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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

Fibonacci Flim-Flam.
The patterns based on the fibonacci numbers, the golden ratio and the golden But part of that rule set is the underlying geometry of the playing field,
http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm

Extractions: f Fibonacci Flim-Flam j by Donald E. Simanek Fibonacci Leonardo of Pisa (1170-1250), nickname Fibonacci, was born in Pisa, Italy. He made many contributions to mathematics, but is best known by laypersons for the sequence of numbers that carries his name: This sequence is constructed by choosing the first two numbers (the "seeds" of the sequence) then assigning the rest by the rule that each number be the sum of the two preceding numbers. This simple rule generates a sequence of numbers having many surprising properties, of which we list but a few: This is but one example of many sequences with simple recursion relations.
THE FIBONACCI CONCEPT OF CREATION NUMBERS
Not knowing much about fibonacci except for the first few numbers, I was forcedto go on a web SACRED geometry GIZA PYRAMID COMPARED TO THE HUMAN FORM
http://www.greatdreams.com/fibonaci.htm

Extractions: THE FIBONACCI CONCEPT OF CREATION NUMBERS Leonardo Pisano Fibonacci Born: 1170 in Pisa, Italy Died: 1250 in Pisa, Italy This page is the result of a series of dreams I had about the fibonacci concept of numbers. In the first dream, I was shown a picture of space. It was purple and full of stars, and amongst the stars was a golden line that went through the space like a fibonacci spiral. In dream two, I was sewing on an ear with short diagonal lines in a spiral shape. In a previous vision earlier this week, I had been told that we are all 'color, sound, and light' so that rather fit with it. Following this dream, I saw a web page search box. I put the word 'spiral' into it and the first page that came up was titled "TIMELINE". Not knowing much about Fibonacci except for the first few numbers, I was forced to go on a web search for the word. This, once again proves to me, that we can learn a lot from dreams. Below you will find some magnificent links to explain what this concept is. ART OF SRI YANTRA DIVISION IN MEAN AND EXTREME RATIO FIBONACCI LIFE AND TIMES FIBONACCI - NUMBERS AND NATURE ... FIBONACCI SPIRAL AT GIZA According to Edgar Cayce, the famous psychic, the opening to the Halls of Records, which hold the history of the Earth, will be found in the right shoulder of the Sphinx. This has been clearly marked geometrically. Looking at the figure below, if you bisect the golden mean rectangle that fits around the spiral at the Giza plateau, it passes exactly through the headdress of the Sphinx. Also, a line extended from the southern face of the middle pyramid and the line that bisects the golden mean rectangle, forms a cross that marks a very specific spot on the right shoulder of the Sphinx.
NEW VISUAL PERSPECTIVES ON FIBONACCI NUMBERS
NEW VISUAL PERSPECTIVES ON fibonacci numbers. other than the rudiments ofalgebra and geometry, so the book may be used as a source of enrichment
http://www.worldscibooks.com/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.
Geometry And Algebra Of The Divine Proportion
In an analogous way, each number in the fibonacci series is the sum of the 1/(f^2) approximates to 3/8, two numbers belonging to the fibonacci series.
http://www.dace.co.uk/proportion_phi.htm

Extractions: The snail must grow in such a way that its body remains more or less the same size in relation to its shell, otherwise the shell would become too heavy to drag around. Also the snail has to fit its body into the big end of the shell. Thus the most recent addition to the shell will maintain a more or less constant relationship with all the shell that has gone before. In an analogous way, each number in the Fibonacci series is the sum of the previous two numbers, and the side of each square in our diagrams is the sum of the sides of the previous two squares. The relationship between any number in the Fibonacci series and the one immediately preceding it can be expressed as a ratio, which you can work out as a decimal on a calculator. Thus we have 1/1, 2/1, 3/2, 5/3, 8/5 ... etc. (for example, 5/3=1.666..., and 8/5=1.6). If you work them out, you will find that these ratios jump back and forth on either side of a hidden number, getting closer and closer to it.
Appendix 4: Programs Of The Courses
geometry of the Golden Section. Pentagon. Decagon. Golden triangles. Golden rhombuses . Vorob ev, NN fibonacci numbers. Moscow, Publisher Nauka , 1978.
http://www.goldenmuseum.com/2101ResolutionAp4_engl.html

