Golden+Ratio

This page will help you learn and understand about the golden ratio Posted by: pineapplepoptart Wow. Lotsa copy and pasting. O_o.

Phi or golden ratio, is around 1.61803 3988. The Golden Ratio is represented by a Greek symbol. Phi was theorized to be first understood by ancient mathematicians in Egypt, several thousand years ago. Since then, the symbol has been appearing in many works of art, such as the statue of Athena in Greece.

posted by: **smokey-dice1212**

Phi (//Golden Ratio//) as a mysterious number has been discovered in many places,. such as art, architectures, humans, and plants.two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is around 1.61803 3988. You might wonder where and when Phi first appeared? Who was the discoverer? According the history of mathematics, Phi was first understood and used by the ancient mathematicians in Egypt, two to three thousand years ago, due to its frequent appearance in Geometry. Phidias (500BC-432 BC), a Greek sculptor and mathematician, studied Phi and used the Phi in many designs of his sculptures, such as the statue of the goddess Athena in Athena, and the state of god Zeus in Olympiad. And Euclid Alexandria (365BC-300BC) had once described the Phi as "dividing a line in the extreme and mean ratio" in his Book VI of Elements. The name "//Golden Ratio//" appears in the form sectio aurea (Golden Section in Greek) by Leonardo da Vinci (1452-1519) who used this the Golden ratio in many of his masterpieces, such as The Last Supper and Mona Lisa. In 1900s, an American mathematician named Mark Barr, represented the //Golden Ratio// by using a greek symbol ϕ some examples of golden ratio in nature are: sunflowers, pine cones, snails, pineapples some examples of golden ratio in nature in the humans body are: that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618. Some other golden proportions in the average human body are:   The distance between the finger tip and the elbow / distance between the wrist and the elbow, The distance between the shoulder line and the top of the head / head length, The distance between the navel and the top of the head / the distance between the shoulder line and the top of the head, The distance between the navel and knee / distance between the knee and the end of the foot. Some people used to think that all these things in nature having the golden ratio was just a coincidence, other people thought it was a sign from god    this video might help explain: http://ca.youtube.com/watch?v=085KSyQVb-U

__Posted by:__ **LemonMPie** Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:

[The Golden Ratio is a universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

Posted by: Salinakung The golden ratio is an irrational mathematical constant, about 1.61803 39887 49894 84820..... In math and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. Posted by: Salinakung Two quantities (positive numbers) //a// and //b// are said to be in the golden ratio: This equation defines //ϕ// (phi). The right equation shows that a //= b ϕ,// which can be replaced in the left part: Cancelling //b// leaves: Multiplying both side by //ϕ// and rearranging the terms leaves: Solution:  <span class="Apple-style-span" style="font-size: 11px; color: rgb(34,34,34); line-height: 15px; font-family: Verdana;"><span class="Apple-style-span" style="font-size: 12px; color: rgb(0,0,0); line-height: normal; font-family: Geneva;"><span style="color: rgb(22,24,22);"> Posted by:jchen727 Some examples of the Golden Ratio in nature. Pine Cone: 8 scales in one direction and 5 in the other (or 8 in one direction and 13 in the other). The ratio 8:5 is 1.6 near the Golden Ratio. Pinapple: 13 scales in one direction and 8 in the other. The ratio 13:8 is even nearer to the Golden Ratio. Sunflower: 55 florets in one direction and 34 in the other, 89 florets in one direction and 55in the other, 144 florets in one direction and 89 in the other. The ratios in order are <span style="color: rgb(0,0,0);">**1.61765**, **1.61819**, and **1.61798**  that is extremely close to the Golden Ratio Scientists speculate that plants that grow in spiral formation do so in Fibonacci numbers because this arrangement makes for the perfect spacing for growth. So for some reason, these numbers provide the perfect arrangement for maximum growth potential and survival of the plant. <span style="color: rgb(22,24,22);">  <span style="color: rgb(22,24,22);">

Posted by : **DummyDerrick**

Meaning of the "Golden Ratio"

<span style="font-size: 14px; font-family: Georgia,serif,Arial,'; font-color: #000000';"> 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. In modern times there has been much interest in the Golden Proportion, Section or Mean. Since the Renaissance it has been used extensively in art and architecture, it figures in the Venetian Church of St. Mark built early in the 16th century, and has become a standard proportion for width in relation to height as used in facades of buildings, in window sizing, in first story to second story proportion, at times in the dimensions of paintings and picture frames. There is something "satisfactory" about the relationships of the Greek "divided lines" proportion, which some have felt to be modern acculturation since the Renaissance. In the l930's the Pratt Institute of New York did a study on various rectangular proportions laid out as vertical frames, and asked several hundred art students to comment on which seemed the most pleasing. The ratio of 1 : 2 was least liked, while the Golden Ratio was favored by a very large margin, which seemed to point to the actual dimensions as generating a pleasing response by their size. 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.