portable massage table), The Athlete (solid portable massage table for the average athlete) and Stadium Flat Top (super strong stationary massage table), The Power Lift Flat Top (the ultimate electric-lift exam and treatment table for any size or weight of athlete) and the Field Chair (on-site massage chair built on a 15% larger frame than our QuickLite massage chair). See our catalog order page to order information.
10000 Decimal Golden Ratio
First 10,000 Digits of the golden ratio. This is the first publication of the GoldenRatio to 10,000 digits. If you know of an earlier one, please let me know.
http://www.wwu.edu/~stephan/webstuff/ratio.digits.html

Extractions: 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
Proportion And The Golden Ratio - Mathematics And The Liberal
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
WNYC - Reading Room
WNYCmail Berlex FAQ. The golden ratio The Story of Phi, the World'sMost Astonishing Number By Mario Livio Random House Copyright
http://www.wnyc.org/books/11057

Extractions: ISBN: 0-7679-0815-5 Available for purchase at Amazon.com Chapter One PRELUDE TO A NUMBER Numberless are the world's wonders.-Sophocles (495-405 b.c.) Less known than pi is another number, phi (f), which is in many respects even more fascinating. Suppose I ask you, for example: What do the delightful petal arrangement in a red rose, Salvador Dali's famous painting "Sacrament of the Last Supper," the magnificent spiral shells of mollusks, and the breeding of rabbits all have in common? Hard to believe, but these very disparate examples do have in common a certain number or geometrical proportion known since antiquity, a number that in the nineteenth century was given the honorifics "Golden Number," "Golden Ratio," and "Golden Section." A book published in Italy at the beginning of the sixteenth century went so far as to call this ratio the "Divine Proportion." In everyday life, we use the word "proportion" either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term "proportion" is used to describe an equality of the type: nine is to three as six is to two. As we shall see, the Golden Ratio provides us with an intriguing mingling of the two definitions in that, while defined mathematically, it is claimed to have pleasingly harmonious qualities.
Golden Ratio Lesson
They have established that the golden ratio can be found in the humanskeleton. Objectives. The student will find patterns in data.
http://www.ruf.rice.edu/~winkler/goldenratio.html

Extractions: This page is still under construction. Last updated August 6, 1996 Patricia Winkler DeBakey High School for Health Professions Houston, TX 77021 This is a lesson intended to give students to opportunity to explore the claims of American researcher, Jay Hambridge, and others. They have established that the Golden Ratio can be found in the human skeleton. The student will: find patterns in data. use deductive reasoning to compare the data from all groups of students and draw general conclusions based on the data. determine sample size. discuss the effect of sample size on the conclusions drawn. develop statistical strategies for improved sample size. determine whether or not there is any data which will skew the results and why the data was inconsistent with the sample. This lesson can be conducted with any age student who can read a tape measure with accuracy. The student should be able to: accurately measure an object to the nearest millimeter or fraction of an inch. compute ratios of related body measures.
Golden Ratio, The
Euclid defined this curious mathematical relationship, widely known as the GoldenRatio, more than 2,000 years ago because of its crucial role in the
http://www.sciencenewsbooks.org/goldenratio.html

Extractions: The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras, who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as one of the greatest treasures of geometry; such medieval thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Debussy, Le Corbusier, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist. More than 2,000 years ago, Euclid defined the number phi-1.6180339887-and linked it to the construction of the pentagram. Since then, this never-ending, never-repeating, irrational number has been called the Golden Number or the Golden Ratio and linked to patterns ranging from the petal arrangement of roses to the composition of the "Mona Lisa." Astrophysicist Livio takes an invigorating look at this ubiquitous number by examining the mathematical, aesthetic, and metaphysical qualities attributed to phi over the centuries. He also profiles the scientists and artists who have harnessed phi for a variety of purposes. He gives special attention to phi's appearance in natural patterns in the world. from Science News
Golden Ratio
A quick description of the golden ratio The golden ratio is often representedby Phi. Its approximate value it 1.61803 Links on the golden ratio
http://www.math.uiuc.edu/~gfrancis/math306/math306web/GoldenRatioPeteWintermute.