Extractions: Appendix 4 Program of the course "Mathematics of the Golden Section" Introduction into Pascal triangle. Pascal triangle and its properties. A little of history. Pascal operation. Binomial coefficients. A number of parts of the given set. Connection with factorials. Generalized Pascal triangles and generalized binomial coefficients. Pascal pyramids and trinomial coefficients. Multinomial coefficients and Pascal hyperpiramids. Introduction to Fibonacci numbers. Fibonacci numbers in the Pascal triangle. Lucas and Catalan numbers. A little of history. Relationships for Fibonacci and Lucas numbers. Application of Fibonacci numbers. Generalized Fibonacci numbers and their algebraic properties. Golden Section. Connection of Fibonacci numbers to the Golden Section. Algebraic properties of the Golden Section. Geometry of the Golden Section. Pentagon. Decagon. Golden triangles. Golden rhombuses. Platonic solids. Golden brick. Golden masonry. Generalized Golden Sections. Fibonacci matrices. The Fibonacci Two by Two Matrix, the Q -matrix, and its generalizations, the Fibonacci (
The Life And Numbers Of Fibonacci
Phi also occurs surprisingly often in geometry. The fibonacci numbers arestudied as part of number theory and have applications in the counting of
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 ... posters! September 1997 Features 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".
Golden Ratio
The explorations with a spreadsheet demonstrate fibonacci numbers and the The mathematical connections between geometry and algebra can be stressed by
http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Meredith/GoldenRatio/in

Extractions: Some Explorations with the Golden Ratio by Sue Meredith An exploration with the Golden Ratio offers opportunities to connect an understanding of ratio and proportion to geometry as well as to introduce historic and aesthetic elements to a mathematical concept. The explorations with a spreadsheet demonstrate Fibonacci numbers and the ratio between each pair. The GSP exploration demonstrates how to construct the ratio geometrically. The use of calculators to prove the definition and properties of this ratio are also appropriate. from Web Page by Ned May(nedmay@Moonstar.com) History The name Golden Ratio or Golden Number , named "phi" by the Greeks for the Greek scultor Phidias. However, this ratio can be found in art and architecture long before the Greeks. The Great Pyramid of Giza built around 2560 BC is one of the earliest examples of the use of this ratio. The length of each side of the base is 756 feet, and the height when built was 481 feet. 756/481=1.571725572. from http://ce.eng.usf.edu/pharos/wonders/Gallery/pyrami.jpg The Parthenon in Athens Greece, built in 440BC, examplifies the use of the golden rectangle in many of the dimensions. The spaces between the columns, height to width, are in proportion to the golden ratio as are most of the exterior dimensions.
EMAT 6680 Class Page Forwrite-up5
Some Golden geometry. 1. Golden Rectangle. A Golden Rectangle is a rectangle withproportions that are two consecutive numbers from the fibonacci sequence.
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Youn/EMAT6690/Essay 1/Some golen

Extractions: If we rotate the shorter side through the base angle until it touches one of the legs, and then, from the endpoint, we draw a segment down to the opposite base vertex, the original isosceles triangle is split into two golden triangles. Aslo, we can find that the ratio of the area of the taller triangle to that of the smaller triangle is also 1.618. (=Phi)
Course Notes
12.1 fibonacci numbers 12.2 Coin Change 12.3 Matrix Chain Multiplication 12.4Longest Common Substring 15 Computational geometry 15.1 Turning Directions
http://www.cse.ohio-state.edu/~gurari/course/cse693s04/cse693s042.html
Sacred Geometry And The Mayan Calendar
In his study of sacred geometry, he found that all forms and The two sequencesmeet as numbers 21 (fibonacci) and 32 (binary) in the 4 Jaguar energy
http://www.experiencefestival.com/a/Sacred_Geometry_and_the_Mayan_Calendar/id/23

Extractions: var amazon_search = 'Sacred+Geometry+and+the+Mayan+Calendar'; Articles Archives Start page News Contact Community Experience Festival World University General Newsletter Contact information Site map Most recommended Search the site Archive Photo Archive Video Archive Articles Archive ... Site map Mayan Calendar Researcher Ian Xel Lungold explains the connection between the Mayan Calendar and Sacred Geometry . One essential point is the pattern displayed by the Mayan calendar . It is built upon a particular ratio, 13 : 20. These proportions are the true basis of all sacred geometry . In his study of sacred geometry , he found that all forms and understandings boil down to one ratio: one-third to two-thirds. This is in no way an exact number system, but a general pattern that Creation naturally follows. Our consciousness is based on this ratio. Ian Xel Lungold Mayan Calendar Sacred Geometry , pattern, venus transit awakening spiritual , what is sacred geometry I have been studying the Mayan calendar since 1996. It became evident very quickly that
Read This: Fibonacci Numbers
The fourth chapter, fibonacci numbers and geometry, discuss the familiar topicsof the golden rectangle and triangle and the pentagram.
http://www.maa.org/reviews/vorobievfibonacci.html

