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Golden Ratio: more books (52)

Numbers, Forbidden and Superstitious: An entry from Macmillan Reference USA's <i>Macmillan Reference USA Science Library: Mathematics</i> by Harry J. Kuhman, 2002
The Golden Mean or Ratio[(1+sqrt(5))/2] by Unknown, 2007-01-22
A Mathematical History of the Golden Number by Roger Herz-Fischler, 1998-01-29
How a Second Grader Beats Wall Street: Golden Rules Any Investor Can Learn by Allan S. Roth, 2009-03-09
Palynology and organic-carbon isotope ratios across a terrestrial Palaeocene/Eocene boundary section in the Williston Basin, North Dakota, USA [An article ... Palaeoclimatology, Palaeoecology] by G.J. Harrington, E.R. Clechenko, et all
Mastering The Void - Research Notes of a Heretic by Manuel Tanase, 2010-04-10
The Fibonacci Number Series by Michael Husted, 2009-07-31
On Cutting Off a Ratio by of Perga Apollonius, 1987-06
On cutting off a ratio: An attempt to recover the original argumentation through a critical translation of the two extant medieval Arabic manuscripts by Apollonius, 1988

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Photography Composition Articles: Golden Ratio, Rule Of Thirds, Golden Mean, Gol
What is the Composition, Golden Section, Golden Mean, Golden Spiral, golden ratio. How to use them in the photography. Rules of Composition.
http://photoinf.com/Golden_Mean/
E-Book On The Golden Ratio
Loosely related to the rule of thirds is the golden ratio also referenced The golden ratio has purportedly been a profound influence since ancient times
http://photoinf.com/Golden_Mean/Edwin_Leong/Camera_Hobby_-_e-Book_on_the_Golden_

Extractions: General Photography Resource, Articles and Reviews for the Amateur Photographer at Camera Hobby 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.
Contents
The Golden Section appears repeatedly in growth patterns in nature. Throughout history, The golden ratio has fascinated mathematicians and artists for
http://milan.milanovic.org/math/english/contents.html

Extractions: Lucas Numbers and the Pascal Triangle ... Blaise Pascal Blaise Pascal was born at Clermont on June,19,1623. In 1653 Pascal invented the arithmetical triangle.Pascal`s arithmetical triangle is the basic number formula in nature. As a mathematician Pascal is best known in connection with his correspondence with Fermat in 1654. in which he laid down the principles of the theory of probabilities. He died in Paris on August 19, 1662... Formulae for Fib(x-n) and Fib(x+2n) It is possible to derive operator`s equations for Fibonacci numbers . There are systems of equations where all the coefficients are the numbers of the Pascal Triangle ... Now available in PDF format for which you will need the ACROBAT READER .
Golden Section In Art And Architecture
once more began to construct their compositions on the golden ratio. The proportions of Michelangelo s David conform to the golden ratio from the
http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm

Extractions: Click on the various animation buttons to access an animation of related slides. The ancient Egyptians were the first to use mathematics in art. It seems almost certain that they ascribed magical properties to the golden section (golden ratio, divine proportion, phi) and used in the design of their great pyramids. pyramids, Giza If we take a cross section of the Great Pyramid , we get a right triangle, the so-called Egyptian Triangle. The ratio of the slant height of the pyramid (hypotenuse of the triangle) to the distance from ground center (half the base dimension) is 1.61804 ... which differs from phi by only one unit in the fifth decimal place. If we let the base dimension be 2 units, then the sides of the right triangle are in the proportion 1:sqrt(phi):phi and the pyramid has a height of sqrt(phi). Visit the web page Cairo and the Pyramids of Giza for more details.
Activities
The golden ratio in Everyday Objects. The golden ratio in Architecture. The golden ratio in Art. Constructing a Golden Rectangle (Method One)
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
ENC Online: ENC Features: Classroom Calendar: Golden Mean Day (Grades 7-12)
The Golden Mean, or golden ratio, is 1.6180339 rounded off to eight digits, or approximately These ratios steadily approach the Golden Mean as a limit.
http://www.enc.org/features/calendar/unit/0,1819,152,00.shtm

