Mineral Growth Spirals illustrating Phi Fibonacci numbers and Phi are related to spiral growth in nature If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number. This property results in the Fibonacci spiral, based on the following progression and properties of the Fibonacci series: 12 + 12 + 22 + 32 + 52 = 5 x 8 12 + 12 + . . . + F(n)2 = F(n) x F(n+1) A Golden spiral is very similar to the Fibonacci spiral but is based on a series of identically proportioned golden rectangles, each having a golden ratio of 1.618 of the length of the long side to that of the short side of the rectangle: The Fibonacci spiral gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively) Golden spiral in human ear Be Sociable.
Time and Quantum Physics relationships to Phi The Golden Ratio seems to be appearing in several places in the Quantum Physics Model. Because an electron has an electric charge and an intrinsic rotational motion, or spin, it behaves in some respects like a small bar magnet and is said to have a magnetic moment. Because the electron also has mass, it behaves in some respects like a spinning top, and is said to have spin angular momentum. The g factor of the electron is defined as the ratio of its magnetic moment to its spin angular momentum. Mathematically, the electron g-factor is approximately: gfactore = -2 / sin (Ø) and the proton g-factor is approximately: gfactorp = 2Ø / sin (1/Ø) Thus it appears that the Golden Ratio, or Phi, is a constant produced by time. The National Institute of Standards and Technology (NIST) states these gfactor constants as per the table below. Such constants are adjusted as new measuring techniques and better materials become available. Insights on this page were contributed by David W. Be Sociable.
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Complex numbers This is an introduction to complex numbers. It includes the mathematics and a little bit of history as well. It is intended for a general audience. The necessary background in a familiarity with ordinary real numbers (all positive and negative numbers and zero) and algebra. In one section some background in trigonometry is needed as indicated with the symbol. I. 1. Solution of quadratics, solution of cubics 2. 3. The Fundamental Theorem of Algebra proved! II. 4. Notation, arithmetic operations on C, parallelogram rule, addition as translation, negation and subtraction 5. The unit circle, the triangle inequality 6. Multiplication done algebraically, multiplying a complex number by a real number, multiplication and absolute value, powers of i, roots of unity, multiplying a complex number by i, a geometric interpretation of multiplication 7. 8. Reciprocals done geometrically, complex conjugates, division 9. Powers, roots, more roots of unity
Math Mysticism: is Hurricane Shape a Fibonaci Spiral? Please beware of the golden ratio math mysticism spreading online. Many people are sharing this image online. It is true that the hurricanes have the general shape of a spiral. In particular, it's a equiangular spiral (aka logarithmic spiral), which is the shape that many (but not all) natural growth takes (⁖ Seashells), but saying that hurricane has golden-spiral is completely baseless, it's like seeing Jesus face on Mars. The so-called “Fibonacci spiral” (aka golden-spiral) you see in the pic is made up of parts of circles, stacked on blocks of squares. (equiangular spiral is a family of spirals. Also, hurricane does not really have a definite shape, we can only say it's roughly that of equiangular spiral based partly on physics and appearance. If you don't have a degree in math, you might be all confused about this Fibonacci number , the Golden ratio, and the various spirals. For highschool-level explanation of the equiangular spiral, see: Equiangular Spiral.
Fibonacci Leonardo Bonacci (c. 1170 – c. 1250)[2]—known as Fibonacci (Italian: [fiboˈnattʃi]), and also Leonardo of Pisa, Leonardo Pisano, Leonardo Pisano Bigollo, Leonardo Fibonacci—was an Italian mathematician, considered as "the most talented Western mathematician of the Middle Ages.".[3][4] Fibonacci introduced to Europe the Hindu–Arabic numeral system primarily through his composition in 1202 of Liber Abaci (Book of Calculation).[5] He also introduced to Europe the sequence of Fibonacci numbers (discovered earlier in India but not previously known in Europe), which he used as an example in Liber Abaci.[6] Life[edit] Fibonacci was born around 1170 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, the consul for Pisa. Guglielmo directed a trading post in Bugia, a port in the Almohad dynasty's sultanate in North Africa. Fibonacci travelled with him as a young boy, and it was in Bugia (now Béjaïa, Algeria) that he learned about the Hindu–Arabic numeral system.[2] Legacy[edit]
Biographie : Leonardo Fibonacci (1170 [Pise] - 1245 [Pise]) Leonard de Pise, plus connu sous le nom de Fibonacci, est le premier grand mathématicien de l'ère chrétienne du monde occidental. D'assez nombreux détails de sa jeunesse nous sont connus par les propos qu'il tient lui-même dans la préface d'un de ses livres, le Liber abaci. Né à Pise vers 1170, il rejoint très jeune son père à la colonie de Bujania, en Algérie, où ce dernier est responsable du bureau des douanes pour le compte de l'ordre des marchands de Pise. Voulant faire de son fils un marchand, il l'initie à l'art du calcul indo-arabe. Fibonacci apprendra en outre les savoirs et algorithmes orientaux grâce à ses nombreux voyages en Syrie, en Grèce, en Egypte. Fibonacci vivait avant l'invention de l'imprimerie, ce qui signifiait que pour avoir plusieurs exemplaires du même ouvrage, il fallait le travail entièrement manuel d'un copiste. Un autre des plaisirs de l'empereur était les défis mathématiques qu'un membre de sa cour posait à la communauté des scientifiques.
