# Ppt on fibonacci numbers nature

##### Lecture 4,5 Mathematical Induction and Fibonacci Sequences.

Lecture 4,5 Mathematical Induction and Fibonacci Sequences Mathematical induction is a powerful, yet straight- forward method of proving statements whose domain is a subset of the set of integers. Usually, a statement that is proven by induction is based on the set of natural numbers. This statement can often be thought of as a function of a number n, where n = 1, 2, 3/

##### Biography ( ) Fibonacci is a short for the Latin "filius Bonacci" which means "the son of Bonacci" but his full name was Leonardo of Pisa, or Leonardo.

A spiral drawn in the squares, a quarter of a circle in each square. PASCAL‘S TRIANGLE Nature One of the most fascinating things about the Fibonacci numbers is their connection to nature. the number of petals, leaves and branches spiral patterns in shells spirals of the sunflower head pineapple scales Flowers Nautilus Sun Flower Pineapple in the introduction to Europe Conclusion The greatest European/

##### Fibonacci By Andréa Rivard.

is equal to all of the rabbits who are more than two months old Fibonacci’s rabbits never die http://www.quabbinqualitypetsupplies.com/sitebuildercontent/sitebuilderpictures/rabbits.jpg Fibonacci Spiral The spiral appears in nature, often in the way leaves grow or the way seeds grow They usually have consecutive Fibonacci numbers 8 clockwise, 13 counter-clockwise, 13 counter- clockwise, 21 clockwise, etc. Minimize space and/

##### Phi, Fibonacci, and 666 Jay Dolan. Derivation of Phi “A is to B as B is to C, where A is 161.8% of B and B is 161.8% of C, and B is 61.8% of A and C is.

years Ancient Egyptian Pyramids Notre Dame Cathedral, Paris Greek Parthenon, Athens United Nations Building, New York The Fibonacci Sequence Leonardo Pisano Fibonacci discovered it in about 1202 A.D. He lived from 1170-1250 He first used the sequence to / 13, 21, 35, 56… Each number is found by adding up the two previous numbers. The Fibonacci sequence is found in nature and is still used to predict growth patterns. The sequence in Nature: Flower Petals 123 5 813 2134 Fibonacci in the Real World “A group of/

##### SPS Lecture: Dynamics of Population Growth Exponential Growth and the Fibonacci Sequence – solutions of simple linear differential and recurrence equations.

this model is unstable and leads to population explosion or decay. References Hahn, Werner, Symmetry as a Developmental Principle in Nature and Art, World Scientific, Singapore, 1998 Lawton, Wayne, Kronecker’s theorem and rational approximation of algebraic numbers, The Fibonacci Quarterly, volume 21, number 2, pages 143-146, May 1983 Thompson, D’Arcy Wentworth, Growth and Form, Vol. I and II, Cambridge University Press/

##### Leonardo Fibonacci By: Cullen Schoen. Picture of Leonardo.

Roman Numeral system. He introduced us to the bar we use in fractions, previous to this, the numerator has quotations around it. More Facts It has been said that the Fibonacci numbers are Natures numbering system and apply to the growth of living things, including cells, petals on a flower, wheat, honeycomb, pine cones and much more. He wrote many books like, Liber Abbaci/

##### He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with.

number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 0 . He was the son of Guilielmo and a member of the Bonacci family. Fibonacci /Mathematicsbrief biography of FibonacciThe art of Algebra from from al-Khwarizmi to Viéte: A Study in the Natural Selection of Ideas The Autobiography of Leonardo Pisano R E Grimm, in Fibonacci Quarterly vol 11, 1973, pages 99-104. Leonard of Pisa and the New Mathematics of the /

##### FIBONACCI NUMBERS AND RECURRENCES Lecture 26 CS2110 – Spring 2016 Fibonacci (Leonardo Pisano) 1170-1240? Statue in Pisa Italy.

