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Fibonacci Numbers in Nature & the Golden Ratio
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<blockquote data-quote="Minita" data-source="post: 64014" data-attributes="member: 3802"><p><a href="http://www.world-mysteries.com/" target="_blank">World-Mysteries.com</a></p><p> </p><p><span style="font-family: 'Times New Roman'">The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.</span></p><p><strong><span style="font-family: 'Times New Roman'">Golden</span></strong><strong><span style="font-family: 'Times New Roman'"> Ratio & Golden Section : : Golden Rectangle : : Golden Spiral</span></strong></p><p style="text-align: center"></p> <p style="text-align: center"><strong><span style="font-family: 'Times New Roman'">Golden Ratio & Golden Section</span></strong></p><p><span style="font-family: 'Times New Roman'">In mathematics and the arts, two quantities are in the <strong>golden ratio</strong> 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.</span></p><p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.jpg" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">Expressed algebraically:</span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span></p><p><strong><span style="font-family: 'Times New Roman'">The golden ratio</span></strong><span style="font-family: 'Times New Roman'"> is often denoted by the Greek letter phi (Φ or φ). </span></p><p><span style="font-family: 'Times New Roman'">The figure of <strong>a golden section</strong> illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately <strong>1.6180339887.</strong></span></p><p style="text-align: center"></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span></p> <p style="text-align: center"><em><span style="font-family: 'Times New Roman'">Successive points dividing a golden rectangle into squares lie on </span></em></p> <p style="text-align: center"><em><span style="font-family: 'Times New Roman'">a logarithmic spiral which is sometimes known as the golden spiral. </span></em></p> <p style="text-align: center"><em><span style="font-family: 'Times New Roman'">Image Source: <a href="http://mathworld.wolfram.com/GoldenRatio.html" target="_blank"><span style="color: blue">http://mathworld.wolfram.com/GoldenRatio.html</span></a> </span></em></p> <p style="text-align: center"></p> <p style="text-align: center"><strong><span style="font-family: 'Times New Roman'">Golden Ratio in Architecture and Art</span></strong></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the <strong>golden rectangle</strong>, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [<em>Source:</em> Wikipedia.org]</span></p> <p style="text-align: center"><strong><span style="font-family: 'Times New Roman'">Here are few examples:</span></strong></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.jpg" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span><span style="font-family: 'Times New Roman'"></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>Parthenon, Acropolis, Athens.</em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>This ancient temple fits almost precisely into a golden rectangle.</em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>Source: <a href="http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm" target="_blank"><span style="color: blue">http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm</span></a> </em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span><span style="font-family: 'Times New Roman'"></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci</em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>We can draw many lines of the rectangles into this figure. </em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>Then, there are three distinct sets of Golden Rectangles: </em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em>Each one set for the head area, the torso, and the legs.</em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><em><a href="http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html" target="_blank"><span style="color: blue">Image Source >></span></a></em></span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">Leonardo's <em>Vetruvian Man</em> is sometimes confused with principles of "golden rectangle", however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square's center and center of the moved circle:</span></p> <p style="text-align: center"><strong><span style="font-family: 'Times New Roman'">Fibonacci Numbers</span></strong></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">The sequence, in which <strong>each number is the sum of the two preceding numbers</strong> is known as the <strong>Fibonacci series:</strong> 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).</span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">The ratio of successive pairs is so-called <strong>golden section</strong> (GS) - 1.618033989 . . . . . </span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.</span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">The <strong>Fibonacci sequence,</strong> generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,</span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">is well known in many different areas of mathematics and science. </span></p> <p style="text-align: center"><strong><span style="font-family: 'Times New Roman'">Pascal's Triangle and Fibonacci Numbers</span></strong></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'">The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.</span></p> <p style="text-align: center"><span style="font-family: 'Times New Roman'"><img src="http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif" alt="" class="fr-fic fr-dii fr-draggable " style="" /></span></p></blockquote><p></p>
[QUOTE="Minita, post: 64014, member: 3802"] [url="http://www.world-mysteries.com/"]World-Mysteries.com[/url] [FONT=Times New Roman]The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.[/FONT] [B][FONT=Times New Roman]Golden[/FONT][/B][B][FONT=Times New Roman] Ratio & Golden Section : : Golden Rectangle : : Golden Spiral[/FONT][/B] [CENTER][B][FONT=Times New Roman] [/FONT][/B][/CENTER] [CENTER][B][FONT=Times New Roman]Golden Ratio & Golden Section[/FONT][/B][/CENTER] [FONT=Times New Roman]In mathematics and the arts, two quantities are in the [B]golden ratio[/B] 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.[/FONT] [CENTER][FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.jpg[/IMG][/FONT][/CENTER] [CENTER][FONT=Times New Roman]Expressed algebraically:[/FONT][/CENTER] [CENTER][FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG][/FONT][/CENTER] [B][FONT=Times New Roman]The golden ratio[/FONT][/B][FONT=Times New Roman] is often denoted by the Greek letter phi (Φ or φ). The figure of [B]a golden section[/B] illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately [B]1.6180339887.[/B][/FONT] [CENTER][COLOR=#3399ff][/COLOR][/CENTER] [CENTER][FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif[/IMG][/FONT][/CENTER] [CENTER][I][FONT=Times New Roman]Successive points dividing a golden rectangle into squares lie on a logarithmic spiral which is sometimes known as the golden spiral. Image Source: [URL='http://mathworld.wolfram.com/GoldenRatio.html'][COLOR=blue]http://mathworld.wolfram.com/GoldenRatio.html[/COLOR][/URL] [/FONT][/I] [COLOR=#3399ff][/COLOR] [B][FONT=Times New Roman]Golden Ratio in Architecture and Art[/FONT][/B] [FONT=Times New Roman]Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the [B]golden rectangle[/B], in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [[I]Source:[/I] Wikipedia.org][/FONT] [B][FONT=Times New Roman]Here are few examples:[/FONT][/B] [FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.jpg[/IMG][/FONT][FONT=Times New Roman] [I]Parthenon, Acropolis, Athens. This ancient temple fits almost precisely into a golden rectangle. Source: [URL='http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm'][COLOR=blue]http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm[/COLOR][/URL] [/I][/FONT] [FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG][/FONT][FONT=Times New Roman] [I]The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci We can draw many lines of the rectangles into this figure. Then, there are three distinct sets of Golden Rectangles: Each one set for the head area, the torso, and the legs. [URL='http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html'][COLOR=blue]Image Source >>[/COLOR][/URL][/I][/FONT] [FONT=Times New Roman]Leonardo's [I]Vetruvian Man[/I] is sometimes confused with principles of "golden rectangle", however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square's center and center of the moved circle:[/FONT] [B][FONT=Times New Roman]Fibonacci Numbers[/FONT][/B] [FONT=Times New Roman]The sequence, in which [B]each number is the sum of the two preceding numbers[/B] is known as the [B]Fibonacci series:[/B] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).[/FONT] [FONT=Times New Roman]The ratio of successive pairs is so-called [B]golden section[/B] (GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.[/FONT] [FONT=Times New Roman]The [B]Fibonacci sequence,[/B] generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1, is well known in many different areas of mathematics and science. [/FONT] [B][FONT=Times New Roman]Pascal's Triangle and Fibonacci Numbers[/FONT][/B] [FONT=Times New Roman]The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám.[/FONT] [FONT=Times New Roman][IMG]http://paranormalis.com/file:///C:/Users/welcome/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif[/IMG][/FONT][/CENTER] [/QUOTE]
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