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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 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 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. The Golden section in Nature

2. The Fibonacci Numbers
   page is a directory of material related to the fibonacci numbers. As it is a preliminary version and The geometry Center at the University of Minnesota used to host a  
   http://math.holycross.edu/~davids/fibonacci/fibonacci.html
   
   Extractions: This page is a directory of material related to the Fibonacci numbers. As it is a preliminary version and a work in progress, please be patient. If you have questions or comments, please send me e-mail: davids@math.holycross.edu During the spring semester of the 1994-1995 academic year I taught a course called "The Fibonacci Numbers". Some materials from that course , including the syllabus and some lecture notes, are available over the Web. I will probably teach the course again in the spring semester of the 1996-1997 academic year. Bob Devaney (a Holy Cross grad) of Boston University has some marvellous material on the Fibonacci numbers' stunning appearance in the Mandelbrot set In case you want them, here are the first 500 Fibonacci numbers, in blocks of 100: I have compiled a lengthy list of factorizations of Fibonacci numbers, which I am converting to HTML as time allows. Some of the entries in the lists are hotlinks to factorizations. Many of these computations were performed with the assistance of the symbolic computation package Maple This section will be worked on during February 1996. Please check back.

3. E-z Geometry Project Topics
   fibonacci numbers and the Golden Section Surrey. fibonacci numbers. fibonacci numbers in Nature - Surrey Mona Lisa the Golden Rectangle. Some Golden geometry. The Golden Spiral  
   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

4. 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. a Golden Rectangle, however, is an interesting exercise in the geometry of the Golden Ratio, as seen at right  
   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

5. Sacred Geometry Home Page
   Sacred geometry is an ancient art and science which reveals the nature of our relationship to the cosmos. Its study unfolds the principle of oneness underlying all creation in its myriad Interestingly, 5 (Platonic) and 13 (Archimedean) are both fibonacci numbers, and 5, 12 and 13 form a perfect right  
   http://www.intent.com/sg
   
   Extractions: In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances. They are also symbolic of the underlying metaphysical principle of the inseparable relationship of the part to the whole. It is this principle of oneness underlying all geometry that permeates the architecture of all form in its myriad diversity. This principle of interconnectedness, inseparability and union provides us with a continuous reminder of our relationship to the whole, a blueprint for the mind to the sacred foundation of all things created. (charcoal sketch of a sphere by Nancy Rawles) Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. There is no point of view given greater or lesser importance, and all points on the surface are equally accessible and regarded by the center from which all originate. Atoms, cells, seeds, planets, and globular star systems all echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm. The circle is a two-dimensional shadow of the sphere which is regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere and other higher dimensional forms of these we might imagine.

6. 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: daily diet. 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.

7. The Life And Numbers Of Fibonacci
   fibonacci, famous for the EM fibonacci sequence /EM , also introduced the decimal system into Europe. Phi also occurs surprisingly often in geometry. For example, it is the ratio of the side of  
   http://pass.maths.org/issue3/fibonacci
   
   Extractions: Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 3 September 1997 Contents Features Coding theory: the first 50 years Mathematics, marriage and finding somewhere to eat Dynamic programming: an introduction Decoding a war time diary ... The life and numbers of Fibonacci Career interview Student interview - Sarah Hudson Career interview - Meteorologist Regulars Plus puzzle Pluschat Mystery mix Letters Staffroom A-Levels: a post-mortem IT and Dearing Travel bursary for conference reports News from September 1997 ... poster! September 1997 Features by R.Knott, D.A.Quinney and PASS Maths

8. The Golden Section - The Number And Its Geometry
   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: The icon means there is a Things to do investigation at the end of the section. The icon means there is an interactive calculator in this section. More 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. 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:

9. 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 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. The Golden section in Nature

10. 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

11. Geometry Forum: Winter 95 Outposts - III
    Forum Outposts. The geometry Forum Newsletter. Winter 1995, page 3. Now, about theGolden Mean. The Golden Mean is the ratio of successive fibonacci numbers.  
    http://mathforum.org/Outposts/W95p3text.html
    
    Extractions: Look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), and notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero. Again, we would want the limit of this sequence to be 1/0, but looking at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative infinity. So which would we assign to 1/0? Negative infinity or positive infinity? Instead of just assigning one of these willy nilly, we say that infinity isn't a number, and that 1/0 is indeterminate. I hope this helps. Ken "Dr." Math Dear Dr. Math: My name is Erica Anderson and I need to know examples of where the Fibonacci sequence is found in nature and how that relates to the Golden Mean. This is for Pre Calculus class. Thanks for your help. Hey Erica, I love this problem. The Fibonacci sequence happens all the time in nature, so much it is amazing. I am having trouble not just sitting here and listing all the occurrences that I can think of, but I will try to resist sending you a seven-page e-mail message. A few examples: Think about a pine cone. Have you ever noticed that the petals spiral up in two directions? Well, the number of petals it takes to get once around is almost always a Fibonacci number.

12. 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 plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). 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

13. Fibonacci Numbers
    relating the fibonacci Sequence to the Golden Mean and thereafter to Sacred geometry.In order to initiate this relationship, we divide each of the numbers in  
    http://www.halexandria.org/dward093.htm
    
    Extractions: We begin our whirlwind tour of F Lo Sophia and Sacred Geometry by first stopping in Pisa, Italy, where in the year 1202 A.D. (or as currently written, C.E. for “Current Era”), a mathematician and merchant, Leonardo da Pisa wrote a book, Liber Abaci (The Book of Computation). Born in 1179, Leo had traveled during the last years of the 12th century to Algiers with his father, who happened to be acting as consul for Pisan merchants. From the Arabs the young Leonardo Bigollo discovered the Hindu system of numerals from 1 to 9, and from the Egyptians an additive series of profound dimensions. Leo promptly shared his illumination with Europeans by writing his book and offering to the intelligentsia (the small minority who could read) an alternative to the reigning, clumsy system of Roman numerals and Greek letters. Books on mathematics are not normally among the best sellers of any era. Leo’s book, nevertheless, had the effect of convincing Europe to convert its unromantic, Romanized numeral system to the one known today as the Hindu-Arabic numeral system. Leo also introduced to the Western World what has become known as the Fibonacci Series.