bee Male bee has only a mother Female bee has mother and father The number of ancestors at any level is a Fibonnaci number Fibonacci in nature 8 The artichoke uses the Fibonacci pattern to spiral the sprouts of its flowers. topones.weebly.com/1/post//2012/10/the-artichoke-and-fibonacci.html The artichoke sprouts its leafs at a constant /

##### Sequences defined recursively. A Sequence is a set of numbers, called terms, arranged in a paticurlar order. Example (1) please find the first five terms.

occurs in nature as the shape of a nail shell or some sea shells. (show the shell I picked on the seaside.) Some similar curves appear in pine cones （松果）. There are closewise spirals and couter-clockwise spirals on pine cones. If you have a good study on pine cones, you can find that the numbers of seeds on sprials are also Fibonacci numbers. The/

##### Chapter 6 - Methods Outline 6.1Introduction 6.3 Math Class Methods 6.4Methods 6.5Method Definitions 6.6Java API Packages 6.7Random Number Generation 6.8Example:

fibonacci 34 { 35 long number, fibonacciValue; 36 37 number = Long.parseLong( num.getText() ); 38 showStatus( "Calculating..." ); 39 fibonacciValue = fibonacci( number ); 40 showStatus( "Done." ); 41 result.setText( Long.toString( fibonacciValue ) ); 42 } 43 44 // Recursive definition of method fibonacci 45 45 public long fibonacci/performance (iteration) and good software engineering (recursion) Recursion usually more natural approach 6.14Method Overloading Method overloading –Methods with same name and /

##### PRESENTED BY: DAWN DOUGHERTY AND EUNETHIA WILLIAMS EDU 528 MAY 2012 THE FIBONACCI SEQUENCE.

understand how the Fibonacci sequence is expressed in nature and be able to identify and recreate Fibonacci spirals. WHAT IS THE FIBONACCI SEQUENCE? WHO WAS LEONARDO FIBONACCI? WHAT WAS GOING ON HISTORICALLY AT THIS TIME IN ITALY AND EUROPE? MATHEMATICAL DEVELOPMENTS AT THIS TIME FIBONACCI’S TRAVELS AN EXCITING BOOK IS PUBLISHED! FIBONACCI’S FAMOUS RABBIT POPULATION PROBLEM THE NUMBERS RECEIVE A NAME FIBONACCI SEQUENCE IN NATURE FIBONACCI SPIRALS FIBONACCI IN OUR/

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

A formula to find a term in a Fibonacci numbers without generating previous terms Jacques Binet in 1843 - known to Euler and Bernoulli 100 years before Fibonacci numbers are actually a combo of two geometric progressions/=(φ^n/sqrt(5)) - (τ^n/sqrt(5)) (two geometric progressions) now for the Fibonacci term 1000 is F1000= (φ^(10000) - τ^(10000)) / sqrt(5) = 43466557686937456... (209 digits) The Fibonacci Sequence in Nature http://www.youtube.com/watch?v=ahXIMUkSX X0 Application : The Towers of Hanoi The Rules/

##### FIBONACCI Lecture 23 CS2110 – Spring 2015 Fibonacci (Leonardo Pisano) 1170-1240? Statue in Pisa Italy And recurrences.

bee Male bee has only a mother Female bee has mother and father The number of ancestors at any level is a Fibonnaci number Fibonacci in nature 4 The artichoke uses the Fibonacci pattern to spiral the sprouts of its flowers. topones.weebly.com/1/post//2012/10/the-artichoke-and-fibonacci.html The artichoke sprouts its leafs at a constant /

##### Lecture 10 Outline Monte Carlo methods Monte Carlo methods History of methods History of methods Sequential random number generators Sequential random.

function.theory of probabilitydistribution inversecumulative distribution function Exponential Distribution: An exponential distribution arises naturally when modeling the time between independent events that happen at a constant average rate/Carlo methods  Numerical integration  Simulation Random number generators Random number generators  Linear congruential  Lagged Fibonacci Summary (2/3) Parallel random number generators Parallel random number generators  Manager/worker  Leapfrog  Sequence /

##### Fibonacci Number man. Fibonacci bunnies 1.At the end of the first month, they mate, but there is still one only 1 pair. 2.At the end of the second month.