Extractions: by Nicolai N. Vorobiev This is a wonderful book with a huge amount of information on Fibonacci numbers. First published in the 1950s for high school students enrolled in a mathematical circle at the Leningrad State University, the book only assumes knowledge of high school mathematics, but the reader will need the sophistication of a good upper-level mathematics major. The warmth and enthusiasm of the late author comes through Mircea Martin's English translation. Given the popular awareness that the best selling novel The Da Vinci Code has given to the Fibonacci numbers and the golden ratio, the book under review comes at an opportune time for the mathematician interested in learning a few new nuggets of information about this topic. The first chapter, "The Simplest Properties of Fibonacci Numbers," discusses the usual properties and identities, as well as some not-so-simple properties. For example, in the discussion of the sums of the reciprocals of the Fibonacci numbers, Vorobiev states that a closed expression for the sum is unknown, but that the sum was recently shown to be irrational. He gives a combinatorial interpretation of some Fibonacci identities using the "Bunny Problem," involving the number of ways a bunny can jump along a lane divided into cells. He does not, however, resort to combinatorial arguments very often, relying more often on induction and on results derived from previous results. Readers interested in more combinatorial proofs of Fibonacci identities should investigate
Read This: Briefly Noted, May 2003
The book does not require more knowledge than high school geometry and is not a The authors of New Visual Perspectives on fibonacci numbers define
http://www.maa.org/reviews/brief_may03.html

Extractions: Search MAA Online MAA Home John de Pillis's Mathematical Conversation Starters is a book of quotations, primarily on mathematical subjects, but also extending to mathematics' links with such matters as faith, belief, art, beauty, music and so on. It also includes a few dialogues exploring various mathematical themes. Here are some quotes I enjoyed: True science is restrictively the study of useless things. For the useful things will get studied without the aid of scientists. To employ rare minds on such work is like running a steam engine by burning diamonds. (C.S. Peirce) Logic is the art of going wrong with confidence. (Morris Kline) That student is best taught who is told the least. (R.L. Moore) Part of the charm of a book of quotations is the pleasure of coming across familiar favorite quotations and the other part is in encountering new quotations to add to one's collection of favorites. Every reader will of course have a different set for the first category, but I am sure the book will provide enough for everyone in both the categories. Undoubtedly a book to dip into again and again and to cherish. [Ramachandran Bharath] Greg N. Frederickson, a Professor of Computer Science at Purdue University, has written two books on dissections, both published by Cambridge University Press: one in 1997
Investigating The Golden Rectangle And The Fibonacci Sequence
and the fibonacci Sequence. a WebQuest for geometry and Algebra students The fibonacci numbers and the Golden Section is hosted at the Department of
http://www.scs.k12.tn.us/STT99_WQ/STT99/Cordova_HS/franklinp1/webquest_folder/Fi

Extractions: Introduction The Golden Ratio is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The Task This WebQuest is designed to lead you to connections between the Golden Ratio and the Fibonacci sequence through the use of algebraic and geometric concepts. You will be absolutely amazed at the number patterns that exist in real-world situations! You will also be asked to integrate Art,Biology, or Music into your final project: creating your own lesson plan. By the end of this WebQuest, you will know the answers to the following questions (Some of you will become experts on this topic!!): Who was Fibonacci?
Taurus: Fibonacci Numbers And The Golden Section. New Age Art. Sacred Geometry.
Inner Garden Art New Age Art, Intuitive Art, Sacred geometry, Meditation,Magical Places and more
http://innergardenart.com/taurusS.html

Extractions: When Laura Volimte paints, she does not worry so much about the exact geometric nature or sequence of the work she is creating. She follows her intuition and picks colors without over analyzing the color scheme. Laura is very carefree and allows her inner child to play while she enjoys the unfolding process. She allows the art to come from within . Laura's creativity workshops help others enter into that same creative space where the inner child is free from constraints, expectations, and the need for outer approval.
LESSON PLANET - 30,000 Lessons And 13 Lesson Plans For Fibonacci Sequences
They identify the pattern among the fibonacci numbers, the fibonacci sequenceand its unexpected relationship with the geometry of the regular pentagon
http://www.lessonplanet.com/search/Math/Geometry/Fibonacci_Sequences

