Geometry.Net - Math_Discover: Golden Ratio

NPR : 'The Golden Ratio'
But in his new book i The golden ratio /i , author Mario Livio examines the mysteriesof pi's lesserknown cousin, phi a number that has both counfounded
http://www.npr.org/display_pages/features/feature_1191723.html

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Golden Ratio
is a golden triangle; it is an isosceles triangle with the ratio of the longer side(1 + Ö5 units) to the shorter side (2 units) equalling the golden ratio.
http://www.maths.uq.edu.au/~infinity/Infinity_13/Polygons.html
Vision Engineer - Golden Ratio
golden ratio Article by Duane Bong The golden ratio is some timesreferred to as the Golden Mean. Its value is approximately 1.618
http://www.visionengineer.com/ref/golden_ratio.shtml
Fibonacci Numbers And The Golden Ratio
Fibonacci numbers and the golden ratio. They called any such rectanglea Golden Rectangle, and so became known as the golden ratio.
http://www.math.umass.edu/~dhayes/math597a/ugcd2/node3.html

Extractions: Next: The Euclidean algorithm stops Up: Analyzing the efficiency of Previous: The worse case: Fibonacci The ancient Greek geometers believed that a rectangle with sides in the ratio was the most pleasing to the eye. They called any such rectangle a Golden Rectangle , and so became known as the Golden Ratio . Naturally, the Greeks defined the Golden Ratio geometrically instead of writing it as a number. Definition 792 A Golden Rectangle is a rectangle R having the following property. If a square on one of the shorter sides of R is removed from R , then the remaining rectangle is similar to R In other words, if a square on one side of a Golden Rectangle is removed from the rectangle, the rectangle that remains is also golden! If u and u v are, respectively, the short and the long side of the Golden Rectangle above, then clearly which shows that or . Solving, we find , demonstrating that the numerical and geometric definitions are the same. The default window in which 2-dimensional graphics are drawn in the Mathematica software package is a Golden Rectangle.
ThinkQuest Internet Challenge Library : Golden Ratio
ThinkQuest Internet Challenge Library. Index Math \ Geometry \ golden ratio= Finalist Index Math \ Geometry \ golden ratio Search The Library.
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GOLDEN RATIO--TOP--
back top. SCIENCE PUZZLE golden ratio. LEVEL1 LEVEL2 LEVEL3 CHALLENGE.
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Golden Ratio
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Construction By Golden Ratio
which is 51deg50'. With such a base angle, the ratio of the hypotenuseto half the base is phi, the golden ratio. The design was
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Extractions: The base angle of the largest trinalges of most representations of Sri Yantra are about 52 degrees, close to the base angle of the Great Pyrmaid of Cheops , which is 51deg50'. With such a base angle, the ratio of the hypotenuse to half the base is phi, the Golden Ratio The design was investigated with the largest two triangles having this base angle. The construction continues similar to the 7x7 grid method. The result is striking, for apart from the fact that the error (at Z and W) is not noticeable, the overall shape is very simlar to examples from the literature . The error is only .3616% of length of the diameter. A small change in base angle results in large error - a 53 degree angle has an error of 9.125%! 1. Draw a rectangle of 2 x 1 units (ABCD) for construction of the large pyramid- angled triangles on base DC. The centre of DC is E. 2. Draw the diaganol DA. It's length is sqr-root 5, by Pythagoras. 3. Draw a circle of radius one centered at point A. Extend the diagonal. 4. Bisect the extended diaganol, creating two line segments of length sqr-root 5 plus 1, divided by 2. This length is phi!
Golden Ratio
golden ratio in 3, 6-fold Geometry. I thought this. To show this isexactly the golden ratio, consider the following Figure. The
http://www.rwgrayprojects.com/Universe/phi/phi01.html

Extractions: Golden Ratio in 3-, 6-fold Geometry I thought that the Golden Ratio only showed up in Geometry once you introduce the 5-fold symmetry of the pentagon or pentagram (in 2-dimensions) or when you introduce the Icosahedron or the regular Dodecahedron in 3-dimensional geometry. But a friend of mine e-mailed to me the following link which shows the Golden Ratio occurs when a triangle is surrounded by a circle: http://www.intent.com/sg/inscribedtrianglephi.html Here I introduce my own description and Figures to explain this. To show this is exactly the Golden Ratio, consider the following Figure. The point "O" is the center of the circle and the triangles. All lengths of the little equilateral triangle (A-C-E) are 1/2 of the corresponding lengths of the big equilateral triangle. Let the segment AB have a length equal to 2. Then DE = 1. AB = OB cos(30) = OB sqrt(3)/2 Which gives us AB = 2 = OB sqrt(3)/2. Which means OB = 4/sqrt(3). Note that OF is the same length (radius of the circle) as OB. We also have OA = OB sin(30) = OB/2 = 2/sqrt(3) Now, OD = OA/2 = 1/sqrt(3).
ENC: Curriculum Resources: The Golden Ratio (ENC-016300, Full Record)
The golden ratio. The use of the golden ratio is shown in works of art and in thearchitecture of structures, including the Parthenon and the Pyramids of Giza.
http://www.enc.org/resources/records/full/0,1240,016300,00.shtm