Pisa The Pattern …….. 0+1=1 2+1=3 1+1=2 3+2=5 5+3=8 8+5=13 13+8=21 etc. etc. etc. Where does Fibonacci Fit in Nature T h e F i b o n a c c i n u m b e r s a r e N a t u r e s/ n a l l o f m a n k i n d. Pineapple Fibonacci The Fibonacci pattern fits into pineapple like this: It fits into a pinecone like this: Fibonacci daisies Plants do not know about this - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and/

##### Geometry: Similar Triangles. MA.912.D.11.5 Explore and use other sequences found in nature such as the Fibonacci sequence and the golden ratio. Block.

Explore and use other sequences found in nature such as the Fibonacci sequence and the golden ratio. Block 30 Fibonacci sequence In mathematics, the Fibonacci numbers are the numbers in the following sequence: 1, 1, 2, 3, 5, 8, 13, 21 … By definition, the first two Fibonacci numbers are 1 and 1, and each remaining number is the sum of the previous two. Fibonacci sequence: recurrence In mathematical terms, the/

##### Inductive Proofs. Mathematical Induction A powerful, rigorous technique for proving that a predicate P(n) is true for every natural number n, no matter.

Inductive Proofs Mathematical Induction A powerful, rigorous technique for proving that a predicate P(n) is true for every natural number n, no matter how large. Essentially a “domino effect” principle. Based on a predicate-logic inference rule: P(0)  n  0 (P(n)  P(n+1))  n /! :≡ 2  3  …  n. – Base case: F(0) :≡ 1 – Recursive part: F(n) :≡ n  F(n-1). F(1)=1 F(2)=2 F(3)=6 The Fibonacci Series The Fibonacci series f n≥0 is a famous series defined by: f 0 :≡ 0, f 1 :≡ 1, f n≥2 :≡ f n−1 + f n−2 Leonardo/

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

seen from its base where the stalk connects it to the tree. Pine Cones On many plants, the number of petals is a Fibonacci number: Buttercups have 5 petals; Lilies and Iris have 3 petals; some Delphiniums have 8; sorne Marigolds /have 13 petals; some Asters have 21 and Daisies can be found with 34, 55 or even 89 petals. Flower Petals Two days from now…  You will choose a pattern from nature/

##### Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci.

In the XIII century an Italian mathematician Leonardo Fibonacci discovered a series of numbers that have very intresting properties. This is definitely one of the most interesting mathematical sequences. The Fibonacci Sequence is closely related with the Golden Ratio. Both of them are equally common in nature. Definition and properties of the Fibonacci Sequence This is the kind of recurrent sequence in which every following/

##### Ever since ancient times, math and numbers were not only seen as a logical an rational process trought which reality could be descrived as postuleted.

a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13,..., each new number being the sum of the previous two. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture, as his pattern is directly linked to many natural entities as the pattern of the “sunflower”, which gives it its supernatural reputation. The/

##### presents……….. Introduction : the golden ratio Origins with the Fibonacci sequence Discovered by Leonardo Fibonacci Born- 1175 AD, Pisa (Italy) Died –

the most appealing one. That’s because, naturally the golden ratio is the most appealing number in the universe That’s why we see things arranged in golden numbers (Fibonacci numbers) Let’s study this by natural and artificial objects! A Golden Tree! The/ Golden Spiral Examples of the Golden Spiral in Nature.... Phyllotaxy Your Beauty What has Φ got to do with/

##### KOLAM DESIGNS BASED ON FIBONACCI NUMBERS S. Naranan 30 January 2008 Copyright: Prof. S. Naranan, Chennai, India.

SANSKRIT PROSODY GOLDEN RATIO GOLDEN RATIO (contd.) CONTINUED FRACTION EXPANSION OF φ FIBONACCI NUMBERS AND THE G.C.D. ALGORITHM GOLDEN RATIO IN NATURE FIBONACCI NUMBERS (MISC) φ IN GEOMETRY VARIANTS OF FIBONACCI RECURSION VARIANTS (contd) KOLAM DESIGNS Some Small Popular Kolams KOLAMS BASED ON FIBONACCI NUMBERS GROUND RULES BASIC EQUATIONS BASIC EQUATIONS (contd) Square Fibonacci Kolam 5 x 5 (1 2 3 5) 3 x 3 (1 1 2 3/

##### Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.

