81. Golden Section The golden Section is a special ratio that was derived during 5th century BC, also known as the golden Age of Greece. During this time, the Greeks were http://milan.milanovic.org/math/english/golden/golden.html

Fibonacci Numbers and the Pascal Triangle Search only in Mathematics Golden Section The Golden Proportion Contents "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 gold; the second we may name a precious jewel." Johannes Kepler [1571-1630] The Parthenon for these reasons, has become to be known as the "perfect building." These unusual mathematics properties are key reasons for the building’s aesthetic beauty and attention received for ages. The Golden Section relationship asserts a natural balance, a dynamic symmetry. Another interesting part of the Golden Mean is its presence in the proportions in nature, specifically the human body. For example, the length of the first finger joint to the length of the next two joints is equal to the length of the two joints to the length of the whole finger. Similarly, the length of the middle finger to the length of the palm is equal to the length of the palm to the length of the whole hand. Lastly, the length of the hand to the length of the forearm is equal to the length of the forearm to the length of the finger to the elbow. This amazing discovery is the main reason for the beauty of the human figure and the patterns commonly found in nature, art, and architecture. The Golden Proportion Golden Section and the human body The Golden Section in Ancient Egypt The Golden Section in Greek art ... Radoslav Jovanovic translated: D.Filipovic created: January 2003.

82. Spirals And Snails The golden mean possesses the following unique and exceptional property. for which the ratio of the sides a/b is equal to the golden mean (fig1) http://www.popmath.org.uk/rpamaths/rpampages/snail.html

Spirals and Snails Nautilus Shell The Golden Mean has value: The golden mean possesses the following unique and exceptional property. If one constructs what is called a golden rectangle, that is to say, a rectangle for which the ratio of the sides a/b is equal to the golden mean (fig1) and if a square is removed from it, one obtains again a golden rectangle (the ratio c/d is still equal to the golden mean), as in fig 2. In other words, this new rectangle has the same proportions as the initial rectangle. It is important to note that this characteristic is only true for the golden rectangle. For example, if a square is removed from the arbitrary rectangle of fig 3, one does not at all obtain a new rectangle having the same proportions as the initial one. Let's return to the golden rectangle and to the passage from fig 1 to fig 2. Since the process is true for all golden rectangles, then one can repeat it and remove a new square from rectangle "cd": one obtains once more a smaller golden rectangle (fig 4). This process can be repeated to infinity to obtain in this way a series of nested rectangles, one inside the other in the form of a spiral (fig 5).

83. Photography Techniques The golden Mean was a major guideline for many artists/painters so it is Now the square can be transformed to a rectangle with a proportion ratio of 58 http://www.photozone.de/4Technique/compose/goldenmean.htm

YOUR PHOTO TECHNIQUE TUTORIAL BASIC HANDCRAFT Light Metering Exposure Compensation BASIC TECHNIQUES The Golden Mean Rule of the Thirds Framing Cross Lines PERSPECTIVE Depth-of-Field Object Isolation by DOF Object Isolation by Contrast Layout: Compression ... Layout: Layering USING LIGHT Spot Light Hi-Key / Low-Key Backlit Monochrome Light ... Colors SPECIAL EFFECTS Skyline Long-time Exposures Panning Dramatic Perspectives ... Reflections THE GOLDEN MEAN Let's start with an introduction of a technique that is well known for many centuries now: The "Golden Mean" (sometimes called "Golden Section") is a geometric formula by the ancient Greeks. A composition following this rule is thought to be "harmonious". The principal idea behind it is to provide geometric lines which can be traversed when viewing a composition. The Golden Mean was a major guideline for many artists/painters so it is certainly worth to have in mind for modern day photographers as well. Theory - Part I Well, let's begin with some words about the theory. The formula starts with a perfect square (marked blue in illustration A). Now we devide the base of the square into two equal parts. We take point x as the middle of a circle with a radius of the distance between point x and y. Thereafter we expand the base of the square till it hits the circle at point z. Now the square can be transformed to a rectangle with a proportion ratio of 5:8. The ratio of A to C is the same as the one from A to B. Luckily the 5:8 ration fits pretty close to the ratio of the 35mm format (24x36mm = 5:7.5).