Extractions: What does it mean to be Golden Welcome to the wonderful world where everything is Golden!! presented to you by Pete Welcome! I am currently enrolled in a college course at the University of Illinois called The History of the Calculus and I thought for one of my course projects it would be very helpful for my fellow scholars to have a webpage that would help explain what it means to be "Golden" (that is in the mathematical sense). The topic of Golden is one of the more exciting topics in mathematics and one people should be familiar with. The goal of this annotated webpage is to list some very helpful websites that will give an overview of Golden with examples and some sites that contain some superb activities and lesson plans involving the concept of Golden for teachers. A quick description of the Golden Ratio: The Golden Ratio is often represented by Phi. Its approximate value it 1.61803... but more accurately is represented by (sqrt.of 5 + 1) / 2. As you notice Phi is an irrational number and has some very interesting properties and is often seen in the real world. To find out more about Phi please look at some of the intriguing webpages below. Links on the Golden Ratio: Fibonacci Numbers and the Golden Section - by Dr. Ron Knott. This is a very detailed webpage with a lot of information on the Golden section and Fibonacci Sequence but on numerous pages. Be ready to maneuver yourself through this page to find what you are looking for, but don't worry it is probably there. There is an overview on what Golden is; there are examples of the Golden section in nature, art, architecture, and music; Golden section occurring in Geometry and Trigonometry; constructions; you name it.
Golden Ratio Design
Local doctor designs medical database applications for the Palm, as well as providing other Web design Category Regional North America Dansville Business and Economy LMRP Tool. The golden ratio or Golden Mean is a number revered since antiquity thatappears with suprising frequency in natural designs and great works of art.
http://www.tonywitte.com/

Extractions: The Golden Ratio or Golden Mean is a number revered since antiquity that appears with suprising frequency in natural designs and great works of art. At Golden Ratio Design we believe well-designed computer applications and web tools are recognizable for their simplicity and elegance. We bring this concept to web-based and Palm OS applications for health care clinical desion making and productivity. The Palm OS is the dominant operating system for handheld computing, and for good reasons: simplicity, utility, dependability and a wide range of available software. As a physician dealing with volumes of critical information on a regular basis, the value of handheld computing has been very apparent to me. I have created a few applications for the Palm OS of use to the medical community, and am at work on other applications placing needed databases literally in the palm of the clinician's hand. LabCode NCD - Medical Necessity coding tool using latest Medicare medical necessity requirements included in the 23 lab test National Coverage Decisions. LabCode LMRP - Medical Necessity coding tool using latest local medical review policy medical necessity requirements.
IAHE - Shop
Flash Cards, Gift Certificate, golden ratio Products, Gymnastic Balls, Herbal Pillows,Journals, golden ratio Products, From 1 to 10 of 95, Next 10 ». Name, Price, Qty.
http://www.iahe.com/controller/IaheProductList?category=Golden Ratio Products
The Golden Ratio In Probability
The golden ratio has been lurking in Probability. Results reach a perfectbalance when phi, f, the golden ratio, is the natural fulcrum.
http://home.ozinet.aunz.com/~mervp/

Extractions: Not 0% either. What % then? What does nature say is the correct balance of success over failure? Not surprisingly, it is arithmetic and algebra and calculus that have been holding the answers to these questions all along. They just needed bringing out into the open for all to see. Results reach a perfect balance when phi, f , the Golden Ratio, is the natural fulcrum Mathematics predicts that favourites will win 38.2% of races, with an average of 38.2% of the people actually choosing that starter. In fact, the percentage of people who choose any starter is a very good indicator of its chances of success. This discovery is a major breakthrough in itself, but many things follow. You will find the Golden Ratio is also involved in election results. It is even at the footie, balancing how often the goal kicker will be successful. The theory was recently published. If you would like to know more, email
ARS Resources - Exclusive Educational Article
What is the golden ratio? Its a little known marketing technique thatsay that, as long as the longest side on any oblong is 1.6
http://www.arsresources.com/articles/golden-ratio.shtml
Golden Ratio
golden ratio. see also golden ratio. Borissavliévitch, Miloutine. The Golden Number.Colman, Samuel. $61.95. Gardner, Martin. ``Phi The golden ratio.''Ch.
http://www.ericweisstein.com/encyclopedias/books/GoldenRatio.html

Extractions: see also Golden Ratio The Golden Number. Colman, Samuel. Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form. Cook, Theodore Andrea. The Curves of Life, Being an Account of Spiral Formations and their Application to Growth in Nature, to Science, and to Art: With Special Reference to the Manuscripts of Leonardo da Vinci. New York: Dover, 1979. 479 p. $11.95. Coxeter, H.S.M. ``The Golden Section, Phyllotaxis.''Ch. 11 in Introduction to Geometry. New York: Wiley, 1989. $61.95. Gardner, Martin. ``Phi: The Golden Ratio.''Ch. 8 in The Second Scientific American Book of Mathematical Puzzles and Diversions: A New Selection. New York: Simon and Schuster, pp. 89-103, 1961. 253 p. 2nd Scientific American collection. $14.95. Ghyka, Matila Costiescu. The Geometry of Art and Life. New York: Dover, 1977. 174 p. $6.95. Herz-Fischler, R. A Mathematical History of the Golden Number. New York: Dover, 1998. 224 p. $14.95. Huntley, H.E. The Divine Proportion. New York: Dover, 1970. 186 p. $5.95. Pacioli, Luca.
Golden Ratio
golden ratio. golden ratio and Fibonacci numbers. Phi Page Golden Section Ratio. Constructionby golden ratio. Egypt Cheops' pyramid. Alta Vista golden ratio.
http://web.hep.uiuc.edu/home/karliner/golden.html
Nature's Golden Ratio, Alaska Science Forum
May 20, 1985. Nature's golden ratio Article 716. Crosssection of nautilus shellshowing the growth pattern of chambers governed by the golden ratio.
http://www.gi.alaska.edu/ScienceForum/ASF7/716.html