5 F 5 · F 6 Pattern: F n · F n+1 © 2008 Pearson Addison-Wesley. All rights reserved 5-4-9 The Golden Ratio Consider the quotients of successive Fibonacci numbers and notice a pattern. These quotients seem to go toward 1.618. In fact, they approach Which is known as the golden ratio. © 2008 Pearson Addison-Wesley. All rights / to connect the vertices of the squares formed. This curve is a spiral. © 2008 Pearson Addison-Wesley. All rights reserved 5-4-13 Example of Spiral in Nature: Shell of Chambered Nautilus

##### Fibonacci Numbers, Polynomial Coefficients, and Vector Programs.

(1 –  X)(1- (-  -1 X) Power Series Expansion of F  n+1 – (-  - 1) n+1 √5√5√5√5  n=0.. ∞ = X n+1 The i th Fibonacci number is: Leonhard Euler (1765) J. P. M. Binet (1843) A de Moivre (1730) ( 1 + aX 1 + a 2 X 2 + … + a n X n + ….. ) ( 1 + bX/in the choice tree for the cross terms has n choices of exponent e 1, e 2,..., e n ¸ 0. Each exponent can be any natural number. Coefficient of X k is the number of non-negative solutions to: e 1 + e 2 +... + e n = k What is the coefficient of X k in the expansion of/

##### Recursion Chapter 12. 2 12.1 Nature of Recursion t Problems that lend themselves to a recursive solution have the following characteristics: –One or more.

stopping case –Eventually the problem can be reduced to stopping cases only. Easy to solve 3 Nature of Recursion Splitting a problem into smaller problems Size n Problem Size n-1 Problem Size /i++) factorial *= i; return factorial; } 21 Fibonacci.cpp // FILE: Fibonacci.cpp // RECURSIVE FIBONACCI NUMBER FUNCTION int fibonacci (int n) // Pre: n is defined and n > 0. // Post: None // Returns: The nth Fibonacci number. { if (n <= 2) return 1; else return fibonacci (n - 2) + fibonacci (n - 1); } 22 GCDTest.cpp ///

##### MATH CHALLENGE Challenge your mind!  Math is all around you in …  Nature  Art  Games  The following slides have links and puzzles for you.  Select.

sizes.” For more information, go to www.fractalus.com www.fractalus.com Previous Index Next Fibonacci The Fibonacci Sequence repeatedly appears in nature. Previous Index Next 1170-1250 A.D. 1,1,2,3,5,8,13,21,… Fibonacci Previous Index Next  Find out more at Fibonacci Numbers and Nature Fibonacci Numbers and Nature Art and Math Artists can use mathematics to show perspective in their paintings. Check out this/

#####  2012 Pearson Education, Inc. Slide 5-5-1 Chapter 5 Number Theory.

3 · F 4 F 4 · F 5 F 5 · F 6 Pattern: F n · F n+1 Example: A Pattern of the Fibonacci Numbers  2012 Pearson Education, Inc. Slide 5-5-9 Consider the quotients of successive Fibonacci numbers and notice a pattern. These quotients seem to go toward 1.618. In fact, they approach Which is known as the golden ratio. The/ a smooth curve to connect the vertices of the squares formed. This curve is a spiral. Spiral  2012 Pearson Education, Inc. Slide 5-5-13 Example of Spiral in Nature: Shell of Chambered Nautilus

##### Learning objective: To recognise and explain a number pattern.

the rule? You add the last two numbers together to get the next number! This number sequence is called Fibonacci numbers. Ok, so how does this link to sunflowers and nature? http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#plants On many plants, the number of petals is a Fibonacci number and the seed distribution on sunflowers has a Fibonacci spiral effect. Activity: Put a line/

##### Recursion COMP171 Fall 2006. Recursion / Slide 2 Recursion * In some problems, it may be natural to define the problem in terms of the problem itself.