84. FUSION Anomaly. Golden Mean The golden mean is a ratio, discovered by the Greeks, that is selfmirroring. Also known as the golden Mean, the internal link Magic ratio, http://fusionanomaly.net/goldenmean.html

Telex External Link Internal Link Inventory Cache Golden Mean This nOde last updated January 20th, 2004 and is permanently morphing... (9 Ik (Wind) / 10 (Muan ( Owl Phi Divine Proportion The Golden mean is a ratio, discovered by the Greeks, that is self-mirroring. It is approximately .6 to 1. If you divide a straight line so that about 61% of it is one one side and 39% on the other, you will find that the ration of the large portion to the small is the same as the ratio of the overall line to the large portion. Rectangles made with these proportions can be subdivided endlessly. This self-mirroring proportion was essential to the art and architecture of the Greeks; it is very pleasing to the mind's eye and was used extensively, including in the design of the Parthenon. In the 15th century an Italian named Fibonacci discovered that if you add 1 to itself, then 2 to 1, then the sum 3 to 2, and the sum 5 to 3, etc., you end up with a series of numbers 1,1,2,3,5,8,13, etc. The ratios of these, one to another, dance around and approach more and more closely the golden mean of .6 to 1. These ratios describe the most efficient way of packaging spirals about themselves IN TWO DIMENSIONS; you will see them in the center of a sunflower. If you count the spirals going one way and they add up to13, there will be either 8 or 21 spirals going the other way. Moving your perspective in or out to about 2/3rds of the original size will move you to the next level of spirals.

85. Emergent Behavior In Gambling Markets Reveals Golden Section Scaling For centuries a special ratio called the golden Section has captivated artists, The golden section ratio that is used to model the amount of money http://www.acadjournal.com/2002/v7/part3/p1/

86. Golden Mean The golden Mean can be construed as the basis of philosophy and Sacred Geometry, These artists and architects discovered that by utilizing the ratio 1 http://www.halexandria.org/dward101.htm

Golden Mean The Golden Mean can be construed as the basis of philosophy and Sacred Geometry , one of the Transcendental Numbers , [*] and is typically derived from Fibonacci Numbers [* Technically, from a strictly mathematical standpoint as has been pointed out by several readers the Golden Mean, Phi, is the solution to a polynomial equation and thus not mathematically transcendental. However, if one looks at the other definitions of transcendental, most would agree that it qualifies.] According to Robert Lawlor [1], “Ancient geometry rests on no a priori axioms or assumptions. Unlike Euclidian and the more recent geometries, the starting point of ancient geometric thought is not a network of intellectual definitions or abstractions, but instead a meditation upon a metaphysical Unity, followed by an attempt to symbolize visually and to contemplate the pure, formal order which springs forth from this incomprehensible Oneness. It is the approach to the starting point of the geometric activity which radically separates what we may call the sacred from the mundane or secular geometries. Ancient geometry begins with One , while modern mathematics and geometry being with Zero Other authors have noted that “Both the ancient Greeks and the ancient Egyptians used the Golden Mean when designing their buildings and monuments.” “Artists as diverse as Leonardo da Vinci and George Seurat used the ratio when constructing their paintings. These artists and architects discovered that by utilizing the ratio 1 : 1.618..., they could create a feeling of order in their works.

87. Golden Section - Pentagon And Pentagram - Dodecahedron This division of the line segment is called the golden section. The ratio AB/AP is called the golden number and often represented by the greek character http://cage.rug.ac.be/~hs/polyhedra/dodeca.html

The Golden Section The Golden Triangle The Regular Pentagon and the Pentagram The Dodecahedron On this page we will derive some relations between a fundamental number and two famous geometrical figures, the one planar, the other spatial. As a result we will understand how a regular dodecahedron can be constructed starting from a cube by putting suitable "roofs" on it, as Euclid already did... The Golden Section The starting point is the division of a line segment in mean and extreme ratio A point on a line segment divides this segment in mean and extreme ratio if the ratio of the whole segment to one part is the same as the ratio of that part to the remaining part of the line segment. Given a line segment AB, the point P divides AB in mean and extreme ratio if AB/AP = AP/PB This division of the line segment is called the golden section . The ratio AB/AP is called the golden number and often represented by the greek character capital PHI. A special isosceles triangle or "golden triangle" We consider an isosceles triangle with a top angle measuring 36 degrees. Both base angles then measure 72 degrees. BD is the bisector of the angle in B. From the fact that the triangles ABD and BDC are isosceles triangles it follows that BC=BD=AD.

