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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

1. Fibonacci Numbers, The Golden Section And The Golden String
   fibonacci numbers and the golden section in nature, art, geometry, architecture, music, geometry Category Science Math Specific numbers fibonacci numbersfibonacci numbers and the golden section in nature, art, geometry, architecture,music and even for calculating pi! Puzzles and investigations.  
   http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
   
   Extractions: a sequence of 0s and 1s which is closely related to the Fibonacci numbers and the golden section. There is a large amount of information at this site (more than 200 pages if it was printed), so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature. The rest of this page is a brief introduction to all the web pages at this site on 10 February 2003 Fibonacci Numbers and Nature 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. All involve the Fibonacci numbers - and here's how and why.

2. The Golden Ratio And The Fibonacci Numbers
   The Golden Ratio and The fibonacci numbers The Golden Ratio ( ) is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one = 1/ + 1. an interesting exercise in the geometry of the Golden Ratio, as seen  
   http://www.friesian.com/golden.htm
   
   Extractions: The Fibonacci Numbers The Golden Ratio ) is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one: . Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one: . Since that equation can be written as , we can derive the value of the Golden Ratio from the quadratic equation, , with a = 1 b = -1 , and c = -1 . The Golden Ratio is an irrational number, but not a transcendental one (like ), since it is the solution to a polynomial equation. This gives us either or . The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal. The Golden Ratio can also be derived from trigonometic functions: = 2 sin 3 /10 = 2 cos ; and = 2 sin /10 = 2 cos 2 . The angles in the trigonometric equations in degrees rather than radians are o o o , and 72 o , respectively. The Golden Ratio seems to get its name from the Golden Rectangle , a rectangle whose sides are in the proportion of the Golden Ratio. The theory of the Golden Rectangle is an aesthetic one, that the ratio is an aesthetically pleasing one and so can be found spontaneously or deliberately turning up in a great deal of art. Thus, for instance, the front of the Parthenon can be comfortably framed with a Golden Rectangle. How pleasing the Golden Rectangle is, and how often it really does turn up in art, may be largely a matter of interpretation and preference. The construction of a Golden Rectangle, however, is an interesting exercise in the geometry of the Golden Ratio

3. The Golden Section - The Number And Its Geometry
   Simple definitions; exact value and first 2000 decimal places; finding the golden section; continued Category Science Math Recreations Specific numbers phi the golden section using geometry (compass and ruler); a new form of fractions (continuedfractions) and the golden section lead back to the fibonacci numbers!  
   http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html
   
   Extractions: A bit of history... Links on Euclid and his "Elements" ... More What is the golden section (or Phi)? We will call the Golden Ratio (or Golden number) after a greek letter, Phi ) here, although some writers and mathematicians use another Greek letter, tau ). Also, we shall use phi (note the lower case p) for a closely related value. A simple definition of Phi There are just two numbers that remain the same when they are squared namely and . Other numbers get bigger and some get smaller when we square them: Squares that are bigger Squares that are smaller is 4 is 9 is 100 One definition of Phi (the golden section number) is that to square it you just add 1 or, in mathematics: Phi = Phi + 1 In fact, there are

4. LycosZone Directory > Homework > Math > Geometry And Trigonometry > Fibonacci Nu
   A list of the first 100 fibonacci numbers, with factorization for each of them. Grade Level 68, or you can Check out other geometry and Trigonometry  
   http://www.lycoszone.com/dir/Homework/Math/Geometry and Trigonometry/Fibonacci N

5. Fibonacci Numbers, The Golden Section And The Golden String
   A site about fibonacci numbers in nature, art, geometry, architecture and music.  
   http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
   
   Extractions: a sequence of 0s and 1s which is closely related to the Fibonacci numbers and the golden section. There is a large amount of information at this site (more than 200 pages if it was printed), so if all you want is a quick introduction then the first link takes you to an introductory page on the Fibonacci numbers and where they appear in Nature. The rest of this page is a brief introduction to all the web pages at this site on 10 February 2003 Fibonacci Numbers and Nature 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. All involve the Fibonacci numbers - and here's how and why.

