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Fibonacci Numbers Geometry:     more detail

The Golden Ratio and Fibonacci Numbers by R. A. Dunlap, 1998-03 The Fabulous Fibonacci Numbers by Alfred S. Posamentier, Ingmar Lehmann, 2007-06-21 Fibonacci Numbers by Nicolai N. Vorobiev, 2003-01-31 1001 Fibonacci Numbers: The Miracle Begins with Unity and Order Follows by Mr. Effectiveness, 2010-01-13 Geometry of Design: Studies in Proportion and Composition by Kimberly Elam, 2001-08-01 Recursion: Function, Parent, Ancestor, Fibonacci Number, Fractal, Fractal-Generating Software, Shape, Differential Geometry, Integral

lists with details

81. 8
    are expressed by the hyperbolic functions and the geometry itself is apply mathematicalmethods of continues mathematics to develop the fibonacci numbers theory  
    http://www.uem.mz/faculdades/ciencias/informat/docentes/stakhov/8.htm

82. Schoolzone
    and Golden section in Nature fibonacci numbers and the golden section in nature,art, geometry, architecture, music, geometry and even for calculating pi!  
    http://www.schoolzone.co.uk/teachers/news/newsletter/issues/2002/November2002sec
    
    Extractions: Word version for easier printing Maths Learning Resources Online animal weigh in Measures: add masses to balance the objects. Easy levels: just numbers, harder ones involve conversions. Good KS3 games. bathroom tiles Shape: Pythagoras gets out of his bath to help you with your maths. Reflection, Translation and Mirroring are covered in three stages in this fun game. The Fibonacci Numbers and Golden section in Nature Fibonacci numbers and the golden section in nature, art, geometry, architecture, music, geometry and even for calculating pi! Puzzles and lots of other things to do!

83. The Life And Numbers Of Fibonacci
    Plus Online Maths Magazine Feature Article  
    http://pass.maths.org.uk/issue3/fibonacci/alt.html
    
    Extractions: Issue 23: Jan 03 Issue 22: Nov 02 Issue 21: Sep 02 Issue 20: May 02 Issue 19: Mar 02 Issue 18: Jan 02 Issue 17: Nov 01 Issue 16: Sep 01 Issue 15: Jun 01 Issue 14: Mar 01 Issue 13: Jan 01 Issue 12: Sep 00 Issue 11: Jun 00 Issue 10: Jan 00 Issue 9: Sep 99 Issue 8: May 99 Issue 7: Jan 99 Issue 6: Sep 98 Issue 5: May 98 Issue 4: Jan 98 Issue 3: Sep 97 Issue 2: May 97 Issue 1: Jan 97 by R.Knott, D.A.Quinney and PASS Maths MCMXCVII The Roman numerals were not displaced until the 13th Century AD when Fibonacci published his Liber abaci which means "The Book of Calculations". Leonardo Fibonacci c1175-1250. Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175AD. He was the son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:

84. E-z Geometry Project Topics
    Cryptography Fermat's Last Thm fibonacci 's Figurative Pi Polyhedra Prime NumbersPythagorus, Similarity Non Euclidean geometry NonEuclid geometry Software;  
    http://www.e-zgeometry.com/links/plinks1.htm
    
    Extractions: e-zgeometry Project Topics A to F G to M N to R S to Z Networks: Go to Top Non Euclidean Geometry: Go to Top Number Theory: Go to Top Optical Illusions: Go to Top Orgami: Go to Top Pascal's Triangle: Go to Top Pentominoes: Go to Top Pi: Go to Top Prime Numbers: Go to Top Pythagorean Theorem: Go to Top Similarity: still searching.......

85. Math Forum - Ask Dr. Math Archives: High School Fibonacci Sequence/Golden Ratio
    path? Consecutive fibonacci numbers Relatively Prime 11/17/2001Prove two consecutive fibonacci numbers are relatively prime.  
    http://mathforum.org/library/drmath/sets/high_fibonacci-golden.html

86. Scheme Of Work For Maths Six Of The Best Cards - From Learn.co.uk
    rectangle geometrically; appreciate the link between numbers and geometryvia fibonacci numbers and the golden ratio; discover the  
    http://www.learn.co.uk/preparation/maths/fibonacci/default.htm
    