88. Did Mozart Use The Golden Section? The ratio of the two divided parts is the golden Section, which equals 0.618. The ratio 0.5 seems neater! If Mozart used the golden Section http://www.science-frontiers.com/sf107/sf107p14.htm

Science Frontiers ONLINE No. 107: Sep-Oct 1996 Issue Contents Other pages Home Page Science Frontiers Online All Issues This Issue Sourcebook Project Sourcebook Subjects Did Mozart Use The Golden Section? Does the brilliance of Mozart's compositions derive entirely from his musical genius, or did he rely in part upon a mathematical construct: the famous Golden Section or Golden Ratio? The Golden Section is a mathematical formula for dividing into two parts: a geometrical line, a musical composition, or anything else possessing the property of length. The ratio of the two divided parts is the Golden Section, which equals 0.618.* For some artists, musicians, architects, the Golden Section is the most esthetic way of dividing the length of anything For humans, the history of the Golden Section goes back at least as far as Euclid in 300 BC. For nature, it began eons ago: The shapes of pine cones, starfish geometry, and other dimensions of living things incorporate the Golden Section. The questions we address here are: (1) Did Mozart consciously make use of this ratio, 0.618, in his music? (2) Why is the Golden Section esthetically pleasing? It is not well known that Mozart was fascinated by mathematics as well as music. He even jotted down equations in the margins of some of his compositions. Chances are excellent that he knew of the Golden Section and its reputation for conferring elegance on structures even musical compositions.

89. Bonsai And The Golden Section The naturally occurring golden Section proportion/ratio reoccurs throughout The golden Section ratio also shows up in the socalled Fibonacci Series. http://www.bonsai4me.com/AdvTech/ATGolden Section.html

BONSAI BASICS ADVANCED TECHNIQUES SPECIES GUIDES ARTICLES ... HOME APPLYING THE PRINCIPLES OF ART TO BONSAI DESIGN There has been much debate in recent years as to the relationship between Bonsai and Art. Is the practice of Bonsai purely horticultural or does it have strong artistic aspects as well? Many enthusiasts initially approach bonsai for it's horticultural practices and it is only when they then attempt to style and form their trees, that the artistic and creative side of the practice comes into play. There are in fact many basic artistic principles at play when we design our trees; or, at least there should be. Many of these principles are already provided for us in the commonly applied 'Rules of Bonsai'. This article deals with one of the most important and basic principles known as 'The Golden Section'. THE GOLDEN SECTION The Golden Section is a Law of Proportionality. It is a Law that occurs frequently in nature and it's use is particularly useful in Art. First developed by Vitruvius, it is most famously known from Leonardo Da Vinci's 1509 drawing 'The Divine Proportion' (see above).

90. Fibonacci Numbers, The Golden Section And The Golden String Fibonacci numbers and the golden section in nature, art, geometry, architecture, music and even for calculating pi! Puzzles and investigations. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci Numbers and the Golden Section This is the Home page for Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, the Golden section and the Golden string. The Fibonacci numbers are add the last two to get the next The golden section numbers are The golden string is a sequence of 0s and 1s which is closely related to the Fibonacci numbers and the golden section. There is a large amount of information at this site (more than 200 pages if it was printed), so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature. The rest of this page is a brief introduction to all the web pages at this site on Fibonacci Numbers the Golden Section and the Golden String together with their many applications What's New? - the FIBLOG 16 August 2005 Fibonacci Numbers and Golden sections in Nature Fibonacci Numbers and Nature Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements. All involve the Fibonacci numbers - and here's how and why. The Golden section in Nature Continuing the theme of the first page but with specific reference to

91. Fibonacci Numbers And The Golden Section In Art, Architecture And Music The golden section and Fibonacci numbers in art, architecture, poetry and music; for schools and teachers or just for recreation! http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html

Fibonacci Numbers and The Golden Section in Art, Architecture and Music This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio in architecture, art and music. Contents of this page The icon means there is a Things to do investigation at the end of the section. The golden section in architecture The Parthenon and Greek Architecture Modern Architecture The Eden Project's new Education Building ... Links to Art sources including Contemporary Artists Fibonacci and Phi in fashioning Furniture Fibonacci in Films Fibonacci and Poetry ... More The Golden section in architecture The Parthenon and Greek Architecture The ancient Greeks knew of a rectangle whose sides are in the golden proportion (1 : 1.618 which is the same as 0.618 : 1). It occurs naturally in some of the proportions of the Five Plato nic Solids (as we have already seen) . A construction for the golden section point is found in Euclid 's Elements. The golden rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in

92. Golden Rectangle -- From MathWorld Defines the golden Rectangle based on phi and shows some formulars and drawings of the results. http://mathworld.wolfram.com/GoldenRectangle.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Constants Golden Ratio ... Rectangles Golden Rectangle Given a rectangle having sides in the ratio , the golden 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. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length and construct with this length. Now complete the rectangle , which is golden since Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1986, p. 39; Livio 2002, p. 119). The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.