6. Pi Fibonacci Numbers
   geometry of arctan(1/8) = arctan(1/13) + arctan(1/21). One can represent as the sumof an arbitrary number of terms involving fibonacci numbers by continuing in  
   http://www.geom.umn.edu/~huberty/math5337/groupe/fibonacci.html

7. Fibonacci
   the introduction of the fibonacci numbers and the fibonacci sequence for which fibonacci is best contains a large collection of geometry problems arranged into eight chapters with  
   http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html
   
   Extractions: Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller. As stated in [1]:- Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was? Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father, recognising and the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

8. E-z Geometry Project Topics
   Native American geometry. fibonacci Music; Access Indiana Math in Music; fibonacci Music Mayan Math Mayan numbers - Washington; Mayan Math - Math Forum Fair.  
   http://www.e-zgeometry.com/links/plinks.htm
   
   Extractions: e-zgeometry Project Topics A to F G to M N to R S to Z Binary Numbers: Go to Top Cartography: Go to Top Centers of Triangles: Go to Top Constructions: Go to Top Cyrptography: Go to Top Fermat's Last Theorem: Go to Top Fibonacci Numbers: Go to Top Figurative Numbers: Go to Top Four Color Theorem: Go to Top Fractals: Cool Math - Fractal Gallery Earl's Fractal Art Gallery Fract-Ed Fractal FAQ ... Dr. Math - Fractals

9. The Math Forum - Math Library - Golden Ratio/Fibonacci
   history; Phi to 2000 decimal places; Phi and the fibonacci numbers Another definition alsobe called the Great Golden Pyramid, because its geometry is that  
   http://mathforum.org/library/topics/golden_ratio/
   
   Extractions: Information about the Fibonacci series, including a brief biography of Fibonacci, the numerical properties of the series, and the ways it is manifested in nature. Fibonacci numbers are closely related to the golden ratio (also known as the golden mean, golden number, golden section) and golden string. Includes: geometric applications of the golden ratio; Fibonacci puzzles; the Fibonacci rabbit binary sequence; the golden section in art, architecture, and music; using Fibonacci bases to represent integers; Fibonacci Forgeries (or "Fibonacci Fibs"); Lucas Numbers; a list of Fibonacci and Phi Formulae; references; and ways to use Fibonacci numbers to calculate the golden ratio. more>> The Fibonacci Series - Matt Anderson, Jeffrey Frazier, and Kris Popendorf; ThinkQuest 1999 Aesthetics, dynamic symmetry, equations, the Divine Proportion, the Fibonacci sequence, the Golden rectangle, logarithmic spirals, formulas, links to other MathSoft pages mentioning the Golden Mean, and print references. Also available as MathML more>> Golden Ratio, Fibonacci Sequence - Math Forum, Ask Dr. Math FAQ

10. Fibonacci Numbers & The Golden Ratio Link Web Page - Recommended Reading
    JeanBernard Condat. The Golden Ratio and fibonacci numbers Richard A. Dunlap.Golden Section HE Huntley. The geometry of Art and Life Matila Costiescu Ghyka.  
    http://pw1.netcom.com/~merrills/books.html
    
    Extractions: Divine Proportion Luca Pacioli The Four Books on Architecture Andrea Palladio, Robert Tavernor (Translator), Richard Schofield (Translator), Richard Scholfield (Translator) Four Books of Architecture Andrea Palladio, Adolpf K. Placzek (Designer), Isaac Ware (Editor) Palladio James S. Ackerman The Palladian Ideal Joseph Rykwert, Roberto Schezen Andrea Palladio: The Architect in His Time Bruce Boucher, Paolo Marton The Mathematics of the Ideal Villa and Other Essays Colin Rowe The Villa: From Ancient to Modern Joseph Rykwert, Roberto Schezen Great Villas of the Riviera Shirley Johnston, Roberto Schezen Ten Books on Architecture Vitruvius Pollio, Morris H. Morgan (Translator) The Seven Lamps of Architecture John Ruskin Elements of Dynamic Symmetry Jay Hambidge Architecture and Geometry in the Age of the Baroque George L. Hersey The Baroque: Architecture, Sculpture, Painting Rolf Toman (Editor), Achim Bednorz (Photographer) Perceptible Processes: Minimalism and the Baroque Claudia Swan (Editor), Jonathan Sheffer, Paolo Berdini (Contributor)

11. Fibonacci Numbers & The Golden Ratio Link Web Page
    A long list of links to pages about fibonacci and his numbers, the golden ratio and applications in Category Science Math Recreations Specific numbers presentation of the relationship between the Golden Ratio and the fibonacci numbers. straightforward analysis of the mathematics and geometry deriving phi.  
    http://pw1.netcom.com/~merrills/fibphi.html
    