COMP171 Fall 2006 Recursion / Slide 2 Recursion * In some problems, it may be natural to define the problem in terms of the problem itself. * Recursion is useful for /2: Fibonacci numbers * Fibonacci numbers: Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... where each number is the sum of the preceding two. * Recursive definition: n F(0) = 0; n F(1) = 1; n F(number) = F(number-1)+ F(number-2); Recursion / Slide 13 Recursion / Slide 14 Example 3: Fibonacci numbers //Calculate Fibonacci numbers using /

##### Basic Practice of Statistics - 3rd Edition

Basic Practice of Statistics - 3rd Edition Lecture 4 1. Fibonacci Sequence 2. Golden ratio Chapter 2 Basic Practice of Statistics - 3rd Edition Fibonacci Sequence Recap: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Finding Patterns in Nature that exhibit numbers in the sequence: your project. Chapter 2 Basic Practice of Statistics - 3rd Edition Plants White Calla Lily One Petal Euphorbia/

##### Quit Ratio Golden Ratio Fibonacci QuitRatio Ratio is a way of showing the connection between two or more numbers. A ratio can be written as a fraction,

. Start with 1, 1. Every number is calculated by adding the two previous numbers together. Quit 1 1 2 3 5 8 Fibonacci Series Quit Compute the ratio of Fibonacci numbers: 2 ÷ 1 = 3 ÷ 2 = 5 ÷ 3 = 8 ÷ 5 = 13 ÷ 8 = 21 ÷ 13 = Golden Ratio 2 1·51·5 1·666… 1·6 1·625 1·615… Quit Fibonacci in Nature Quit 21 Spirals 11 2 3/

##### Fibonacci Sequence by Lydia Bliven & Ethel Jones.

will there be in twelve months time?  from Fibonnaci’s book Liber abaci The Fibonacci Sequence 1 1 2 3 5 8 13 21 34 55 89 184 273… is formed by adding the latest two numbers to get the next one, starting from 0 and 1: 0 1 --the /3 5 8 13 21 34 55 89 184 273… Fibonacci in Nature Flowers 3 brown carpals, 5 green stamens, 2 sets of 5 green petals 55 spirals to right, 34 spirals towards center Fibonacci in Nature Vegetables & Fruits Fibonacci in Nature Pinecone The Fibonacci Sequence 1 1 2 3 5 8 13 21 34 55/

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

. 1,1,2,3,5,8,13,21,… 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/

##### MATHLETES 10-24-12. Fibonacci Numbers and “The Golden Ratio”

Quotient It also satisfies the following equations: φ ≈ 1.618034 The most “aesthetically pleasing” rectangle has a length to width ratio of φ:1 Fibonacci Numbers and the Golden Ratio in art and nature The Parthenon in Greece Fibonacci Numbers and the Golden Ratio in art and nature Count the number of spirals in the sunflower Stars! (At least their drawings) A Formula connecting the Golden Ratio to the/

##### Fibonacci Numbers, Vector Programs and a new kind of science.

after their second month 3.each month every fertile pair begets a new pair, and 4.the rabbits never die Inductive Definition or Recurrence Relation for the Fibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(0) = 0; Fib (1) = 1 Inductive Rule For n>1, Fib(n) =/ tree for the cross terms has n choices of exponent e 1, e 2,..., e n ¸ 0. Each exponent can be any natural number. Coefficient of X k is the number of non-negative solutions to: e 1 + e 2 +... + e n = k What is the coefficient of X k in /

##### The Fibonacci Sequence The sequence of numbers 1,1,2,3,5,8….in which each successive number is equal to the sum of the two preceding numbers.

