# Ppt on fibonacci numbers and the golden

##### Are We Golden? Investigating Mathematics in Nature

Mathematics in Nature What do our skeletons, the Parthenon, Greek statues, and the Fibonacci Sequence have in common? Do our bodies have mathematical relationships in common with nature? The Golden Ratio (Phi or the golden number) The Golden Ratio can be found: Greek Statues, urns, and artwork The Parthenon Leonardo da Vinci’s artwork All around us …. Windows, playing cards, book covers, nature, and buildings Leonardo da Vinci’s Vitruvian Man What do we/

##### Fibonacci Sequence and the Golden Ratio Robert Farquhar MA 341.

,… Has intrigued mathematicians for centuries. Shows up unexpectedly in architecture, science and nature (sunflowers & pineapples). Sunflower Seedhead and the Fibonacci Sequence Fibonacci Sequence Cont’d Has useful applications with computer programming, sorting of data, generation of random numbers, etc. Fibonacci Sequence and the Golden Ratio A remarkable property of the sequence is that the ratio between two numbers in the sequence eventually approaches theGolden Ratio” as a limit. 1/1=1 2/1=2 3/

##### HANNAH WIKUM & BRIAN LAUSCHER Pascal’s, Fibonacci’s Numbers, Algebraic Expansions & Combinations.

’s Triangle has many patterns Use Pascal’s Triangle to solve Combinations and Algebraic expansion Example: (x+y) 4 = x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4 Fibonacci Numbers are known as the natural numbers (0, 1, 1, 2, 3, 5, 8, 13, etc.) Ratio of adjacent Fibonacci Numbers equals the Golden Ratio Picture Bibliography http://recycle.lbl.gov/apac2007/Blaise_pascal.jpg http://goitaly/

##### The power of logarithmic computations Jordi Cortadella Department of Computer Science.

© Dept. CS, UPC25 Fibonacci numbers // Pre: n  0 // Returns the Fibonacci number of order n. int fib(int n) { if (n (2, 1)); A[1][1] = 0; M2x2 Fn = power(A, n - 1); // Complexity O(log n) return Fn[0][0]; } Introduction to Programming© Dept. CS, UPC26 Fibonacci numbers and golden ratio Introduction to Programming© Dept. CS, UPC27 Fibonacci numbers and golden ratio Introduction to Programming© Dept. CS, UPC28 Fibonacci numbers and golden ratio Introduction to Programming/

##### The Golden Mean The Mathematical Formula of Life Life.

13,... (add the last two to get the next) The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13,... (add the last two to get the next) The golden ratio and Fibonacci numbers relate in such that sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci numbers. The golden ratio and Fibonacci numbers relate in such that sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci numbers.shell Have/

#####  Objectives: 1.) To express numbers with exponents 2.) To evaluate expressions with exponents.

://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#golden  Technology ◦ Nanometer ◦ Megabyte ◦ Gigabyte ◦ Terabyte 555 2 x x x 3a 2 b y 1 = y 2,345 1 = ? 2626 64 2525 2424 2323 2 4 2121 2020 3636 729 3535 243 3434 3 3232 9 3131 3030 Copy and fill in the table. Do you find a pattern? How many times/

##### The Golden Ratio By :Marissa Murphy. What is The Golden Ratio? The Golden ratio which is represented by the Greek letter phi is a special number approximately.