    Extractions: Getting Started The Life and Numbers of Fibonacci Brief history and a quick walk through the concepts, this web site addresses the basic and more advanced issues elegantly and concisely. Written by Dr. Ron Knott and D. A. Quinney. Who was Fibonacci? Dr. Ron Knott's excellent resources at our disposal again, describing the man and his contributions to mathemtatics. Also be sure to visit his other pages, specifically his Fibonacci Numbers and the Golden Section page. Relation between the Fibonacci Sequence and the Golden Ratio Dr. Math's discussion of the Golden Ratio, Rectangle and Fibonacci sequence. Simple layout and concise graphics aid the initial learning experience. Ask Dr. Math Another Dr. Math web site, this one containing all the questions gathered pertaining to Fibonacci and Golden Ratio. Rabbit Numerical Series Ed Stephen's page has some cute rabbits and quickly describes the derivation of the Fibonacci Series and Golden Ratio.

12. NEW VISUAL PERSPECTIVES ON FIBONACCI NUMBERS
    fibonacci sequences and the wellknown fibonacci numbers. various generalizationsof the fibonacci recurrence relation rudiments of algebra and geometry, so the  
    http://www.wspc.com/books/mathematics/5061.html
    
    Extractions: This book covers new ground on Fibonacci sequences and the well-known Fibonacci numbers. It will appeal to research mathematicians wishing to advance the new ideas themselves, and to recreational mathematicians, who will enjoy the various visual approaches and the problems inherent in them. There is a continuing emphasis on diagrams, both geometric and combinatorial, which helps to tie disparate topics together, weaving around the unifying themes of the golden mean and various generalizations of the Fibonacci recurrence relation. Very little prior mathematical knowledge is assumed, other than the rudiments of algebra and geometry, so the book may be used as a source of enrichment material and project work for college students. A chapter on games using goldpoint tiles is included at the end, and it can provide much material for stimulating mathematical activities involving geometric puzzles of a combinatoric nature.

13. Fibonacci Number -- From MathWorld
    Quart. 1, 60, 1963. Brousseau, A. fibonacci numbers and geometry. Fib. Quart.10, 303318, 1972. Clark, D. Solution to Problem 10262. Amer. Math.  
    http://mathworld.wolfram.com/FibonacciNumber.html
    
    Extractions: for n = 3, 4, ..., with . The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, ... (Sloane's ). As a result of the definition (1), it is conventional to define . Fibonacci numbers are implemented in Mathematica as Fibonacci n The Fibonacci numbers give the number of pairs of rabbits n months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa in his book Liber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61-62 and 65). The ratios of successive Fibonacci numbers approaches the golden ratio as n approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratios of alternate Fibonacci numbers are given by the convergents to , where is the golden ratio , and are said to measure the fraction of a turn between successive leaves on the stalk of a plant ( phyllotaxis ): 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called

14. Phyllotaxis -- From MathWorld
    have 34, 55, or 89 petalsall fibonacci numbers H. and Guy, R. K. Phyllotaxis. In The Book of numbers. 11 in Introduction to geometry, 2nd ed. New York Wiley  
    http://mathworld.wolfram.com/P/Phyllotaxis.html
    
    Extractions: The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclockwise. Surprisingly, these numbers are consecutive Fibonacci numbers . The ratios of alternate Fibonacci numbers are given by the convergents to , where is the golden ratio , and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). A similar phenomenon occurs for daisies , pineapples, pinecones, cauliflowers, and so on. Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals; corn marigolds have 13 petals; asters have 21 petals; and daisies have 34, 55, or 89 petalsall Fibonacci numbers Daisy Fibonacci Number Spiral

15. Lesson Planet - Math,Geometry,Fibonacci Sequences Lesson Plans
    Home/Math/geometry fibonacci Sequences (2). Lesson Plans (12 of 2) 1. The fibonaccinumbers and the Golden Section - The fibonacci numbers and the Golden  
    http://lessonplanet.teacherwebtools.com/search/Math/Geometry/Fibonacci_Sequences
    
    Extractions: Grade K - 2 higher ed Search from over 20,000 online lesson plans by keyword and grade! Membership Log In User Name: Password: Apr. 07, 2003 13:50 PST Resources Other Teacher Resources Education Clip Art Grant Information Curriculum Tools Collaborative Projects ... Link To Us Teacher Discussions Click to discuss teaching topics with your peers! Pre K-6 Elem. Discussion 7-12 Sec. Discussion Ed. Tech. Discussion privacy Found lessons and other resources for ' fibonacci sequences. Web Site Books Software Maps ... Videos Find 'fibonacci sequences' books Supplies Online Courses Category matches for: ' fibonacci sequences Home/Math/Geometry Fibonacci Sequences (2) Home Math Geometry ... Fibonacci Sequences Lesson Plans (1-2 of 2): The Fibonacci numbers and the Golden Section - The Fibonacci numbers and the Golden Section Grades: Add to Learninglinks Tell a friend! AMOF : the Amazing Mathematical Object Factory (formerly known as ECOS) - This World Wide Web (WWW) site, maintained by Canada's SchoolNet, is part encyclopedia and part calculator for combinatorial objects such as sets, combinations, partitions, magic squares, and Fibonacci sequences. The Amazing Mathematical Object Factory (...