93. Graphic Product Design Software: Tweak Your UI And Graphic Design - Atrise Golde A transparent screen grid for computerbased design, it allows to control and correct sizes and proportions in design projects using the golden section http://www.atrise.com/golden-section/

Search Entire Web Site Atrise Everyfind Atrise HTMLock Atrise Golden Section Other Advanced Search Powered by Atrise Everyfind AGS Menu Overview What is Golden Section? Screenshots FAQ ... User's Manual Quick Info Version: Released: Jul 13, 2005 Price: OS: Windows 2000/XP/2003 Our awards Home Atrise Golden Section Overview The program is an adjustable graphic design screen grid for the web, graphic, logo and user interface (UI) design. It allows you to easy control and correct the sizes and proportions in your design product using the golden section proportion. Golden Section Proportion and our Design Software From the earliest times, human cultures have striven to define beauty. A study of nature, architecture, and art reveals that a common principle - that of pleasant proportion – is an universal characteristic that contributes to defining beauty. The proportion that is pleasing to the eye is the Golden Section, or simply the ratio of one relationship to another, or 1 to 0.618. Now you can use this same ratio in your design projects. Atrise Golden Section allows you to quickly and easily check the sizes and proportions in your on-screen design projects by using the ratio of the Golden Section. It generates a resizable grid that overlays your work and displays the Golden Section ratio. Beauty and proportion can now be discovered as easily as moving your mouse.

94. Cynthia Lanius' Lesson: Make An Approximation To The Golden Rectangle Mathematics lessons for elementary, middle, and high school including geometry. http://math.rice.edu/~lanius/Geom/building.html

Cynthia Lanius Building a Golden Rectangle Golden Ratio 1.61803 Golden Ratio Table of Contents Introduction Find golden rectangles. Build golden rectangles. Confirm the ratio using algebra. Let's build an approximation of the golden rectangle using square grid paper . At each step we will calculate the ratio of length to width. Remember the Golden Ratio is approximately 1.61803. Step 1: Start with a square 1 by 1. Ratio of Sides Step 2: Add another 1 by 1 square. Ratio of Sides Step 3: Add a 2 X 2 square. Ratio of Sides Step 4: Add a 3 X 3 square. Ratio of Sides Step 5: Add a 5 X 5 square. Ratio of Sides Step 6: It's Your Turn! Build the next approximation on square grid paper. Then observe the pattern and answer the following: Length n Answer width of n+1 Length n + width n Answer Length of n+1 The lengths of the various rectangles are 1, 2, 3, 5,..., Notice that each term is found by adding the 2 previous ones, the sequence known as the Answer Fibonacci Sequence Every step of the rectangle will produce a ratio closer and closer to the golden ratio of approximately Answer Golden Ratio Table of Contents Introduction Find golden rectangles.

95. The 'Phinest' Source To The Golden Section, Golden Mean, Divine Proportion, Fibo Information on the golden Section, Divine Proportion, Fibonacci series and phi. http://goldennumber.net/

GoldenNumber.net Home Contact Meet the Phi Guy "Phriends" in Phi Site Search ... News Dedicated to providing you with the phinest information on The Golden Section , Ratio or Mean The Divine Proportion The Fibonacci Series and the golden number, Phi ( NEW PhiMatrix Graphic Design Software Free Download Site Sections: Overview Design/Composition Life Mathematics ... Other Resources Welcome to GoldenNumber.Net - The Phi Source GoldenNumber.Net exists to share information on the pervasive appearance of Phi in life and the universe. Its goal is to present a broad sampling of phi related topics in an engaging and easy-to-understand format and to provide an online community (aka The Phi Nest), in which new findings about Phi can be shared. Select an area of interest below or START HERE . Enjoy the 'phi nomenon'! SITE CONTENTS Overview Design/Composition Life Mathematics ... Universe News: The Da Vinci Code Dan Brown's book "

96. Golden Section The Divine Proportion, or golden Section, has many unique, intriguing relationships and properties found in nature, art, music and mathematics. http://goldennumber.net/goldsect.htm