Extractions: Article #716 by Larry Gedney This article is provided as a public service by the Geophysical Institute, University of Alaska Fairbanks, in cooperation with the UAF research community. Larry Gedney is a seismologist at the Institute. Cross-section of nautilus shell showing the growth pattern of chambers governed by the golden ratio. What do the chambers of a nautilus shell have in common with the Parthenon and playing cards? It turns out that their forms are examples of a standard proportion. There is a fundamental ratio found over and over again in nature that seems to please human perceptions. Geometrically, it can be defined as the ratio obtained if a line is divided so that the length of the shorter segment is in the same proportion to that of the longer segment as the length of the longer segment is to the entire line. Mathematically, these ratios are such that the longer segment is 1.618054 times the length of the shorter segment, while the shorter is 0.618054 times the longer. These are remarkable numbers. Not only are the figures after the decimal point identical in both, but each is the reciprocal of the other (that is, the number 1 divided by either yields the other). These are the only two numbers that demonstrate this property. Unlike pi, another fundamental constant in which the decimals extend to infinity (3.14159. . .), these factors are exact after the first six decimals.
Nature's Golden Ratio, Part II, Alaska Science Forum
Alaska Science Forum. June 17, 1985. Nature's golden ratio, Part II Article 720. Theearlier column told only half the story of the golden ratio, however.
http://www.gi.alaska.edu/ScienceForum/ASF7/720.html

Extractions: Article #720 by Larry Gedney This article is provided as a public service by the Geophysical Institute, University of Alaska Fairbanks, in cooperation with the UAF research community. Larry Gedney is a seismologist at the Institute. Daisy head reveals two sets of opposing spirals formed by individual florets. The clockwise spiral contains 21 arms; the counter-clockwise spiral contains 34. These are two adjacent numbers in the Fibonacci series. Seldom has an article appearing in this space generated the volume of reader response as did last month's column on the Golden Ratio. The interest shown seems to justify a sequel. To recapitulate briefly, the Golden Ratio consists of the two numbers 1.618034 and 0.618034, each of which is the reciprocal of the other. Rectangles with sides proportioned 0.618034 to 1 (or 1 to 1.618034) are often the shape taken by such commonplace items as picture frames and playing cards. Thus, the shape seems to be subliminally pleasing to the human eye, as witnessed by the many ways in which it is used in art and in construction. It is also found in nature, reflected in essentially every spiral form from a snail shell to the arms of a galaxy. The earlier column told only half the story of the Golden Ratio, however. Historically, credit for recognition of the peculiar mathematical properties of this ratio must go to a 13th century Italian known as Fibonacci. The "Fi" part of his name meant "son of." The Bonacci part meant "simpleton."
Golden Ratio Prehistory
drawing of phi rectangle and its construction. A brief pre history of the goldenratio. .gif version = xxx Kb. view vector version xxx Kb. info on vector viewer.
http://www.recoveredscience.com/constgoldenprehistory.htm

Extractions: recoveredscience .com We offer surprises about and numerals and their ancient religious uses in our e-book Ancient Creation Stories told by the Numbers by H. Peter Aleff Site Contents NUMERALS Numerals Introduction Horus Eye Fractions Creation by numerals ... Reader responses Visit our other Sections: Prime Patterns Board Games Astronomy Medicine Store Stuff Home Page
Golden Ratio Properties
The mathematics of Genesis 1. in the layout of the Jerusalem Temple. You areon page. Some properties of the golden ratio phi. Goldmea2.gif (24451 bytes).
http://www.recoveredscience.com/const305goldenproperties.htm

Extractions: recoveredscience .com We offer surprises about and numerals and their ancient religious uses in our e-book Ancient Creation Stories told by the Numbers by H. Peter Aleff Site Contents NUMERALS Numerals Introduction Horus Eye Fractions Creation by numerals ... Number perceptions Golden ratio properties Golden ratio prehistory Woman Wisdom Constant e ... Reader responses Visit our other Sections: Prime Patterns Board Games Astronomy Medicine Store Stuff Home Page

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