# Ppt on fibonacci numbers and nature

##### Math and the Mona Lisa A Study of the Golden Ratio By: M. Mendoza and C. Ginson.

and the Golden Mean. His most notable contribution to mathematics was a work known as Liber Abaci, which became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. Leonardo Fibonacci Leonardo Fibonacci Fibonacci numbers/. Dodecahedrons consist of 12 pentagons, which exhibit phi relationships in their proportionsChristDodecahedronsChristDodecahedrons In Nature: "Nature hides her secrets because of her essential loftiness, but not by means of ruse/

##### 1 Tirgul no. 8 Topics covered: H Recursion: Fibonacci Factorial, GCD Backtracking – N-Queens and Knight moves.

If we now draw quarter circles in each of the rectangles: H This is a spiral (the Fibonacci Spiral). H A similar curve to this occurs in nature as the shape of a snail shell or some sea shells Fibonacy Rectangles and Shell Spirals 11 H Fibonacci numbers can also be seen in the arrangement of seeds on flower heads H The picture here is/

##### Ontologies Reasoning Components Agents Simulations Rule-Based Reasoning: Constraint Solving and Deduction Jacques Robin.

Application Domain Easier to reflect frequent policy changes than imperative code Semi-Natural Language Syntax for Business Rules  Associate key word or key phrase/ rules over simpagation rules and simpagation over propagation rules  Preferring simplification and simpagation rules with highest number of heads  Preferring propagation rules with lowest number of heads  Preferring /N,Y), fib(W,X), plus(X,Y,V). Example Term Rewriting as CHR  Solving:fibonacci a)plus(X,0)  X b)plus(X,suc(Y))  suc(plus(X,Y))/

##### FIBONACCI NUMBERS 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393,

is found by adding together the squares of the inner two numbers (here 22=4 and 32=9 and their sum is 4+9=13). [SIDE 3] c = (Fn+1 )² x (Fn+2 )² FIBONACCI’S NUMBERS IN NATURE Fibonacci spiral found in both snail and sea shells. A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral created by drawing circular arcs connecting the opposite corners/

##### Fibonacci Sequence A Mathematics Webquest

search on your own and summarize two other places in nature where you can find the Fibonacci Sequence. Process Movies/Literature… Perhaps you’ve read the novel The Da Vinci Code by Dan Brown, or have seen the movie based on the novel. In the story, Robert Langdon knows that a code is being used because he recognizes the Fibonacci Numbers in a jumbled order/

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

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 (The Book of Calculation), 1202 (1228) Practica Geometriae (The Practice of/

##### Fibonacci Numbers and The Golden Section 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987... Thomas J. Hill Kristi Selkirk Melissa Zale Amber Ballance.

Fibonacci Numbers and The Golden Section 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987... Thomas J. Hill Kristi Selkirk Melissa Zale Amber Ballance Who was Fibonacci? Born: 1170 in (probably) Pisa (now in Italy) Died: 1250 in (possibly) Pisa (/: 1).... Golden Section in Art A B C D AC = CD AD AC AND DB = BA DA DB . Golden Section In Nature Nature Continued… BIBLIOGRAPHY http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. New/

##### 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.

= 53+ 5 = 85+ 8 = 13 8+ 13 = 24 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 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/

##### 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)

in nice snails of the inner ear - a body which is hardly accidental form of logarithmic spiral Fibonacci numbers are undoubtedly part of the natural harmony that is pleasant to feel, nice looks and even sounds nice. The music is based on 8-speed octave as 1, 3, and 5th notes create the basis of all chords. Euphonious, harmonious chords are not random. The most/

##### RECURSIONL27-L28. To learn and the following concepts To design a recursive algorithm To solve problems using recursion To understand the relationship.

function is a function that invokes/calls itself directly or indirectly. Useful programming technique Enables you to develop a natural, straightforward, simple solution to a problem that would otherwise be difficult to solve.  Useful for many tasks, like sorting and mathematical functions like factorial, fibonacci, exponentiation, GCD, Tower of Hanoi, etc. Recursion - Introduction Recursive Thinking Recursion Process A child couldnt sleep, so her/