    Extractions: Key question What are Fibonacci numbers and how are they linked to the golden ratio? How do Fibonacci numbers work, where do you find them and what can you do with them? Where the unit fits in Pupils explore the Fibonacci numbers and the golden ratio, looking at the mathematics behind these ideas and their links to nature and architecture. This unit can be tied into sequences, functions and graphs and properties of rectilinear shapes Expectations At the end of this unit most pupils will use knowledge and understanding of sequences to explain the properties of Fibonacci numbers explore the wide occurrence of Fibonacci numbers and the golden ratio in nature explore the geometry of the golden ratio and Fibonacci numbers explore Fibonacci numbers in music, puzzles and games

87. Fibonacci Numbers
    The fibonacci numbers pop up in a multitude of places in mathematicsand nature. A very comprehensive site covering a variety of  
    http://www.cut-the-knot.com/arithmetic/Fibonacci.shtml
    
    Extractions: Consider the counting problem How many sequences of 1's and 2's sum to 14? To get a feeling for the problem, let's do the counting when the sum is small. Let f(n) be the number of sequences of 1's and 2's which sum to n. f(1) = 1 because there is only one way that a sequence of 1's and 2's sum to 1, namely, 1 = 1. f(2) = 2 because there are exactly two ways a sequence of 1's and 2's can sum to 2, namely, 1+1 = 2 and 2 = 2. f(3) = 3 because 1+1+1 = 1+2 = 2+1 = 2. As we go on computing f(n) for small values of n, no obvious formula emerges. However, we do notice a relationship between f(n) and f(n-1) and f(n-2). For, suppose we split the sequences of 1's and 2's that sum to n into two groups, those that end in 1 and those that end in 2. For those sequences that end in 1, their sums, not counting the last 1 is a sequence of 1's and 2's that sum to n-1, and all such actually. Hence the number of sequences of 1's and 2's that sum to n and end in a 1 is f(n-1). Similarly, the number of sequences of 1's and 2's which sum to n and end in a 2 is f(n-2). Hence f(n) = f(n-1)+f(n-2), for n>2, a so-called recursion formula for f(n). This is of great help in solving our counting problem. Repeatedly applying this recursion formula, we get f(14) = 610, that is, there are 610 sequences of 1's and 2's which sum to 14.

88. A Problem Including Fibonacci Numbers
    (It's easy to see that the numbers thus obtained are 1 less than the Fibonaccinumbers.) Each iteration takes looking into 1 bank account.  
    http://www.cut-the-knot.com/ctk/Sharygin.shtml
    
    Extractions: Recommend this site Solution Alexander Bogomolny Let's first think of the ministers as sitting on a bench, say along a wall, and identify each of them with the number of his seat. Let f(x) denote the amount of money in the x's bank account. An ordered k) triple of ministers (n, m, k) is called suitable or S-triple if and . Let's finally define the size of a triple (n, m, k) as The problem is equivalent to finding an S-triple of size 2. The solution is based on two observation. Assume for example that Take any t between n and m. Then one of the two. Either , or, In the former case, (t, m, k) is suitable. In the latter case, hence (n, t, m) is suitable. In both case, the size of the triple is obviously less than More than that, for any S-triple, there are ways to reduce its size. Continuing in this manner we shall necessarily reach a solution. But can we always find a suitable triple to start from? Yes, we can. Think again of the ministers as sitting by the round table. Pick any 3 - a, b, c - following a certain direction around the table. Of the three numbers f(a), f(b), f(c) one is the largest. To introduce an ordering of ministers start counting with a minister with a smaller amount and move toward the one with the largest bank account. As we'll see, this is not the best way to start the process but it does show that the problem is solvable at least in principle. There always exists an S-triple of size 2. Now, let's relate to any S-triple

89. The Educational Encyclopedia, Mathematics
    homework help and math word problems, algebra, geometry, trigonometry, calculus fibonaccinumbers and nature fibonacci, golden section, golden mean, golden ratio  
    http://users.pandora.be/educypedia/education/mathematics.htm
    
