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/

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/

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/

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/

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/

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/

**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 /

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 /

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/

**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 /

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/

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/

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 /

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 /

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/

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 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/

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**/

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/

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/

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/

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/

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

(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/

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 ///

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/

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

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/

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 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/

. 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/

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/

. 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/

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/

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 /

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/

**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/

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

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/

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/

/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/

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/

– 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/

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 /

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/

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**/

-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/

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**/

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 /

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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