:Marissa Murphy What is The Golden Ratio? The Golden ratio which is represented by the Greek letter phi is a special number approximately equal to 1.618. The golden ratio appears in geometry, art, architecture, and in nature etc. The Golden ratio is said to be pleasing to the eye. To understand what the golden ratio is you have to understand the Fibonacci sequence. The Fibonacci sequence is the sequence of numbers 0, 1, 1, 2, 3, 5, 8, 13/

##### Dr Nazir A. Zafar Advanced Algorithms Analysis and Design Advanced Algorithms Analysis and Design By Dr. Nazir Ahmad Zafar.

used as inputs in Euclid’s algorithm. As n approaches infinity, the ratio F(n+1)/F(n) approaches the golden ratio:  =1.6180339887498948482... Applications of Fibonacci Sequences Dr Nazir A. Zafar Advanced Algorithms Analysis and Design Fibonacci sequences The Greeks felt that rectangles whose sides are in the golden ratio are most pleasing The Fibonacci number F(n+1) gives the number of ways for 2 x 1 dominoes to cover a 2/

##### THE GOLDEN RATIO AS A NEW FIELD OF ARTIFICIAL INTELLIGENCE - THE PROPOSAL AND JUSTIFICATION Ilija TANACKOV, Jovan TEPIĆ University of Novi Sad, Faculty.

relations between Fibonacci and Lucas numbers with hyperbolic functions, as well as the function of the golden section function in the probabilistic golden section of the elementary exponential distribution, intuitively leads us to the connection between these two significant constants. One of the possible relations is defined by this theorem. Theorem: For sufficiently large n and ch Lucas numbers in continuous domain, exponential relationship between consecutive ch Lucas numbers and the ratio of/

##### The Da Vinci Code: Use of Fibonacci Sequences, Golden Ratio and Cryptography A Nicolet College Library Literature Lecture Series presentation by Gary Britton.

such segments. The number is called the Golden Ratio and is 1.618033989…. or approximately 1.62 as in the diagram. Φ The divine proportion is denoted by the Greek letter phi. (though Dan Brown chooses not to use the symbol for any of his frequent references to the number) The exact value is (1+√5)/2. Also called the golden ratio or golden section. The ratio of successive numbers in the Fibonacci sequence approaches Φ. Φ (continued) In the book Langdon recalls/

##### The Mathematics of Phi By Geoff Byron, Tyler Galbraith, and Richard Kim It’s a “phi-nomenon!”

to as the golden ratio, divine proportion, golden mean, and golden section. American mathematician Mark Barr first used the Greek letter phi to designate the proportion Reasons for choosing Phi: Phi is the first letter of Phidias, who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci. Phi is also the 21st letter of the Greek alphabet, and 21 is one of the numbers in the Fibonacci series/

##### Making Predictions Assume our class has 25 people, and the entire Sophomore class has 100 people –therefore it is 4 times larger. Predict that next Monday,

later, but it is interesting to note that the Fibonacci sequence can create the golden ratio Fibonacci and The Golden Ratio Here is a short series of Fibonacci numbers: To calculate the golden ratio, you divide the number you want by the number before it in the sequence – for example 8 is divided by 5 ( = 1.6) The higher you go, the closer you get to the exact golden ratio Try this on the calculator 0 1Undefined 11 22 31.5 51/

##### The Golden Mean The Mathematical Formula of Life Life.

the last two to get the next) The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13,... (add the last two to get the next) The golden ratio and Fibonacci numbers relate in such that sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci numbers. The golden ratio and Fibonacci numbers relate in such that sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements, all involve the Fibonacci numbers.shell A M B The/

##### GOLDEN MEAN AUKSO PJŪVIS. Definition of the Golden Rectangle The Golden Rectangle is a rectangle that can be split into a square and a rectangle similar.

http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature Fibonacci numbers In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. http://io9.com/5768696/the-fibonacci-series- when-math-turns/

##### Golden ratio Group members: Eisha khattak Novera ijaz Bakhtawar adnan Mashyam shafeeq Group leader: Arooba Ahsan Submitted to: Ma’am khaula (maths teacher)

names — the Golden mean, the Golden section, divine proportion, etc. Historically, the number can be seen in the architecture of many ancient creations, like the Great Pyramids and the Parthenon. Leonardo Fibonacci Around 1200, mathematician Leonardo Fibonacci discovered the unique properties of the Fibonacci sequence. This sequence ties directly into the Golden ratio because if you take any two successive Fibonacci numbers, their ratio is very close to the Golden ratio. As the numbers get higher, the ratio/

##### From Counting to Pascal: A Journey through Number Theory, Geometry, and Calculus NYS Master Teachers March 9, 2015 Dave Brown – Slides.

