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Topic: Fibonacci series

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	Fibonacci Numbers, the Golden section and the Golden String
		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.
		Here is a series that is very similar to the Fibonacci series, the Lucas series, but it starts with 2 and 1 instead of Fibonacci's 0 and 1.
		These General Fibonacci series are called the G series but the Fibonacci series and Phi again play a prominent role in their mathematical properties.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/fib.html (1948 words)

	Modified Fibonacci Series
		Fibonacci Numbers can be easily had by using starting numbers other than 0 and 1.
		Any series which uses the methods of adding the last two numbers to obtain the next in the series, will always have as a limit of the ratio of those last two numbers, F and f.
		Not surprisingly, the general rule is that the nth number of all of the different, modified Fibonacci series, all differ between the adjacent series by the (n-2)th number of the original Fibonacci series.
	www.halexandria.org /dward094.htm (720 words)

	Forex Fibonacci Book. Series of Free Forex ebooks
		Leonardo Fibonacci is a famous Italian mathematician, founder of a simple series of numbers that refer to ratios valid for natural proportions of things on the planet.
		A few examples of Fibonacci numbers are pine cones, sunflowers, pineapples, palm trees, spider webs, snail shells, DNA molecules and millions of other things in the universe.
		Fibonacci retracement and extension levels carry important information for experienced as well as novice Forex traders as they help to identify entry and exit points during the trade.
	www.fibonaccibook.com (323 words)

	Fibonacci Series
		In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.
		After the 40th number in the series, the ratio is accurate to 15 decimal places.
		Perhaps a better way is to consider 0 in the Fibonacci series to correspond to the 1st Fibonacci number where n = 1 for 0.
	goldennumber.net /fibonser.htm (310 words)

	Fibonacci Series
		In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.
		Perhaps a better way is to consider 0 in the Fibonacci series to correspond to the 1st Fibonacci number where n = 1 for 0.
		Divine Proportion, Fibonacci Series and Phi (1.618 0339 887...
	www.goldennumber.net /fibonser.htm (310 words)

	Fibonacci number - Wikipedia, the free encyclopedia
		Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci sequence.
		The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
		Fibonacci numbers are used by some pseudorandom number generators.
	en.wikipedia.org /wiki/Fibonacci_number (3843 words)

	Music and the golden section, divine proportion and Fibonacci series
		Fibonacci and phi relationships are often found in the timing of musical compositions.
		As an example, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song.
		Fibonacci and phi are used in the design of violins and even in the design of high quality speaker wire.
	evolutionoftruth.com /goldensection/music.htm (236 words)

	Fibonacci Series - Search Results - MSN Encarta
		Fibonacci Series, in mathematics, series of numbers in which each member is the sum of the two preceding numbers.
		Fibonacci, Leonardo (1170?-1240?), Italian mathematician, who compiled and supplemented the mathematical knowledge of classical, Arabic, and Indian...
		Sequence and Series, in mathematics, an ordered succession of numbers or other quantities, and the indicated sum of such a succession, respectively.
	encarta.msn.com /Fibonacci_Series.html (120 words)

	Fibonacci sequence
		Fibonacci is perhaps best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.
		It is called the Fibonacci series after Leonardo of Pisa or (Filius Bonacci), alias Leonardo Fibonacci, born in 1175, whose great book The Liber Abaci (1202), on arithmetic, was a standard work for 200 years and is still considered the best book written on arithmetic.
		Fibonacci obtains results which are equivalent to: It is claimed [13] (5*) that this last result should be called "Leonardo's Theorem" as it was not definitely stated in any earlier work.
	www.geocities.com /forex_trade_system/fibonacci-sequence.html (1217 words)

	fUSION Anomaly. Fibonacci
		The Fibonacci SEQUENCE 0, 1, 1, 2, 3, 5, 8, 13, 21, …, in which each term is the sum of the two preceding terms, occurs in higher mathematics in various connections.
		infinite series in which each successive integer is the sum of the two that precede it- for example, 1, 1, 2, 3, 5, 8, 13, 21, 34....
		Fibonacci used this experience to improve on the commercial computing techniques he knew and to extend the work of classical mathematical writers, such as Greek mathematicians Diophantus and Euclid.
	fusionanomaly.net /fibonacci.html (758 words)