Extractions: Skip Navigation You Are Here ENC Home ENC Features Classroom Calendar Search the Site More Options Don't lose access to ENC's web site! Beginning in August, goENC.com will showcase the best of ENC Online combined with useful new tools to save you time. Take action todaypurchase a school subscription through goENC.com Classroom Calendar By Category By Month ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Find detailed information about thousands of materials for K-12 math and science. Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants. August 13 Graphic: Ji Sun An There's Pi Day and Mole Day, so it seems only right that the Golden Mean should have its own day. The Golden Mean, or Golden Ratio, is 1.6180339 rounded off to eight digits, or approximately 1.618. That makes August 13the day that is 61.8 percent of way through the yeara fitting day to proclaim Golden Mean Day. (Of course, if you prefer to celebrate during the traditional school year, you could have students figure out what 61.8 percent of your school year would be, or 61.8 percent of a specific month.) The Golden Mean represents an aesthetically pleasing proportion of height to width: 1 to 1.618 . . . . Like pi, it's an irrational number. It's often represented by the Greek letter Phi.
Golden Ratio
The ratio of lengths AE/AD is called the golden ratio. What is the golden ratio? Express it in two ways with radicals, and as a decimal approximation.
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)
Golden Rectangle In Graphic Design - Golden Ratio Phi And Other Mathematical Mea
Found in nature as well as the work of man, the golden rectangle or golden section is a visually pleasing geometric shape with specific proportions.
http://desktoppub.about.com/od/goldenrectangle/

Extractions: zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') About Desktop Publishing Graphic Design Principles of Design Golden Rectangle Desktop Publishing Essentials Choose Desktop Publishing Software Learn to Use Desktop Publishing Software ... Help zau(256,140,140,'el','http://z.about.com/0/ip/417/C.htm','');w(xb+xb+' ');zau(256,140,140,'von','http://z.about.com/0/ip/496/7.htm','');w(xb+xb); Sign Up Now for the Desktop Publishing newsletter! See Online Courses Search Desktop Publishing Found in nature as well as the work of man, the golden rectangle or golden section is a visually pleasing geometric shape with specific proportions. The measurement of 1.61803398874989..., known as the Golden Mean or Phi, a sequence of numbers known as the Fibonacci Series, and the Golden Rectangle are all mathematically connected. However, for the layperson the primary ratio of interest is 3:5 or 5:3 the Golden Proportion. Explore the mathematics and the art of the Golden Proportions and the Golden Rectangle, especially as it applies to paper sizes, page layout, and graphic design. Alphabetical Recent Up a category Design of Information Includes a discussion of the Golden Mean, the Golden Rectangle, and landscape vs. portrait orientation.
Nature's Golden Ratio, Alaska Science Forum
Article about the golden ratio in nature. They called it the golden ratio. Not understanding why, they knew that it felt good and it looked good,
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.
Investigating The Golden Rectangle And The Fibonacci Sequence
What does the Fibonacci sequence have to do with the golden ratio? 2) Define the golden ratio and the Fibonacci Sequence, including an approximation for
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?
Golden Ratio: Information From Answers.com
golden ratio This article is about the mathematical ratio. For the Aristotelian concept of golden mean see Nicomachean Ethics.
http://www.answers.com/topic/golden-ratio

Extractions: showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping golden ratio Wikipedia golden ratio The golden ratio is an irrational number , approximately 1.618, that possesses many interesting properties. Shapes defined by the golden ratio have long been considered aesthetically pleasing in Western cultures, reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios. The golden ratio is still used frequently in art and design. The golden ratio is also referred to as the golden mean golden section golden number or divine proportion The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry . The golden ratio may have been understood and used by the Egyptians. The discovery of irrational numbers, numbers that cannot be represented as an exact ratio of two
Fibonacci And The Golden Ratio
Discover how this amazing ratio, revealed in countless proportions throughout nature, applies to the financial markets.
http://www.investopedia.com/articles/technical/04/033104.asp

Extractions: There is a special ratio that can be used to describe the proportions of everything from nature's smallest building blocks, such as atoms, to the most advanced patterns in the universe, such as unimaginably large celestial bodies. Nature relies on this innate proportion to maintain balance, but the financial markets also seem to conform to this 'golden ratio'. Here we take a look at some technical analysis tools that have been developed to take advantage of it. Mathematicians, scientists, and naturalists have known this ratio for years. It's derived from something known as the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.). But this sequence is not all that important; rather, it is the quotient of the adjacent terms that possesses an amazing proportion, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. So, why is this number so important? Well, almost everything has dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function for the building blocks of nature.
*Golden Mean*
Now we have established the golden ratio where we see that line BC is to AB, Within this ratio of proportion called The Golden Mean I find the
http://www.asis.com/asis/golden.html

Extractions: We find in the ASIS logo, two methods of constructing the golden proportion using Euclidean tools. The logo represents a system or series of systems that ever resolve themselves in harmony and usefulness. The golden proportion was selected to be an integral part of the logo representing Access Solutions Information Services. The construction is a network of scribed arcs and straight lines that resolve themselves in useful relationship. We are able to arrive, through technological, conceptual, human based rationality at a relationship that nature exudes. Nature can't help herself, it's just how the flower grows. But man can stray and create abominations. So many of us live far removed from natural structure and as we venture into cyber space we are even further removed from order that can keep us sane. Our logo pays homage to nature's wisdom and serves as a reminder that with a little care, we can structure our world in a way that gives us beauty and peaceful resolve. Information requires perspective or context. The relevance of information is based on its usefulness.
BrainPhire?Fractal Golden Ratio Harmonics In BrainWaves As Mechanism Of Euphoria
And this confirms why so many peoples HEART s choose a GOLDEN MEAN Ratio (the original (Insert note here added by Dan Winter 1.618 = GOLDEN MEAN RATIO)
http://www.soulinvitation.com/brainphire/