Will Fibonacci return? Algebraic Implications Try these! Compute the numbers How do these relate to the Golden Ratio? Algebraic Implications Prove that the continued square root (infinite surd) also holds: Trigonometric Implications How would you get this one?! When you see the number 5, what shape do you think of? Trigonometric Implications Compute the lengths of the segments L and M. Trigonometric Implications HINT: Consider the center triangle, ABC. Also, consider the smaller/

##### INTRODUCTION TO THE GOLDEN MEAN … and the Fibonacci Sequence.

. So what is the Golden Ratio?  The golden ratio is 1 to 1.618034.  The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13,... (add the last two numbers to get the next)  The GOLDEN RATIO and FIBONACCI NUMBERS  sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements all involve the Fibonacci numbers.shell The Golden Mean also known as…  The Golden Ratio  The Golden Section  The Golden Rectangle  The Golden NumberThe Golden Spiral  The Divine Proportion Wait… this/

##### Fibonacci The Fibonacci Sequence The Golden Ratio.

Fibonacci The Fibonacci Sequence The Golden Ratio He Loves Me? After picking off those petals, what do you have left? Now try counting the number of spirals Who is Fibonacci? Leonardo of Pisa, or more commonly, Fibonacci Italian mathematician who lived during the 12 th century Spread the use of the hindu- arabic numbers in Europe Used the Fibonacci sequence in his book Liber Abaci And the Sequence? The first two terms of the sequence are 1 and 1 Each succeeding/

##### IBD MEETUP/NORTHRIDGE

relies on this innate proportion to maintain balance, but the financial markets also seem to conform to this golden ratio.‘ USE OF FIBONACCI #S IN TECHNICAL ANALYSIS Fibonacci numbers are commonly used in Technical Analysis with or without a knowledge of Elliot wave analysis to determine potential support, resistance, and price objectives. 38.2% retracements usually imply that the prior trend will continue, 61.8% retracements imply/

##### 6.1 Golden Section 6.2 More about Exponential and Logarithmic Functions 6.3 Nine-Point Circle Contents 6 Further Applications (1)

(61.8%) of the song, as opposed to the middle or end of the song. Note that the numbers 1,2,3,5,8,13 are consecutive terms of the Fibonacci sequence. Fig. 6.14 Further Applications (1) 6 Content P.12 6.1 Golden Section (b) In Nature Number of petals in a flower is often one of the Fibonacci numbers such as 1, 3, 5, 8, 13 and 21. Further Applications/

##### THE FIBONOCCI SEQUENCE IN REAL LIFE BY ANNE-MARIE PIETERSMA, HARRY BUI, QUINN CASHELL, AND KWANGGEUN HAN.

– 1) + F (n – 2) FIBONOCCI AND THE GOLDEN RATIO When you have a square and add a square of the same size, you form a new rectangle. If you continue adding squares whose sides are the length of the longer side of the rectangle; the longer side will always be a successive Fibonacci number. Eventually the large rectangle formed will look like a Golden Rectangle - the longer you continue, the closer it will be. SIMPLE/

##### Engineering Optimization

- basic idea Start at final interval and use symmetry and maximum interval reduction: d << IN IN IN-1 = 2IN IN-2 = 3IN IN-3 = 5IN IN-4 = 8IN IN-5 = 13IN Yellow point is point that has been added in the previous iteration. Fibonacci number Sectioning – Golden Section For large N, Fibonacci fraction b converges to golden section ratio f (0.618034…): Golden section method uses this constant interval/

##### Princeton University COS 423 Theory of Algorithms Spring 2002 Kevin Wayne Fibonacci Heaps These lecture slides are adapted from CLRS, Chapter 20.