##### Comenius Project 2011- 2013 Greece, 16-21 october 2012 The golden ratio “The two highways of the life: maths and English”

we count the spirals in a “roman cauliflower” we can find numbers of the Fibonacci sequence as 8 and 13. Fibonacci sequence and golden ratio in nature Also in a sunflower can be counted spirals in a number equal to the Fibonacci sequence, like 34 and 21. Fibonacci sequence and golden ratio in nature … or in a pine cone: 8 and 13 spirals! Conclusions Mathematics is a science, but it is also a/

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

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/

##### DATA STRUCTURES II UNIT 4 – Heaps SUYASH BHARDWAJ FACULTY OF ENGINEERING AND TECHNOLOGY GURUKUL KANGRI VISHWAVIDYALAYA, HARIDWAR.

not involve deleting an element in O(1) amortized time. Fibonacci Heaps Fibonacci heaps are especially desirable when the number of EXTRACT-MIN and DELETE operations is small relative to the number of other operations. Fibonacci heaps are loosely based on binomial heaps. A collection of trees/top-down approach to get there… So, – Inserting may very likely involve moving a data item around to maintain the sequential nature of the data in a leaf. 214 Node Split – a bit more difficult (1 of 2) Using a top-down/

##### M. Böhlen and R. Sebastiani 9/26/20161 Data Structures and Algorithms Roberto Sebastiani

– fib(2) = 1 – fib(n) = fib(n-1) + fib(n-2), n>2 ● Numbers in the series: – 1, 1, 2, 3, 5, 8, 13, 21, 34,... M. Böhlen and R. Sebastiani 9/26/201648 Fibonacci Implementation fib INPUT: n – a natural number larger than 0. OUTPUT: fib(n), the nth Fibonacci number. fib(n) if n  2 then return 1 else return fib(n-1) + fib/

##### 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 mathematician/

##### By Nicole Age 10 For Mrs. Fischer Grade 4

of the Middle Ages.” His Numbers increase by adding the last two numbers and decrease by subtracting the last two numbers. What Are The Fibonacci Numbers? The Fibonacci numbers are when, starting with 0 you add the last 2 numbers to get the next number, so 0+1=1, 1+1=2, 1+2=3, 3+2=5, and so on and so forth. Fibonacci in Nature Fibonacci numbers appear in nature everywhere! The most obvious things/

##### Fibonacci By Andréa Rivard.

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 energy uses http://upload. wikimedia http://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Mammilaria_gigantea.jpg/

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

2008 Copyright: Prof. S. Naranan, Chennai, India FIBONACCI SERIES 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/

on k] Base cases: F0 = 1  1, F1 = 2  . Inductive hypotheses: Fk  k and Fk+1  k + 1 slightly non-standard definition (definition) (inductive hypothesis) (algebra) (2 =  + 1) (algebra) Fibonacci Numbers and Nature pinecone http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html cauliflower Union Fibonacci Heaps: Union Union. Combine two Fibonacci heaps. Representation. Root lists are circular, doubly linked lists. min min 23 24/

##### Arrays and Strings A way to make oodles of variables, and a deeper look at classes.

use index numbers within the brackets to refer to individual components of an array Assigning values to arrays fibonacci[0] fibonacci[1] fibonacci[2] fibonacci[3] fibonacci[4] 1 1 2 3 5 [0] [1] [2] [3] [4] Arrays and FOR loops It is often useful to use arrays and FOR loops together for assigning values to arrays and for outputting values of arrays int c; int[] naturals; naturals = int[5/

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

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/

##### 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.

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/ ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks/

##### 1 Library Methods and Recursion Instructor: Mainak Chaudhuri

2*v1*v2*Math.cos(angle*Math.PI/180.0)); System.out.println(“Resultant of ” + v1 + “ and ” + v2 + “ at ” + angle + “ degrees is ” + resultant); } 8 Which one /natural numbers is ” + Sum(n)); } public static int Sum(int n) { if (n == 1) return 1;// initial condition return (Sum(n-1) + n); } 14 Fibonacci series A second order recurrence F n = F n-1 + F n-2 for n > 2; F 1 = F 2 = 1 class Fibonacci { public static void main (String arg[]) { int n = 10; System.out.println(n + “th Fibonacci number is ” + Fibonacci/