2,3,5,8,13,21,34,55,89,and on, and on, and on, and on. Where In Nature Is it Found??? The Fibonacci Sequence is found in many ways in nature. EX. The petals on a flower, and the seeds on a sunflower. Petals on a flower Buttercups have 5/marigolds have 13 asters with 21 daises with 21 34 or 89 The End You can probably see that the Fibonacci sequence is important mathematically and in nature. Quiz What are the first 6 numbers in the sequence {Hint the last one is 8 }? Who invented it? {Hint last name is in the/

##### 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.

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, 21, 34, 55, 89, 144….The Fibonacci / for you to understand how the Golden Ratio comes from the Fibonacci sequence. If you take the ratio of any two of the sequential numbers from the sequence you will see that the larger the numbers the closer it gets to 1.618(phi) Understanding What The/

##### SECTION 5-5 The Fibonacci Sequence and the Golden Ratio Slide 5-5-1.

F 3 · F 4 F 4 · F 5 F 5 · F 6 Pattern: F n · F n+1 THE GOLDEN RATIO Slide 5-5-6 Consider the quotients of successive Fibonacci numbers and notice a pattern. These quotients seem to go toward 1.618. In fact, they approach Which is known as the golden ratio. GOLDEN RECTANGLE Slide 5-5-7 A/ Construct the divisions of a (nearly) golden rectangle below. Use a smooth curve to connect the vertices of the squares formed. This curve is a spiral. EXAMPLE OF SPIRAL IN NATURE: SHELL OF CHAMBERED NAUTILUS Slide 5-5-10

##### 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.

have 5 and 8 arms, respectively (or of 8 and 13, depending on the size)- again, two Fibonacci numbers  Scientists speculate that plants that grow in spiral formation do so in Fibonacci numbers because this arrangement makes for the perfect spacing for growth. Where does it occur? Nature – Animals  This very special spiral (called the logarithmic spiral) is exactly that of the nautilus shell and/

##### MATHS IN NATURE AND ARTS FIBONACCI’S SEQUENCE AND GOLDEN RATIO.

is a sequence known in the Western World thanks to Leonardo of Pisa, a XIII century Italian mathematician, also called Fibonacci. It is an infinite sequence of Natural Numbers where the first and the second terms are 1, and the rest of the terms are obtained by adding the /previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Examples in nature: – The number of spirals, of the 3 different typologies, that could be found on almost any sunflower is 21, 34 and 55. – The daisy has seeds/

##### Algorithmic Art Mathematical Expansions –Geometric, Arithmetic Series, Fibonacci Numbers Computability –Turing Fractals and Brownian Motion, CA –Recursive.

/9, a=1/3 Area of Koch triangle is equal to the 8/5 of the area of the base triangle Fibonacci Numbers The Fibonacci numbers form a sequence defined by the following recurrence relation: Where phi is the golden ratio, 1.618033…. Fibonacci Numbers Natures numbering system: the leaf arrangement in plants, pattern of the florets of a flower, the bracts of a pinecone, the scales of/

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

one more than phi? Algebraic Implications Let’s play with the recursive nature: Can we do it again? Have far can this go? Algebraic Implications Continued Fraction Expansion Algebraic Implications Convergents of the continued fraction What do you notice? Algebraic Implications Will Fibonacci return? Algebraic Implications Try these! Compute the numbers How do these relate to the Golden Ratio? Algebraic Implications Prove that/

##### Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Do Now: Aim: What is an arithmetic sequence and series? Find the next three numbers in the sequence 1,

– all other terms are defined by using the previous terms pattern found in nature sequence – a set of ordered numbers Aim: Arithmetic Sequence Course: Alg. 2 & Trig. Fibonacci Patterns Suppose a newly-born pair of rabbits, one male, one female, are/When famous German mathematician Karl Gauss was a child, his teacher required the students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class busy for some time. Gauss gave the answer almost immediately. Can you/