Extractions: Search ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Search Browse About Curriculum Resources 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. URL: http://www.geom.umn.edu/~demo5337/s97b/ This World Wide Web (WWW) site, developed by students as a class project, offers a series of activities designed to lead users to an understanding of how the Golden Ratio and the Fibonacci Sequence are applied to real world situations. The use of the Golden Ratio is shown in works of art and in the architecture of structures, including the Parthenon and the Pyramids of Giza. The mathematical relationships are developed in an interactive format through problems, including a calculator based activity, that allow users to explore the number relationships found in the Golden Ratio and in the Fibonacci Sequence. All answers are included. Also found are links to related mathematical topics. Abstracted 01/00. (Author/JRS) Reviews and Awards: ENC Digital Dozen, December 1999. (
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Camera Hobby - EBook On The Golden Ratio, Chapter 16
Comments on Photography Chapter 16 The golden ratio Loosely related to the ruleof thirds is the golden ratio also referenced to the Golden Rectangle.
http://www.camerahobby.com/E-Book/Ebook-GoldenRatio_Chapter16.htm

Extractions: 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 shape of the Golden Ratio in physical form is supposed to be considered very esthetically pleasing to humans. One of the most interesting graphic forms of the Golden Ratio is the nautilus shell and the way the shell starts from a central origin and then spirals outward around itself until it reaches the horn or opening of the shell where the cephalopod's head is located. The Bowers and Worthington speaker company of Britain, better known as B+W, designed a reference speaker with the nautilus design and called itthe Nautilus. A most intriguing and apparently an excellent sounding speaker.
Aethmogen - November 2001 Golden Ratio Motor Generator
Articles golden ratio Nov 2001. THE golden ratio MOTOR GENERATOR .Dr. Robert Adams October 19 2001. Click for image (size 780k).
http://www.aethmogen.com/wri/radams/200111goldenratio.shtml

Extractions: Click for image (size 780k) A young enthusiastic colleague of mine, currently resident in Auckland, New Zealand, by the name of Andrew Thorpe, phoned me today to advise that David Saltresse of Wales, Great Britain, had contacted him via email to advise him he had worked out the geometric placement of all 4 of the output A.C. generator windings of one of my motor generators which was originally invented in the early seventies. He advised Andrew that the geometric placement of the output generator stator windings was measured with the aid of incorporating the Golden Ratio . "All 4" David said "are found to confirm they correspond directly with the golden ratio geometrics". There was always the remote possibility that out of the many hundreds of people worldwide who would have attempted to replicate my original machines, few, if any , would have taken the advantage of probing deep into the detailed schematic diagram of this particular motor generator which I included in my original motor manual, published by Nexus Magazine, possibly because that schematic diagram appears at first glance unorthodox or a difficult exercise to achieve! As David Saltresse discovered, this is not so, if you look with eyes to see at this schematic diagram. Congratulations go to David Saltresse of Wales.
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.bottomlayer.com/bottom/deutsch/golden.html

Extractions: II. Calculations Based on Standard Values, and the Appearance of the Golden Ratio in the Quantum of Action Part I consists of an investigation into the relationships among basic phenomena at an arbitrary microscopic "natural scale." At this hypothetical natural scale, each can be set to a unitary value of 1 (or -1). In other words, I assume for the purposes of this investigation that each of the phenomena can be "quantized," at least in the sense of having a value to which they will naturally gravitate (using the word in its generic sense) during interactions, regardless of whether there are smaller or larger fractional values that can be imagined, or which may be mathematically necessary for calculating events such as virtual interactions. (In the same way, for example, the speed of light may be considered as invariant regardless of the fact that photons may have amplitudes for travelling at greater or lesser speeds in the course of Q.E.D. interactions.) By manipulating standard (macroscopic) formulas for the interrelationships among these phenomena, it is shown that certain equivalencies appear among them, such that power, energy, action and mass may be reduced to functions of time.
Golden Ratio
golden ratio. The ration ACCB is called the golden ratio. In the terminologyof the early mathematicians AB is divided by C in extreme and mean ratio.
http://www.cis.uab.edu/info/grads/hux/golden.html

Extractions: Golden ratio 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 KEPLER[1571-1630 Let a line AB be divided into two segments by the point C. If C is a point such that AB:AC as AC:CB, C is the golden cut or the golden section of AB. The ration AC:CB is called the golden ratio. In the terminology of the early mathematicians AB is divided by C in extreme and mean ratio. Kepler called it the divine proportion Xiao Hu's home page
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The Golden Mean
The golden Mean is a ratio that is present in the growth patterns of many thingsthespiral formed by a shell or the curve of a fern, for example.
http://galaxy.cau.edu/tsmith/KW/golden.html
Golden-ratio.co.uk
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American Scientist: Science Observer
0.618. That value has been given many names, including the goldenratio, the golden number and even the divine proportion. In the
http://www.americanscientist.org/issues/Sciobs96/Sciobs96-03MM.html

Extractions: March-April 1996 Did Mozart Use the Golden Section? Mike May Considerable evidence suggests that Mozart dabbled in mathematics. According to his sister, Wolfgang "talked of nothing, thought of nothing but figures" during his school days. Moreover, he jotted mathematical equations in the margins of some of his compositions, including Fantasia and Fugue in C Major , where he calculated his odds of winning a lottery. Although these equations did not relate to his music, they do suggest an attraction to mathematics. The structure of Mozart's music attracted the attention of John F. Putz, a mathematician at Alma College. "My sonwho is a composer and pianisttold me that Mozart's piano sonatas are divided into two distinct sections," Putz recalls. "I knew that Mozart's music is highly regarded for its elegant proportions, among other things, so I thought it would be interesting to check whether the divisions Mozart used were very close to golden-section divisions." The golden sectiona precise way of dividing a line, music or anything elseshowed up early in mathematics. It goes back at least as far as 300 b.c., when Euclid described it in his major work, the
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