    Extractions: Science Animals Biology Botany Bouw ... Resources Mathematics Algebra Arithmetric Complex numbers Formulas ... Fractals General overview Geometry Integrals and differentials Miscellaneous Statistics ... Trigonometry General overview Aplusmath this web site is developed to help students improve their math skills interactively, algebra, addition, subtraction, multiplication, division, fractions, geometry for kids Ask Dr. Math Ask Dr. Math a question using the Dr. Math Web form, or browse the archive Calculus tutorial Karl's calculus tutorial, limits, continuity, derivatives, applications of derivatives, exponentials and logarithms, trig functions (sine, cosine, etc.), methods of integration Cut the knot! algebra, geometry, arithmetic, proofs, butterfly theorem, chaos, conic sections, Cantor function, Ceva's theorem, Fermat point, cycloids, Collage Theorem, Carnot's theorem, bounded distance, barycentric coordinates, Pythagorean theorem, Napoleon's theorem, Ford's touching circles, Euclid's Fifth postulate, Non-Euclidean Geometry, Projective Geometry, Moebius Strip, Ptolemy's theorem, Sierpinski gasket, space filling curves, iterated function systems, Heron's formula, Euler's formula, Hausdorff distance, isoperimetric theorem, isoperimetric inequality, Shoemaker's Knife, Van Obel theorem, Apollonius problem, Pythagoras, arbelos, fractals, fractal dimension, chaos, Morley, Napoleon, barycentric, nine point circle, 9-point, 8-point, Miquel's point, shapes of constant width, curves of constant width, Kiepert's, Barbier's

90. SEDL - Math And Science Online Mentoring: Search Results
    Your selection From the topic Patterns, Functions, and Algebra, you selectedthe Q A record concerning fibonacci numbers. fibonacci numbers.  
    http://www.sedl.org/scimast/archives/answers/53.html
    
    Extractions: Question: What are Fibonacci numbers and how are they used in the real world? Answer: From my reading I have found that in 1202 Fibonacci proposed the followingproblem: "Begin with a pair of newborn rabbits that never die. When a pair ofrabbits is 2 months old, it begins producing a new pair of rabbits each month.How many rabbits will there be at a given time?" So if you begin with a pair of newborns, at month 1 there is 1 pair.They don't breed until they are 2 months old, so at month 2 there are still 1pair.At month 3 there is the original pair and the new pair they produced, so 2pair.At month 4 there are those 2 pair plus the new pair the original pair producedthis month = 3.At month 5, there are the 3 pair, plus a new pair produced by the original pairand a pair produced by the second pair, so = 5 pair.The next month the third pair is old enough to produce, so we add 3 pair = 8.etc. So the resulting sequence is 1, 1, 2, 3, 5, 8, ... An easy way to get thenext number is to add the 2 previous numbers. Therefore, f(n+2) = f(n) + f(n+1).

91. Math Education: Newsletter: Math Forum Internet News No. 3.50 (14 Dec 1998)
    THE MATH FORUM INTERNET NEWS. fibonacci Golden Section Knott Geometryin Motion Dragonfly. fibonacci numbers AND THE GOLDEN SECTION - Ron Knott.  
    http://www.math.yorku.ca/Who/Faculty/Monette/MathEd/0074.html

92. Math 6 - Frisbie Middle School
    Math 6. As part of the Rialto Unified School District, Frisbie Middle Schoolis using the Glencoe instructional materials to teach mathematics.  
    http://www.rialto.k12.ca.us/frisbie/math6.html
    
    Extractions: As part of the Rialto Unified School District, Frisbie Middle School is using the Glencoe instructional materials to teach mathematics. Math 6 is a general mathematics course for sixth grade students. Under the Curriculum Links the alignment of the course can be checked against the RUSD Math Matrix, California Framework Standards, and the NCTM Standards. Links are provided to support the Math 6 curriculum for students, teachers and parents. Some of the links are to Internet sites with information, others link to sites with interactive lessons which could be used at home as a supplement or during class and still others could be used as enrichment for students willing to work before or after school. A similar Math 7 and Math 8 page are available for the general mathematics course for all seventh and eighth grade students. As you use these pages, comments and/or suggestions would be appreciated. Please send them to Suzanne Alejandre Math Matrix - RUSD RUSD Course of Study CA Academic Standards Commission California Mathematics Framework

93. Chromatism
    CHROMATISM. The work of Edward S. May. Chromatism is a discipline whichlies at the confluence of mathematics and art. The Amplitude  
    http://www.moonstar.com/~nedmay/Welcome.html

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