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

quartet code for 21 2 is (5 8 13 21), The “Golden rectangles” are 8 x 13, two consecutive Fibonacci Numbers. But here we consider 21 as a “Generalized Fibonacci Number” and the quartet code is (13 4 17 21). The 21 2 is made up of a 13 2 surrounded by/12 7 19 26) Design This square kolam, we name it “a e-kolam”, e representing the Euler’s constant (2.7182818), the base for natural logarithms. The code implies that a 26 2 kolam has a central 12 2 enveloped by four rectangular kolams 7 x 19. The ratio of the /

##### The Fibonacci Sequence and The Golden Ratio By Reed Cogliano.

find the Golden Ratio we divide Fn/Fn+1 =   5/8 = 1.5, 8/13 = 1.625 Fibonacci Sequence in Nature  Spirals let leaves have maximum sunlight Golden Ratio  Angles of leaves The Lucas Numbers  When a plant grows differently it tries to copy the Fibonacci sequence  Some plants like corn grow opposite each leaf  When a plant grows differently it tries to copy/

##### Violin Fibonacci Segments: Neckboard off the instrument to neckboard on the instrument Open strings to remaining portion of unstringed instrument Unboarded.

the larger the interval between notes. This partially disproves Fibonacci, because the notes can’t be thought of as natural numbers in a scale. The interval ratio between the notes /gets larger as the notes get sharper. E.X. C=1x, D=2.1x, E=3.2x, F=4.3x, etc. Wolfgang Amadeus Mozart Phi is most prominent in Mozart’s many sonatas. The Fibonacci Sequence, by Deux- Elles, contains songs specifically attributed as being based in fibonacci numbers/

##### Universalities in Complex Systems: Models of Nature Konrad Hoppe Department of Mathematics.

-However, frequently found in nature (www.goldenratiomyth.weebly.com)(www.fractalfoundation.org) Fibonacci Sequence -The sunflower spirals are created by starting in the middle and placing seeds by rotating about a constant angle of 360/1.618 degrees before placing the next seed -1.618 is a special number, it is the Golden Ratio -Directly related to the Fibonacci numbers -This placing is optimal in/

##### ZERO The story of how it finally came to be a citizen in The Land of Numbers.

was on equal footing with other numbers. 1175 - 1250 The people of Fibonacci‘s time were resistant to change and they had little need for Fibonacci’s new numbers. “Millers showed the kinds and/ amounts of flour in their sacks by the way they knotted the drawstrings; and everyone, merchant and banker, sophisticates and illiterates, knew how to reckon sums up to a million with their fingers.” Quote taken from: Kaplan, R (1999).The nothing that is: A natural/

##### G LDEN NUMBER HALUK YAVUZ Cmpe220 Presentation. WWhat is golden ratio? IIs it a stupid irrational number like pi? WWhy human being likes it soo.

Golden Mean. The higher the numbers you use are, the closer to you get. Since the numbers go on forever, Ф does too. Its the most irrational number. Lets see the result when we divide two consecutive numbers of Fibonacci sequence. 3/2 = 1/and 32-averaged faces were rated significantly higher than individual faces. Langlois explained her findings as being broadly based on natural selection (physical characteristics close to the mean having been selected during the course of evolution), and on "prototype /

##### CS 122 – Jan. 9 The nature of computer science General solutions are better What is a program? OO programming Broad outline of topics: – Program design,

these sequences? 8, 13, 18, 23, 28, … 3, 6, 12, 24, … Well known recursive problems: – Factorial, Fibonacci numbers, Pascal’s triangle Thinking recursively Recursion is a different way of approaching the problem of having to repeat something – It’s accomplishing the same/traversals Important operation is to visit all vertices  iterator – General procedures like BFS and DFS do not take the hierarchical nature of the data into account. They “skip around” too much. 3 ways to traverse a binary tree, each done /

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