14. Sacred Geometry
    the first five books of the Bible being representations of sacred geometry. Giventhat disclaimer, the amazing phenomenon of fibonacci numbers (complete with  
    http://www.halexandria.org/dward095.htm
    
    Extractions: Tradition holds that over the entrance to Plato’s Academy was written the words: Agewmetrhtoz mhdeiz eisitw Only he who is familiar with geometry shall be admitted here (Apparently a knowledge of the Greek language was also important.) This website is not quite so stringent, and will happily admit any she who is interested in entering. (The guys will, of course, still need some geometry.) Alternatively, anyone who can carry a tune may enter in a bucket or by any other means. Music is, after all, based on geometry and mathematics. For anyone who doesn’t think of themselves as being “familiar with geometry,” the words attributed to Plato’s school might be considering insulting, and thereby construed to have little or no validity. Any grapes from the discourteous must inevitably be sour. This circumstance is indeed unfortunate in that anyone who is not familiar with geometry, anyone who does not appreciate the premise that Philosophy is about understanding the meaning of a mathematical ratio (the Golden Mean ), and anyone who adheres exclusively to a literal interpretation of spiritual teachings...

15. The Life And Numbers Of Fibonacci
    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: Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 3 September 1997 Contents Features Coding theory: the first 50 years Mathematics, marriage and finding somewhere to eat Dynamic programming: an introduction Decoding a war time diary ... The life and numbers of Fibonacci Career interview Student interview - Sarah Hudson Career interview - Meteorologist Regulars Plus puzzle Pluschat Mystery mix Letters Staffroom A-Levels: a post-mortem IT and Dearing Travel bursary for conference reports News from September 1997 ... poster! September 1997 Features by R.Knott, D.A.Quinney and PASS Maths

16. Scheme Of Work For Maths Six Of The Best Cards - From Learn.co.uk
    explore the golden rectangle geometrically; appreciate the link betweennumbers and geometry via fibonacci numbers and the golden ratio;  
    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

17. The Golden Rectangle And The Golden Ratio
    I have since found the construction in geometry, by Harold R. Jacobs. Addendum 4 Each denominator is the product of two consecutive fibonacci numbers.  
    http://www.jimloy.com/geometry/golden.htm
    
    Extractions: Go to my home page click here for the alternative Golden Rectangle and Golden Ratio page This diagram shows a golden rectangle (roughly). I have divided the rectangle into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us: a/b = (a+b)/a This fraction, (a+b)/a, is called the golden ratio (or golden section or golden mean). Above I have defined the golden rectangle, and then said what the golden ratio is, in terms of the rectangle. Alternatively, I could have defined the golden ratio, using the above equation. And then a golden rectangle becomes any rectangle that exhibits this ratio. From our equation, we see that the ratio a/b=1/2+sqr(5)/2 -1/2+sqr(5)/2 or 0.61803398875 . . .) is called the golden ratio. Also, other mathematical quantities are called phi. The golden ratio is also called tau. Some people call the bigger one (1.61803398875 . . .) Phi (an uppercase phi) and the smaller one (0.61803398875 . . .) phi.

18. Jim Loy's Mathematics Page
    Dr. Ron Knott s fibonacci numbers and the Golden Section; Integer Jim s Math Squad(another Jim); Cinderella (geometry program); Geometer s Sketchpad (geometry  
    http://www.jimloy.com/math/math.htm
    
    Extractions: Go to my home page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com. "He must be a 'practical' man who can see no poetry in mathematics." - W. F. White. Dedicated to the memory of Isaac Asimov. See the top of my Science pages for comments on Dr. Asimov. My Mathematics Pages were described briefly in the Math Forum Internet News No. 5.48 (27 November 2000) My Mathematics Pages were listed on ENC Online's Digital Dozen for Sep. 2003, as one of the most educational sites on the WWW. ENC is the Eisenhower National Clearinghouse, and is concerned with science and mathematics education. My theorem: There are no uninteresting numbers. Assume that there are. Then there is a lowest uninteresting number. That would make that number very interesting. Which is a contradiction. A number of readers have objected that "numbers" in the above theorem should be "natural numbers" (non-negative integers). My reply to one reader was this: Yes, but I wanted to keep it simple and quotable. And the proof that all numbers are interesting should not be boring. From natural numbers, it can be generalized to rationals, as fractions with interesting numerators and denominators are obviously interesting. And what could be more interesting than an irrational that cannot be formed from any finite combination of rationals? I see that David Wells' book

19. Fibonacci Numbers & The Golden Ratio Link Web Page - Recommended Reading
    The Golden Ratio and fibonacci numbers Richard A. Dunlap. Number Theory in the QuadraticField With Golden Section Unit HE Huntley. The geometry of Art and Life  
    http://pw1.netcom.com/~merrills/books.html
    
    Extractions: Course in the Art of Measurement With Compass and Ruler (German) Albrecht Durer 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)

20. Fibonacci Numbers & The Golden Ratio Link Web Page
    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.

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