)   degree(x) for all nodes x. 42 Golden Ratio Definition. The Fibonacci sequence is: 1, 2, 3, 5, 8, 13, 21,... Definition. The golden ratio  = (1 +  5) / 2 = 1.618… n Divide a rectangle into a square and smaller rectangle such that the smaller rectangle has the same ratio as original one. Parthenon, Athens Greece 43 Fibonacci Facts 44 Fibonacci Numbers and Nature Pinecone Cauliflower 45 Fibonacci Proofs Fact F1. F k   k/

##### THE GOLDEN RATIO: IT’S EASY! Do you have any problem in understanding the famous Golden Ratio wich troubles young students all over the world? A group.

. ALL FIBONACCI NUMBERS. 2) THE RATIO OF THE FOREARM TO HAND OUR HAND CREATES A GOLDEN SECTION IN RELATION TO YOUR ARM, AS THE RATIO OF YOUR FOREARM TO YOUR HAND IS ALSO 1.618, THE DIVINE PROPORTION. 3) YOUR FEET THE FOOT HAS SEVERAL PROPORTIONS BASED ON PHI LINES, INCLUDING: 1- THE MIDDLE OF THE ARCH OF THE FOOT. 2- THE WIDEST PART OF THE FOOT. 3- THE BASE OF THE TOE LINE AND BIG/

##### 1 Presented at Central University of Finance and Economics 中央财经大学 Beijing by 卜若柏 Robert Blohm Chinese Economics and Management Academy 中国经济与管理研究院

.html 13 Early Greek Mathematics and Astronomy (cont.d)  Importance of the two double series: and the Fibonacci numbers “Of the two series which thus begin alike and then part company, the one leads to the square-root of 2 or the hypotenuse of an isosceles right-angled triangle, and the other leads to the Divine or Golden Section. These are the two famous surds or irrational numbers of antiquity, and they are also the two pillars of Euclidian/

##### GOLDEN RATIO GOLDEN SECTION FIBONACCI NUMBERS 1, 1, 2, 3, 5, 8, 13….. The ratio of any consecutive numbers is the golden ratio A pattern found in nature.

with lengths of Fibonacci numbers The spiral made is found in nature GOLDEN SPIRAL MODERN USES TWITTER WHERE IS IT FOUND? Leonardo da Vinci Leonardos famous Mona Lisa reflects the artist’s use of the Golden Section. * The rectangle around her face represents a Golden rectangle. * If you subdivide the rectangle at the eyes the vertical side of the rectangle is divided by the golden ratio. Even in the time of the ancient Greeks, the golden rectangle was considered/

##### From Baeyer Strain Theory to the Golden Section Title Adolph von Baeyer (1835 - 1917)

, 89, 144, 233, 377, 610, 987, 1597,2584, 4181, 6765, 10946… Leonardo Pisano Fibonacci (~1170-1250) mouse over Fibonacci Series A series of numbers in which each number is the sum of the two preceding numbers. Fibonacci Series 2 Fibonacci Series a/b = smaller/larger number b/a = larger/smaller number The Golden Section (Phi) is the limit of the ratio b/a. Fibonacci Spiral Fibonacci Spiral and the Golden Rectangle The sunflower mouse over Leonardo’s Mona Lisa mouse over/

##### The Golden Mean The Mathematical Formula of Life

.618—that has many names. Most often we call it the Golden Section, Golden Ratio, or Golden Mean, but it’s also occasionally referred to as the Golden Number, Divine Proportion, Golden Proportion, Fibonacci Number, and Phi. Have You Seen This? Note that each new square has a side which is as long as the sum of the latest two squares sides. The Golden Rectangle The golden ratio is typically depicted as a single large rectangle formed/