	The Math Forum - Math Library - Fibonacci Sequence
		Fibonacci numbers are closely related to the golden ratio (also known as the golden mean, golden number, golden section) and golden string.
		Fibonacci and his original problem about rabbits that gave the series its name; the family trees of bees; the golden ratio and the Fibonacci series; the Fibonacci Spiral and sea shell shapes; branching plants; flower petal and seed-heads; and the leaf...more>>
		The Fibonacci Series is a series of numbers in which each number is the sum of the two previous numbers...
	mathforum.org /library/topics/fibonacci (1742 words)

	Fibonacci (Site not responding. Last check: )
		It is called the Fibonacci series after Leonardo of Pisa or (Filius Bonacci), alias Leonardo Fibonacci, born in 1175, whose great book The Liber Abaci (1202), on arithmetic, was a standard work for 200 years and is still considered the best book written on arithmetic.
		One of the most spectacular examples of the Fibonacci Series in nature is in the head of the sunflower.
		The Fibonacci series have been observed in phylotaxis and extensively studied in three different spiral arrangements.
	www.goldenmeangauge.co.uk /fibonacci.htm (1388 words)

	The Fibonacci Series - Home
		The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci (fi-bo-na-chee) in 1202.
		Fibonacci mathematics is a constantly expanding branch of number theory, with more and more people being drawn into the complex subtleties of Fibonacci's legacy.
		The Fibonacci Forum is a bulletin board intended for the discussion of all things Fibonacci; Fibonacci number-theory, Leonardo Fibonacci, appearances of the Fibonacci Series in day-to-day life, etc.
	library.thinkquest.org /27890/mainIndex.html (370 words)

	Fibonacci Numbers
		Put all this together with the present wide interest in the Fibonacci series, which was possibly the least important part of his work at the time, and you have an impressive picture of a 12th century mathematical mind at work setting the sub-structure for modern mathematical thinking.
		If the "fault" in the Fibonacci calculation is inexactitude based on its arithmetic calculations, the value of this series is the applicability to situations which deal with whole numbers, like Leonardo's famous calculation for the number of rabbits which can be generated from a single pair in a hypothetical year of breeding.
		A Virtual Library on the Fibonacci Numbers Here is detailed information on almost everything that anyone has at any time written on the Fibonacci Series, assembled by a professional in the fields of mathematics, engineering and mathematical education.
	community.middlebury.edu /~harris/fibonacci.html (1478 words)

	OPB NTTI: Patterns of Fibonacci
		The Fibonacci series is a very dynamic application of mathematical patterns in the real world.
		Fibonacci numbers can be quite complicated and branch into many areas of nature, art and mathematics.
		This lesson focuses on having students recognize the origin of the Fibonacci series and its relationship to the spiral shape.
	www.opb.org /education/ntti/2003/patternsoffibonacci (2752 words)

	Fibonacci
		This sequence is known as the Fibonacci series, and is well known in mathematics.
		Fibonacci numbers 2 3 5 8 13 and 21 are drawn in fl.
		You can find the Fibonacci numbers by counting the helical series of scars on many small coniferous trees, on their cones, and in the helical arrangement of leaves on plants.
	www.branta.connectfree.co.uk /fibonacci.htm (5852 words)

	Math Forum - Ask Dr. Math Archives: High School Fibonacci Sequence/Golden Ratio
		I need examples of where the Fibonacci sequence is found in nature and how it relates to the Golden Mean.
		I have to prove that in the Fibonacci sequence, F(k) is a divisor of F(nk), where n is a natural number (so, F(nk) = A*F(k) where A is a natural number).
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/high_fibonacci-golden.html (666 words)

