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1. 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

2. Golden Ratio Antennas
golden ratio ANTENNAS. In the Borderland Research. It provides considerabledetail on the development of the golden ratio Antenna stickers.

Extractions: GOLDEN RATIO ANTENNAS The following information is adapted from a Nov/Dec 1986 article in the Journal of Borderland Research. It provides considerable detail on the development of the Golden Ratio Antenna stickers. These antennas were designed by Eric Dollard as an improvement on concentric ring antennas for the Lakhovsky Multi Wave Oscillator (MWO). The general theory behind the MWO was to excite a concentric ring antenna which would produce a wide spectrum of radio frequency waves. In this theory it is understood that the RNA-DNA coil in the nucleus of every cell has a resonant frequency within this range. Just as a tuning fork will ring when an identical fork is struck in close proximity, the cells will resonate to their individual frequency pulled from the frequency ocean of the MWO. Eric's design of the Golden Ratio Antenna is based on the mathematics of life, which is a logarithmic function. This form of proportioning can be seen everywhere in nature. This is the sacred geometry used in the architecture of old, Greek and Roman temples, the Great Pyramid, etc. It has also been concluded by certain Orgonomists that 'the Golden Ratio is a basic mathematical property of the orgone energy.' (J. of Orgonomy, V.8, N.2, Rosenblum, The Golden Section)

3. THE GOLDEN RATIO AND FIBONACCI NUMBERS
THE golden ratio AND FIBONACCI NUMBERS by Richard A Dunlap (Dalhousie University,Canada) In this invaluable book, the basic mathematical properties of the
http://www.worldscientific.com/books/mathematics/3595.html

Extractions: In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relationship to the golden ratio are presented. These concepts are applied to algorithms for searching and function minimization. The Fibonacci sequence is viewed as a one-dimensional aperiodic, lattice and these ideas are extended to two- and three-dimensional Penrose tilings and the concept of incommensurate projections. The structural properties of aperiodic crystals and the growth of certain biological organisms are described in terms of Fibonacci sequences. Contents: Basic Properties of the Golden Ratio Geometric Problems in Two Dimensions Geometric Problems in Three Dimensions Fibonacci Numbers Lucas Numbers and Generalized Fibonacci Numbers Continued Fractions and Rational Approximants Generalized Fibonacci Representation Theorems Optimal Spacing and Search Algorithms Commensurate and Incommensurate Projections Penrose Tilings Quasicrystallography Biological Applications Construction of the Regular Pentagon The First 100 Fibonacci and Lucas Numbers

4. Golden Ratio (1985)
golden ratio (1985). This quilt is based on the mathematical constant knownas the golden ratio, also known as the Golden Mean or Golden Section.
http://www.frogsonice.com/quilts/golden-ratio/

Extractions: 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.

5. Web Site Unavailable
A biography of the number phi. The golden ratio The Story of Phi The World sMost Astonishing Number Mario Livio Broadway Books, 320 pp, \$24.95.
http://www.yalereviewofbooks.com/archive/winter03/review03.shtml

6. Investigating The Golden Rectangle And The Fibonacci Sequence
Introduction. The golden ratio is the ratio of the length to the width of whatis said to be one of the most aesthetically pleasing rectangular shapes.
http://www.scs.k12.tn.us/STT99_WQ/STT99/Cordova_HS/franklinp1/webquest_folder/Fi

Extractions: Introduction The Golden Ratio is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The Task This WebQuest is designed to lead you to connections between the Golden Ratio and the Fibonacci sequence through the use of algebraic and geometric concepts. You will be absolutely amazed at the number patterns that exist in real-world situations! You will also be asked to integrate Art,Biology, or Music into your final project: creating your own lesson plan. By the end of this WebQuest, you will know the answers to the following questions (Some of you will become experts on this topic!!): Who was Fibonacci?

7. 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.

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.

8. 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.

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."

9. The Golden Ratio And Fibonacci Modelling Applied To The S&P 500
The golden ratio and Fibonacci Modelling Applied to the S P 500. CAPTAINHOOK. Well,here we are again, about to look at some sticky numbers that apply to what
http://www.gold-eagle.com/editorials_03/captainhook070703.html