##### Some arithmetic problems raised by rabbits, cows and the Da Vinci Code

/lavande.html François Rouvière (Nice) The Golden Number and aesthetics The architectural theory of Luca Pacioli: De divina proportione The Golden Number and aesthetics The architectural theory of Luca Pacioli: De divina proportione Marcus Vitruvius Pollis (Vitruve, 88-26 av. J.C.) Léonard de Vinci (Leonardo da Vinci, 1452-1519) The architectural theory of Luca Pacioli: De divina proportione Music and the Fibonacci sequence Dufay, XVème siècle Roland de/

##### Is the Ratio of Development and Recapitulation Length to Exposition Length in Mozart’s and Haydn’s Work Equal to the Golden Ratio? Ananda Jayawardhana.

form. Introducing a regression model, the author carryout a statistical analysis of possible golden ratio forms in the musical works of Mozart. He also include the master composer Haydn (1732-1809) in his study. Part I Probability and Statistics Related Work Fibonacci (1170-1250) Numbers and the Golden Ratio Golden Ratio http://en.wikipedia.org/wiki/Golden_ratio Construction of the Golden Ratio http://en.wikipedia.org/wiki/Golden_ratio Fibonacci Numbers and the Golden Ratio 1, 1, 2, 3/

##### An excursion through mathematics and its history.

. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The book of calculations. D.The book of sums. E.Peter Rabbit and family. HQ5. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The/

##### 9/14/2015Assoc. Prof. Stoyan Bonev1 COS220 Concepts of PLs AUBG, COS dept Lecture 08 Iteration and Recursion in Procedure Oriented Programming Reference:

/2015Assoc. Prof. Stoyan Bonev38 Fibonacci series 1, 1, 2, 3, 5,... Fibonacci (1170–1250) discovered the numerical series now named after him, which is closely connected to golden ratio (1.618033989).Fibonacci At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio— believing this proportion to/

##### The Golden Ratio. Background Look at this sequence… 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... Can you tell how it was created? SStart with the numbers.

with the numbers 1 and 1. TTo get the next number add the previous two numbers together. Do you recognize this sequence of numbers? It is called the Fibonaccis sequence discovered by a man called none other than Leonardo Fibonacci. What is the Golden Ratio? The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Lets look at some of these ratios: What is the Golden Ratio/

##### Fibonacci Sequence A Mathematics Webquest

was used by Robert Langdon to retrieve the safety deposit box? Find another movie or book that references the Fibonacci numbers. Process d. Art and Architecture… The Parthenon in Greece was built using the Golden Ratio…the ratio of any two consecutive Fibonacci numbers (1:1.618) ‏ Menu ACTION ITEM: Find two other places you can find the Fibonacci Sequence and/or the Golden Ratio in Architecture and Art. Evaluation The rubric for this webquest can be found/

##### Aromaticity, the DaVinci Code and the Golden Section Title.

, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946… Leonardo Pisano Fibonacci (~1170-1250) mouse over Fibonacci Series A series of numbers in which each number is the sum of the two preceding numbers. Fibonacci Series 2 Fibonacci Series a/b = smaller/larger number b/a = larger/smaller number The Golden Section (Phi) is the limit of the ratio b/a. Fibonacci Spiral Fibonacci Spiral and the Golden Rectangle The sunflower Leonardo’s Mona Lisa/

##### SPECIAL KOLAMS The Generalized Fibonacci Numbers permit a wide choice for the rectangles that go into the square designs. With the standard “Fibonacci.

4 11 15) Kolam (13 4 17 21) Design Since 21 is a Fibonacci Number, the canonical quartet code for 21 2 is (5 8 13 21), TheGolden rectangles” are 8 x 13, two consecutive Fibonacci Numbers. But here we consider 21 as a “Generalized Fibonacci Numberand the quartet code is (13 4 17 21). The 21 2 is made up of a 13 2 surrounded by four rectangles 4/

##### Excursions in Modern Mathematics, 7e: 9.4 - 2Copyright © 2010 Pearson Education, Inc. 9 The Mathematics of Spiral Growth 9.1Fibonacci’s Rabbits 9.2Fibonacci.