	2CS21 - FIBONACCI (Site not responding. Last check: )
		Fibonacci (Figure 1) is the name whereby the mathematician Leonardo of Pizza (1170-1250) is commonly known today.
		This corresponds to the fact that each month, the new rabbit births consists of one pair to each of the newly adult pairs plus one pair for each of the earlier adult pairs.
		It turns out that the Fibonacci sequence describes all sort of natural phenomena such as: the spiraling pattern on Nautilus shells, elephant tusks, sheep horns, birds claws; the spiral arrangement of seeds on the face of certain sunflower; the pattern on fir cones and pine apples; and the branching patterns many plants and trees.
	www.csc.liv.ac.uk /~frans/COMP101/week8/fibonacci.html (307 words)

	Use of the Fibonacci Series in the Bassoon Solo in Bartok's Dance Suite
		Perhaps the most significant use of the Fibonacci series in the First Movement can be found in the tonal and rhythmic organization of the melody presented by the bassoons.
		Though the listener (or performer) may not recognize that the Fibonacci series is being used in this passage, nevertheless I believe that the movement from smaller to larger melodic intervals is heard.
		The last two groupings of 4 could be joined as components of a single group of 8, which is the next number in the Fibonacci series after 5.
	idrs.colorado.edu /Publications/Journal/JNL17/JNL17.Ewell.Bartok.html (920 words)

	Science NetLinks: The Fibonacci Sequence
		Tell students that the Fibonacci sequence is a series of numbers developed by Leonardo Fibonacci as a means of solving a practical problem: Fibonacci wanted to determine the rate at which pairs of rabbits would reproduce.
		Fibonacci's Rabbits, found on the Fibonacci Numbers and the Golden Section in Nature website, provides text and an illustration related to the Fibonacci sequence.
		Go to Easier Fibonacci Puzzles for activities in which students manipulate objects such as bricks (substitute blocks and conduct as a hands-on activity), dominoes, and chairs in order to find numerical patterns and solve the puzzles.
	www.sciencenetlinks.com /lessons.cfm?DocID=134 (1090 words)

	Pi Day, Divine Proportion, Golden Number, Fibonacci, Phi
		The ratio between two consecutive numbers in the Fibonacci Series tend to be equal to the golden number: 0.618...
		Fibonacci #10 / Fibonacci #9 = 55 / 34 = 1.61764...
		The longer the Fibonacci series, the closer the ratio of two consecutive numbers to the golden number PHI or the divine ratio / proportion.
	www.saliu.com /bbs/messages/958.html (3274 words)

	Number 89 and the Fibonacci series
		The reciprocal of 89, a Fibonacci number, is based on the Fibonacci series
		This is a little curiousity involving the number 89, the twelfth number in the Fibonacci series.
		If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add them all together, you get 0.011235955..., the same number as the reciprocal of 89.
	www.evolutionoftruth.com /goldensection/Number89.htm (128 words)

	Phi & Fibonacci Anomalies
		Note: The Fibonacci series spiral on the left is slightly different from the perfect spiral generated by Phi (1.61804...) because of the approximations early in the series leading to Phi.
		Plants illustrate the Fibonacci series in the numbers and arrangements of petals, leaves, sections and seeds.
		The Fibonacci series appears in the foundation of many other aspects of art, beauty and life.
	evolutionoftruth.com /div/fibomyst.htm (591 words)

	Fibonacci series on microstructures
		Their work 'Triangular and Fibonacci number patterns driven by stress on core/shell microstructures' was published on the August 5 issue of Science.
		Fibonacci patterns come from a numerical sequence that fascinated the Italian mathematician Leonardo Fibonacci in the early 1200s.
		This is the first time that the triangular tessellation and Fibonacci number patterns have been grown with inorganic materials on a micrometre scale, according to the CAS researchers.
	www.physorg.com /news5895.html (806 words)

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