Extractions: If the Source of Becoming Informed, and of Knowing - Lies in How Much CHARGE Information can Be Contained or Embedded in the Body.. And - Peak Awareness is Peak Charge- Then The Goal of EDUCATION Should be that Peak.. Consider - the Physics of Euphoria and Peak Awareness ( even BLISS) as the Ultimate Educator BrainPhire? Study in Russia Indicates Fractal Golden Ratio Harmonics in BrainWaves as Mechanism of Euphoria / Active Visualization / Bliss? May Support Neurofeedback Solutions to ATTENTION DISORDERS Based on IMPLOSION. notes from Dan Winter and James Barrett, Oct 17,2001, Index Study in Russia (Konstantin Korotkov, Gas Discharge Visualization Group) http://www.heartbeat2000.com/korotkov.htm , kindly reprinted by James Barrett (HeartLink Project), heartbeat2000.com , shows that Bliss / Euphoria - active visualization states, measureably correspond to Golden Mean Fractal Ratio between EEG contained harmonics see BLISS TUNER Above: Dr Korotkov- shows by GDV measurement that the altered state / bliss / euphoria which accompanies increased perception - has consistent qualities. The GDV area (aura size) should correlate to increased heart coherence. "I am reminded of the day Dr.Ed Wilson, Research Director Monroe Institute, decided to quit Bob Monroe and travel to my farm to make our film, when he discovered Fibonacci (Phi directed) audio heterodynes in ear headphones measureably induced (Lexicor's) 'definition' of transcendance in Brain Wave Mapping. He was so excited we had found parallels in the Heart EKG for compassion." Dan Winter
The Golden Ratio
The ratio of the sides of this two regular pentagons is the golden ratio . A rectangle whose length/width ratio is the golden ratio is a golden rectangle.
http://www.ac-noumea.nc/maths/amc/polyhedr/gold_.htm

Extractions: It is interesting to point out that every sequence defined as the Fibonacci sequence by f(n+1)=f(n)+f(n-1) leads to the golden ratio, no matter what the two initial values f(0) and f(1) are: f(n+1)/f(n) -> 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
The Geometry Junkyard: Pentagonal Geometry And The Golden Ratio
The golden ratio in an equilateral triangle. If one inscribes a circle in an ideal Includes sections on the golden ratio, conics, Moiré patterns,
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.
What Is Golden Section? : Atrise Image, Logo And Interface Design Software
golden Section is a proportion, in which a straight line (or a rectangle) that the ratio of the smaller to the greater part is the same as that of the
http://www.atrise.com/golden-section/golden-section.html

Extractions: Home Atrise Golden Section Golden Section is a proportion, in which a straight line (or a rectangle) is divided into two unequal parts in such a way, that the ratio of the smaller to the greater part is the same as that of the greater part to the whole figure (AB:BC=BC:AC) Similar to the mathematical value pi, Golden Section cannot be expressed as a finite number, but an approximation of 0.618 : 1. This proportion has been known since antiquity (Euclid and Vitruvius discussed it) and has been said to possess inherent aesthetic value because of an alleged correspondence with the laws of nature or the universe. The claims have been supported by an immense quantity of data, collected both from nature and arts. The ancient Egyptians were the first to use mathematics in art. It seems almost certain that they ascribed magical properties to the golden section and used it, when designing their great pyramids.
The Golden Mean
A detailed treatment of the golden Mean ratio . 1 to 1.618034
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.
Golden Mean Introduction & Resources
Fibonacci and the golden Mean Some very useable and easily understood explanations of when seen as a ratio, are very close to the golden Mean 89/144=
http://www.spyrock.com/nadafarm/html/gm.html

Extractions: I was entranced by the myth of the artist when I was a teenager. As I began to read about the history of western art I kept bumping into mysterious phrases like " Golden Mean ," " Golden Proportion ," " Golden Rectangle ," and " Golden Section ." I was certain that there must be some great mystery of the universe revealed in this mysterious sounding concept. Why else was there so much talk about it? But even though I heard it frequently mentioned or saw examples of its use in the compositions of the old masters, it was never clear to me just why it was such a big deal. It was not until many years later when I learned about Fibonacci Numbers and their relationship to the Golden Section that it started to come together for me. It was when I was well into the work on the triptychs at NFMOA that I realized how the Golden Proportion is within us and around us everywhere!

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