Extractions: CAPTAINHOOK In an effort to be expedient, I will not delve into the entire background base of knowledge one should possess about Fibonacci's theories and principles necessary for you to be able to fully enjoy the scope of this discussion. Instead, I would encourage you to refer to my previous paper that deals with this same exercise which pertains specifically to the precious metals complex, attached below, in order to familiarize yourself with the basics surrounding the principles we will employ in this treatise, as well. Although you have no reason to think the model constructs that were arrived at in my previous work regarding Fibonacci's principles applied to the precious metals markets could apply directly, and in proportion, to what could be perceived as unrelated markets, but I can assure you, they do. www.gold-eagle.com/editorials_03/captainhook041603.html The first task at hand is to formulate a basic construct for the model we will employ for this exercise. In doing so, we will establish a base from which to gauge our observations, in order to refine the model into a functional predictive tool. In the case of the relationship between the SPX and VIX, it makes a lot of sense to first look at averaged outcomes, or the 'mean' values of the ratio between the two, in order to help establish the trend, dimension range, and volatility characteristics, as it pertains to values and time. i.e. the 'harmonic signature'. Thus, the first element we will examine is the 'mean' in the intermediate trend sequences of the VIX / SPX ratio to see if there is an identifiable harmonic signature that we can formulate into progression and regression coefficient factors. (See Figure 1)

10. Golden Ratio Properties
27, pp. 189 to 217. The mathematics of Genesis 1. in the layout of the JerusalemTemple. 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

11. Golden Ratio Prehistory
Numerals and constants. tell the creations of numbers and world. Abrief prehistory of the golden ratio. Pentagrams before Pythagoras.
http://www.recoveredscience.com/const305goldenprehistory.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

12. Golden Ratio In Arts
golden ratio in arts. golden ratio is often used in different kind of arts.Even the architectures and music composers does use the golden ratio.
http://www.mikkeli.fi/opetus/myk/pv/comenius/kultainen.htm

Extractions: Golden ratio is often used in different kind of arts. Even the architectures and music composers does use the golden ratio. These following things can lead you to understand how the golden ratio is used in geometry and arts. In this picture I'll show you how to write a golden rectangle: Construct a square Then bisect the square Draw a line from one end of the bisecting line to one of the opposite corners. Extend the baseline of the square. Using the diagonal line as the radius, drop an arc from the corner of the square down to the baseline. Draw a line from the point of intersection of the arc and the baseline, perpendicular to the baseline. Extend the top edge of the square to meet this line and form a rectangle. This rectangle is referred to as the golden rectangle. This is structure is in Athen, Greece. As you might see the space between the columns form golden rectangle. This structure was made by a Greek sculptor Phidias. The golden spiral is one thing which is important to know. To begin constructing the first square, draw an arc from one corner of the rectangle down (or up) until it intersects with the adjacent side. Then draw a line perpendicular to the side that is being intersected, from the point of intersection to the opposite side.

13. The Golden Ratio
the golden ratio. A simple knot made The ratio of the sides of this tworegular pentagons is the golden ratio . A rectangle whose length
http://www.ac-noumea.nc/maths/amc/polyhedr/gold_.htm

14. Golden Ratio -- From MathWorld
The golden ratio in the Hierarchy of Time the interesting result that the quantum of action (h) is shown to be the productof two values which appear to be related to each other by the golden ratio. .
http://www.astro.virginia.edu/~eww6n/math/GoldenRatio.html

Extractions: The golden ratio, also known as the divine proportion, golden mean, or golden section, is 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 a Pisot-Vijayaraghavan constant . It also 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

15. Art History Glossary - G - Golden Ratio
golden ratio is a term (with an astounding number of aliases) used to describe aestheticallypleasing proportioning within a piece. golden ratio. Glossary.

16. 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

Extractions: 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

17. Slashdot | The Golden Ratio
The golden ratio. surprised and fascinated me. I thought it was goingto be solely about the golden ratio. Mario Livio does cover
http://books.slashdot.org/books/04/02/05/1836213.shtml

Extractions: raceBannon writes "The book surprised and fascinated me. I thought it was going to be solely about the Golden Ratio. Mario Livio does cover the topic but along the way he throws in some mathematical history and even touches on the idea that math may not be a universal concept spread across the galaxy." Read on for the rest of raceBannon's review. The Golden Ratio author Mario Livio pages publisher Broadway rating reviewer raceBannon ISBN summary Through telling the tale of the Golden Ratio, Livio explains how this simple ratio pops up in all kinds of physical phenomenon and debunks the idea that the ratio is present in many famous man-made structures and art work. Along the way, he provides historical tidbits regarding some of the well-known and not so well-known mathematicians throughout the ages and he tells the story of some of the more famous and not so famous mathematical advances. Finally, he discusses the possibility that mathematics may represent some kind of global truth that exists throughout the cosmos. I have to admit that it is a little spooky to me that this ratio, this irrational number (1.6180339887...), pops up in many varied natural phenomena from how sunflowers grow to the formation of spiral galaxies; not to mention that the Golden Ratio and the Fibonacci Series are related. It makes you want to think that there is a God with a plan.

18. 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

Extractions: 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

19. 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

20. 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

Extractions: 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. 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?_ 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 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.

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