Extractions: Join Lesson Planet Today! First Name: Last Name: E-mail: For only $9.95 a year, gain full access to Lesson Planet's directory of over 30,000 links to lesson plans as a Lesson Planet Silver Member! For only a year ( Back to School Special: $19.95 ), become a Gold Member and gain full access to over 30,000 links to lessons AND our TeacherWebTools suite of online tools (featuring TeacherSiteMaker, Online Storage, NewsletterMaker, LessonMaker and more!) Home Math Geometry Found 13 ' Fibonacci Sequences ' related Lesson Plans. Also for ' Fibonacci Sequences 4 Web Sites * Log in or become a Lesson Planet Member to gain access to lesson plans. Lesson Plans (1 - 10 of 13): All Science NetLinks: The Fibonacci Series - Students explore the Fibonacci series. They identify the patten among the Fibonacci numbers, look for applications of these numbers, and explore the ways that this pattern can be related to objects and shpaes in both the natural and designed world.
Activity 2
The fibonacci Sequence is a mathematical sequence of numbers constructed so thateach number Let Fi represent the ith number in the fibonacci Sequence.
http://homepage.mac.com/efithian/Geometry/Activity-02.html

Extractions: The Golden Ratio The Golden Ratio is a number that occurs in both mathematics and in nature. In this activity you will examine how this ratio occurs aesthetically, geometrically, and mathematically. Examine the five rectangles drawn below from several different views. Choose the one rectangle that is most appealing to you and place an X on it. Use a metric ruler to measure both sides of each rectangle to the nearest millimeter. For each rectangle, divide the length of the longest side by the length of the shortest side and write this ratio on the rectangle. Which rectangle was the most liked in the entire class?_ What is its ratio?_ The Golden Ratio can also be found in the dimensions of the human body. Measure your height to the nearest centimeter and record it below. Measure the distance of your navel from the floor to the nearest centimeter and record that also. Divide your height by your navel height to find the height/navel ratio rounded to two decimal places and record that below. Height = Navel Height = Ratio = The Golden Ratio also appears in the measurements of many common geometric figures. The common pentagram on the next page has the golden ratio hidden in its structure. This star is composed of a regular pentagon with an isosceles triangle on each side. Use a metric ruler to measure the lengths of the segments AB, AC, AD, and DC to the nearest millimeter. Divide to find the ratios of AB to AC; AC to AD; and AD to DC rounded to two decimal places.
Fibonacci
Like a signal locking in on the target, the fibonacci sequence homes in on Ø . Integers, or whole numbers, represent things which can be designed and
http://kjmaclean.com/Geometry/Fibonacci.html
Virtual Library Of Useful URLs - Fibonacci Numbers And The Pascal
Links to the other web site on fibonacci numbers and Pascal triangle. Sacred geometry Home Page. The Sphere, Circle, Point, Square Root of 2 and 3,
http://milan.milanovic.org/math/links/links.html

Extractions: NATURAL SCIENCES AND MATHEMATICS Fibonacci Numbers and the Golden Section A great website by Ron Knott. Article on the famous sequence and the Golden Section. Includes definitions and examples of applications... Museum of Harmony and Golden Section Professor Alexey Stakhov and his daughter Anna Sluchenkova created the idea and now this web site contains extensive pages covering all aspects of the Golden Section in culture, nature, art, math, Fibonacci, classrooms, and clubs. Text available in English and Russian.... The Fibonacci Association Official website of The Fibonacci Association. The Fibonacci Association, incoporated in 1963, focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas... Sloane's On-Line Encyclopedia of Integer Sequences On-line Encyclopedia of Integer Sequences. Enter an integer sequence, separated by commas, and it will search for the sequence in its extensive database.
Fibonacci Numbers
fibonacci numbers and the golden section in nature, art, geometry, A siteabout fibonacci numbers in nature, art, geometry, architecture and music.
http://www.ebroadcast.com.au/dir/Science/Math/Recreations/Specific_Numbers/Fibon
InterMath | Investigations | Algebra | Patterns
patterns, and geometry associated with the fibonacci numbers and golden section . The fibonacci numbers, Pineapples, Sunflowers and The Golden Mean
http://www.intermath-uga.gatech.edu/topics/algebra/patterns/r06.htm

Extractions: Search the Site Investigations Algebra Patterns ... Recommended Investigations Choose two whole numbers (number one and number two). Add them together and form a Fibonacci-like sequence (add number one and number two together to get number three, then add number two and number three together to get number four, etc.). End with a total of ten numbers. Repeat the process by starting with two different numbers.

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