: 9.4 - 31Copyright © 2010 Pearson Education, Inc. This rectangle has l = 8 and s = 5. This is a Fibonacci rectangle, since 5 and 8 Example 9.8Golden and Almost Golden Rectangles are consecutive Fibonacci numbers. The ratio of the sides is l/s = 8/5 = 1.6 so this is not a golden rectangle. On the other hand, the ratio 1.6 is reasonably close to so we will think of this/

##### Excursions in Modern Mathematics, 7e: 9.5 - 2Copyright © 2010 Pearson Education, Inc. 9 The Mathematics of Spiral Growth 9.1Fibonacci’s Rabbits 9.2Fibonacci.

spiral. Example 9.12Spiral Gnomonic Growth Excursions in Modern Mathematics, 7e: 9.5 - 22Copyright © 2010 Pearson Education, Inc. More complex examples of gnomonic growth occur in sunflowers, daisies, pineapples, pinecones, and so on. Here, the rules that govern growth are somewhat more involved, but Fibonacci numbers and the golden ratio once again play a prominent role. Complex Gnomonic Growth

##### Patterns in Nature. Mathematics….& patterns We don’t know all the answers unlike in class! Mathematics is a science which looks for patterns and structure.

middle ages Born in Pisa, Italy, the city with the famous Leaning Tower,~ 1175 AD Major contributions in arithmetic, algebra and number theory Decimal system Nature and Fibonacci White calla lily Nature and Fibonacci Euphorbia Nature and Fibonacci trillium Nature and Fibonacci Black eyed susan More Fibonacci Pinecones and pineapples… Count the number of spirals. Spirals in a pine cone: clockwise and anti-clockwise And more Spirals Golden Ratio Compute the ratio of Fibonacci numbers: 2 ÷ 1 = 3 ÷ 2 = 5 ÷ 3/

##### Do Now: Write a similarity ratio to answer the question. If you have a vision problem, a magnification system can help you read. You choose a level of.

this, sheets of paper and blank canvases are often somewhat close to being golden rectangles. 8.5x11 is not particularly close to a golden rectangle, by the way. The golden ratio is seen in some surprising areas of mathematics. The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13..., each number being the sum of the previous two numbers) approaches the golden ratio, as the sequence gets infinitely long. The sequence is sometimes defined/

##### Introduction to Fibonacci number In mathematics, the Fibonacci numbers are the numbers in the following integer sequence: By definition, the first two.

, 2, 3, 5, 8, 13, 21, and 34. See golden spiral.golden spiral Occurrences in mathematics The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal’s Triangle The Fibonacci numbers can be found in different ways in the sequence of binary strings. The number of binary strings of length n without consecutive 1s is the Fibonacci number F n+2. For example, out of the 16 binary strings of length 4, there/

##### Who was Fibonacci ? Greatest European mathematician of the middle ages Born in Pisa, Italy, the city with the famous Leaning Tower,~ 1175 AD Major contributions.

calla lily Nature and Fibonacci Euphorbia Nature and Fibonacci trillium Nature and Fibonacci Black eyed susan More Fibonacci Pinecones and pineapples… Count the number of spirals. Spirals in a pine cone: clockwise and anti-clockwise And more Pascal’s Triangle Finding the n-th term of a Fibonacci Sequence Summation Formula For the first “n” numbers in the Fibonacci Sequence Fibonacci Sequence As the terms increase, the ratio between successive terms approaches 1.618 This is called thegolden ratio” –Ratio/

##### A bc two other rotations Problem: Compute the other four permutations two other reflections Further discussion: there are 24 permutations of the set {a,b,c,d},