16. Geometry Reconstruction 1.
    We will recreate this amazing geometry on the drawing board and then try to Theserectangles have their equivalents expressed in fibonacci numbers, ie 55 x 144  
    http://members.optushome.com.au/fmetrol/reconstruction/construct1.html
    
    Extractions: The central core of the anthropometric model which emphasises how the curves of the lower two spirals rise up to meet with the poles of the upper two spirals is without doubt the most fascinating aspect of the geometry. We will recreate this amazing geometry on the drawing board and then try to verify it. The formula for phi ) on your calculator is = 1.618033989 ... hence the irrational numbers for the widths of the first 2 rectangles = 4.894829191 ... inches and 3.025170809... inches. On the drawing board accuracy is generally held to the Fibonacci approximate values, ie; 4.895 inches (124.3mm) and 3.025 inches (76.8mm) respectively.

17. Books On The Golden Section, Fibonacci Series, Divine Proportion And Phi
    fibonacci Fun Fascinating Activities With Intriguing numbers by Trudi foundationalresource that goes fairly deep into the mathematics and geometry of phi  
    http://goldennumber.net/books.htm
    
    Extractions: Investor Resources Books, etc. Elliott Wave International For those interested in applying Fibonacci numbers to stock market analysis, Elliott Wave International (EWI) is one of the world's largest providers of market research and technical analysis. Their staff of full-time analysts provides global market analysis via electronic online services to institutional investors 24 hours a day. EWI also provides educational services that include periodic conferences, intensive workshops, video tapes, special reports and books. Their site offers free unlimited access to club articles, a message board and tutorials in the application of Elliott Wave Theory. Pi - The Movie. That's right, the movie, and it really should have been called Phi but the producers probably felt that Pi would have appeal to a broader audience. Released in 1998, this intense and intriguing drama focuses on a mathematical genius who finds patterns in everything. If you like this site, you'll probably like this movie. Available in

18. Links To Other Sites And Information
    geometry. A Little geometry. Sacred geometry Home Page The uniquenumbers in math. Life. fibonacci - The numbers of Life - An Overview.  
    http://goldennumber.net/links.htm
    
    Extractions: and Information Dr. Ron Knott's Fibonacci Numbers and the Golden section Site - One of the best and most complete sites on the Internet on this topic, and a winner of many awards. The Museum of Harmony and Golden Section Fibonacci Quarterly Home Page - Research Magazine The life and numbers of Leonardo Fibonacci The Golden Mean Fascinating Fibonaccis - Mystery and Magic in Numbers The Golden Mean ... The Fib-Phi Link Page - A great page provided by Dawson Merrill with links to a variety of phi and Fibonacci related sites, categorized by topic with brief descriptions of each site. Relationship to Divine Creation The Evolution of Truth Sister s ite to this site) Fibonacci - The LIX Unit - In Astronomy and the Ancient Pyramids Fibonacci numbers and the Golden Section in Art, Architecture and Music

19. The Life And Numbers Of Fibonacci
    An article in PASS mathematics magazine.Category Science Math Specific numbers fibonacci numbers Phi also occurs surprisingly often in geometry. fibonacci to write about the sequencein Liber abaci may be unrealistic but the fibonacci numbers really do  
    http://plus.maths.org/issue3/fibonacci/
    
    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 If X N appears as XN then your browser does not support subscripts or superscripts. Please use this alternative version 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:

20. 20,000 Problems Under The Sea
    Number Theory fibonacci numbers Number Theory Pell numbers arrays NumberTheory sequences floor function Probability geometry dodecahedra Set  
    http://problems.math.umr.edu/keywords/g/ge.html
    
    Extractions: Home Search Tips Power Search Subject Index ... House of Math Keywords beginning with: ge Keywords are listed alphabetically. Following each keyword is a list of all classifications containing that keyword. To perform a search, select one or more checkboxes then click the "Find Problems" button. Select the maximum number of problems to be displayed per screen: generalized Fibonacci sequences

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