GoldenNumber.net Home Contact Meet the Phi Guy "Phriends" in Phi ... News Phi The Golden Number Golden Section SITE SECTIONS: Home Overview Design/Composition Life ... Other Resources IN THIS SECTION: Fibonacci Series Golden Section The Symbol for Phi History Phi to 20,000 places Pronouncing Phi ... Phi Phonetics Jump to any topic Home Phi Overview Acoustics Architecture Art Bible Bucky Balls Color Compositions Cosmology Credit Cards Divine Proportion Energy Fibonacci Series Fibo's 24 Pattern Five (5) and Phi Geometry Golden Section/ History Life Human Hand Human Face Human Body Human Beauty Development Human Heartbeat Human Health Body Temperatures Animals 1 Animals 2 Plants DNA Mathematics Marketing/Design Means Music Orthogons Pascal's Triangle Penrose Tiling Phi Phonetics Phi's Phormula Phi Symbol Phi to 20000 places Poetry Population Growth Powers of Phi Pronouncing Phi Quasi-crystals Quantum Time Quincy Park Solar System Spirals Stock Markets Theology Universe Affliates Do It Yourself!!! Feedback Golden Mean Gauges Use of Images Links to other sites Meet the Phi Guy News 'Phriends' in Phi Phi Logo Merchandise School Projects Search the Site Translate this site WTC Proposal The Golden Section The Golden Section is a ratio based on a phi The Golden Section is also known as the Golden Mean, Golden Ratio and Divine Proportion. It is a ratio or proportion defined by the number

97. The Golden Section Short page describing the golden Section. Includes a description of various constructions and history. http://www.perseus.tufts.edu/GreekScience/Students/Tim/Golden.html

Please note: These papers were prepared for the Greek Science course taught at Tufts University by Prof. Gregory Crane in the spring of 1995. The Perseus Project does not and has not edited these student papers. We assume no responsibility over the content of these papers: we present them as is as a part of the course, not as documents in the Perseus Digital Library . We do not have contact information for the authors. Please keep that in mind while reading these papers. The Golden Section If you don't think math is fun, be gone. For everybody else on the web who enjoys math, here is a beautiful and interesting element of geometry tracing as far back as the Greeks, and maybe even further. Note: I'm only human, and a student at that, so please look kindly upon my humble attempt to verbalize beauty that was never meant for words. WARNING: BORING MATH AHEAD! ONLY THE STOUT OF HEART DARE BRAVE THIS WORLD. What is The Golden Section ? The golden section is a line segment divided into two parts. Point C is positioned such that the ratio of the short half to the long half is equal to the ratio of the long half to the whole. Symbolically: AC-B where CB / AB = AC / AB, or AB^2 = BC x AC

98. The Golden Section A description of the golden Section and its importance in architecture. Includes instruction on how to construct a golden Section starting with a square. http://www.ewersarchitecture.com/golden_section.htm

HOME RESIDENTIAL PROJECTS COMMERCIAL PROJECTS ABOUT US ... Recycling Information The Golden Section T H E G O L D E N S E C T I O N The Ewers Architecture logo is the arc that transforms a square into a Golden Section. The Golden Section has been used extensively by architects, musicians, artists, and mathematicians over the centuries. Ewers Architecture uses the arc to remind us that a simple form such as this can have powerful implications and symbolism. The Golden Section can be constructed from a simple square, as shown here. Note that the arc shown in Step Two is the Ewers Architecture logo. STEP ONE Construct a simple square. STEP TWO Using a line from the midpoint of one side of the square to an opposite corner as the radius, construct an arc that will define the extents of the new rectangle. STEP THREE Using the endpoints of the arc, complete the rectangle. STEP FOUR This is the proportion recognized since antiquity as the Golden Section. A ( v e r y ) S H O R T H I S T O R Y The Golden Section has been referred to as the Divine Proportion, the Golden Rectangle, or the Fibonacci Sequence (after Leonardo Fibonacci of Pisa who pioneered some of the early mathematical phenomena and its connection with nature). This proportion has been demonstrated in the Greek architecture of the Parthenon, the Renaissance architecture of Leon Battista Alberti’s Santa Maria Novella in Florence, and used extensively by Le Corbusier in his quest for modular designs of modern architecture.

99. Museum Of Harmony And Golden Section Everything about Harmony, golden Section, Fibonacci Numbers, Fibonacci computers and latest investigations and developments in this area of science. http://www.goldenmuseum.com/

Welcome to the Museum of Harmony and Golden Section English Russian Our Museum is recognized the best site by the Russian Informational Network "The Best in Internet" All questions and requests please send: Designer

100. Museum Of Harmony And Golden Section http://www.goldenmuseum.com/index_engl.html

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