Leonardo Fibonacci Un article de Wikipédia, l'encyclopédie libre. Leonardo Fibonacci Statue de Léonard de Pise, dans sa ville natale Leonardo Fibonacci (v. 1175 à Pise, Italie - v. 1250) est un mathématicien italien. Il avait, à l'époque, pour nom d'usage « Leonardo Pisano » (il est encore actuellement connu en français sous l'équivalent « Léonard de Pise »), et se surnommait parfois lui-même « Leonardo Bigollo » (bigollo signifiant « voyageur » en italien). Biographie[modifier | modifier le code] Né à Pise en Italie, son éducation s'est faite en grande partie à Béjaïa en Algérie, où son père Guilielmo Bonacci était le représentant des marchands de la république de Pise. Ayant aussi voyagé en Égypte, en Syrie, en Sicile, en Provence pour le compte de son père, et rencontré divers mathématiciens, Fibonacci en rapporta à Pise en 1198 les chiffres arabes et la notation algébrique (dont certains attribuent l'introduction à Gerbert d'Aurillac). De 1202 à 1225, il est occupé par ses différents ouvrages.
Suite de Fibonacci Un article de Wikipédia, l'encyclopédie libre. Elle doit son nom à Leonardo Fibonacci qui, dans un problème récréatif posé dans l'ouvrage Liber abaci publié en 1202, décrit la croissance d'une population de lapins : « Un homme met un couple de lapins dans un lieu isolé de tous les côtés par un mur. Combien de couples obtient-on en un an si chaque couple engendre tous les mois un nouveau couple à compter du troisième mois de son existence ? » Cette suite est fortement liée au nombre d'or, φ (phi). Ce nombre intervient dans l'expression du terme général de la suite. Croissance de population des lapins selon une suite de Fibonacci Présentation mathématique[modifier | modifier le code] Formule de récurrence[modifier | modifier le code] Le problème de Fibonacci est à l'origine de la suite dont le -ième terme correspond au nombre de paires de lapins au -ème mois. Notons le nombre de couples de lapins au début du mois . Plaçons-nous maintenant au mois désigne la somme des couples de lapins au mois et où
Suite de Fibonacci The Fibonacci Sequence is the series of numbers: The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 = 55 It is that simple! Here is a longer list: Can you figure out the next few numbers? Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit neatly together? The Rule The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series). First, the terms are numbered from 0 onwards like this: So term number 6 is called x6 (which equals 8). So we can write the rule: The Rule is xn = xn-1 + xn-2 where: xn is term number "n" xn-1 is the previous term (n-1) xn-2 is the term before that (n-2) Example: term 9 is calculated like this: Golden Ratio And here is a surprise. Using The Golden Ratio to Calculate Fibonacci Numbers
Fibonacci Numbers, the Golden section and the Golden String Fibonacci Numbers and the Golden Section This is the Home page for Dr Ron Knott's multimedia web site on the Fibonacci numbers, the Golden section and the Golden string hosted by the Mathematics Department of the University of Surrey, UK. The Fibonacci numbers are The golden section numbers are 0·61803 39887... = phi = φ and 1·61803 39887... = Phi = Φ The golden string is 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ... a sequence of 0s and 1s that is closely related to the Fibonacci numbers and the golden section. If you want a quick introduction then have a look at the first link on the Fibonacci numbers and where they appear in Nature. THIS PAGE is the Menu page linking to other pages at this site on the Fibonacci numbers and related topics above. Fibonacci Numbers and Golden sections in Nature Ron Knott was on Melvyn Bragg's In Our Time on BBC Radio 4, November 29, 2007 when we discussed The Fibonacci Numbers (45 minutes). listen again online or download the podcast. and phi . The Golden Section
Fibonacci number A tiling with squares whose side lengths are successive Fibonacci numbers In mathematics, the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence: or (often, in modern usage): (sequence A000045 in OEIS). The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling;[3] this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34. By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation with seed values or The Fibonacci sequence is named after Fibonacci. Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. Origins[edit] List of Fibonacci numbers[edit] and