. In addition, the number of times one has circled the stalk will be another Fibonacci number. http://people.bath.ac.uk/ajp24/goldenratio.html#The%20Golden%20Rati o%20and%20The%20Human%20Body The Golden Ratio and The Human Body Careful study of the dimensions of the human body gives numerous examples of the Golden Ratio and also the number 5, which was mentioned before as to be linked to the Golden Ratio. (See "The Golden Ratio and Shapes" - "Pentagon")The Golden Ratio and ShapesPentagon Scientists/

##### Fibonacci Фибоначи. The Fibonacci numbers form a series in mathematics, which is defined recursively as follows: F(0) = 0 F(1) = 1 F(n) = F(n-1)

after him because it was popularized. It turns out that the greater are the numbers of the sequence of Fibonacci, the attitude of the last two numbers are close to the golden mean and border transition (with an infinite number of numbers in a row) becomes equivalent to the golden mean. Often the sequence of Fibonacci is also associated with the following task: A pair of rabbits (males and females) can produce per unit time (eg one month) a/

##### NAME: MUQSIT HAIDER 7T GOLDEN RATIO MIRACLE OF KAABA.

the golden ratio, can see the earth from the space and knows the measurements that are accepted in the future, knows math very well, God. It is the mean point of earth. METHAMETICAL PROOF? Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and: More generally: where above, the/

##### The Golden Mean By Susan Convery Foltz Broward College EPI 003 Technology February 8, 2009.

steps, especially living things, but also in art and architecture. How to Calculate the Golden Mean: The Golden Mean is the ratio most pleasing to the eye. The Golden Mean is the ratio most pleasing to the eye. The ratio is the same as the ratio of the number 1 to Phi Φ (1.6180339887) The ratio is the same as the ratio of the number 1 to Phi Φ (1.6180339887) A Golden Rectangle Fibonacci Numbers and the Golden Mean The 11th century Mathematician, Leonardo of Pisa, derived a/

##### Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman.

II Cross multiplication yields: The Golden Quadratic III Setting Φ equal to the quotient a/b and manipulating this equation shows that Φ satisfies the quadratic equation: The Golden Quadratic IV Applying the quadratic formula to this simple equation and taking Φ to be the positive solution yields: Properties of Φ Φ is irrational Its reciprocal, 1/ Φ = Φ - 1 Its square, Φ 2 = Φ + 1 Φ Is an Infinite Square Root Φ is an Infinite Continued Fraction Φ and the Fibonacci Numbers The Fibonacci numbers {f n } are/

##### MovementandExploration Lessons 3-4. Scientists and Mathematicians…. who have inspired, communicated, and transformed their creativity to change our world.

8 9 0 Order of ordinals matters My exploration of numbers showed patterns in nature. In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. Fibonaccis Golden Ratio Fibonnacii Click the link http://www.world-mysteries.com/sci_17_hand.gif Fibonaccis Golden Ratio Fib Sunflowers Bracts of a pinecone Petals of a/

##### Fibonacci Numbers and Binet Formula (An Introduction to Number Theory)

φ φ=1+1/φ φ2=φ+1 quadratic equation φ=(1+sqrt(5))/2 (only the positive answer) φ=1.618033989... Golden Ratio and practical application most famous and controversial in history - human aesthetics Converting between km and miles 1 mile= 1.6093 km 13 km = 8 miles Fibonacci (1,1,2,3,5,8,13,21...) OK, using Fibonacci numbers, how many miles are in 50 kilometers?? (show your work) Binet Formula/

##### Da Vinci code Can you crack the complex code? complex code?

Da Vinci code Can you crack the complex code? complex code? Fibonacci – Magic numbers in design Golden Ratio is φ (phi) φ = 1 + = 1.6180339 φ = ½(√5+1) = 1.62 (2dp) 1φ Every side in the pentagram is φ times another side. Alpha-Numeric codes Line up your code strips and read off the letters a b c d e f g h i j k l m n o p q/