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

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 series starts like this. /3 and it continues as follows... 1 1 2 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 /

##### The Golden Ratio and Fibonacci Numbers in Nature By: Mary Catherine Clark.

The Golden Ratio and Fibonacci Numbers in Nature By: Mary Catherine Clark  Leonardo Fibonacci was the most outstanding mathematician of the European Middle Ages.  He was known by other names including Leonardo Pisano or Leonard of Pisa.  Little was know about his life except for the few facts given in his mathematical writings.  Fibonacci was born around 1170.  Received his early education from a Muslim schoolmaster.  His/

##### Algorithms and Flowcharts for Programming CFD Dr. Ugur GUVEN.

the Function using the relevant numerical method and the Given Boundary Conditions of the Flow/ Sample Flowchart Sum of first 50 natural numbers Sample Flowchart Here is the sample flowchart/number Sample Flowchart Flowchart for computing factorial N 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... By definition the first two numbers are: Fibonacci(0) = 0 Fibonacci(1) = 1 Fibonacci(2) = 0 + 1 = 1 Fibonacci(3) = 1 + 1 = 2 Fibonacci(4) = 1 + 2 = 3 Fibonacci(5) = 2 + 3 = 5 Fibonacci(6) = 3 + 5 = 8 Fibonacci/

##### Introduction to Computing Using Python Recursion and Algorithm Development  Introduction to Recursion  Recursion Examples  Run Time Analysis  Search.

LFLFLFRFLFLFLFRFLFLFLFRFLFRFLFRFLFLFLFRFLF Pen drawing instructions: We want to develop function koch() that takes a nonnegative integer as input and returns a string containing pen drawing instructions instructions can then be used by a pen drawing app such / rfib(n-2) def rfib(n): returns n-th Fibonacci number if n < 2: # base case return 1 # recursive step return rfib(n-1) + rfib(n-2) There is a natural recursive definition for the n-th Fibonacci number: >>> rfib(0) 1 >>> rfib(1) 1 >>>/

##### 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.

Arabic 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 himself /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 Middle Ages by J and F Gies, Thomas Y Crowell publishers, 1969/

##### Dale Roberts CSCI N305 Functions Recursion Department of Computer and Information Science, School of Science, IUPUI.

+ + Dale Roberts 1/* Fig. 5.15: fig05_15.c 2 Recursive fibonacci function */ 3#include 4 5long fibonacci( long ); 6 7int main() 8{8{ 9 long result, number; 10 11 printf( "Enter an integer: " ); 12 scanf( "%ld", &number ); 13 result = fibonacci( number ); 14 printf( "Fibonacci( %ld ) = %ld ", number, result ); 15 return 0; 16} 17 18/* Recursive definition of function fibonacci */ 19long fibonacci( long n ) 20{ 21 if ( n == 0 || n == 1/

##### Math 409/409G History of Mathematics The Fibonacci Sequence Part 1.

409/409G History of Mathematics The Fibonacci Sequence Part 1 The Fibonacci Problem “A man put one pair of rabbits in a certain place entirely surrounded by a wall. How many rabbits can be produced from that pair in a year if the nature of these rabbits is such that / we will look at some of the properties of the Fibonacci sequence, but until then I’d like you to think about the following puzzle which is based on the Fibonacci numbers F 4  3, F 5  5, F 6  8, and F 7  13. This ends the lesson on Part/

##### 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.

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 = 8 ÷ 5 = 13 ÷ 8 = 21 ÷ 13 = Another/

##### From Mathematics to Generic Programming Course Slides – Part 2 of 3 Version 1.0 October 5, 2015 Copyright © 2015 by Alexander A. Stepanov and Daniel E.

matrix equation goes from one Fibonacci number to the next: So the nth Fibonacci number is obtained by : 133 Fibonacci Computation = Raising Matrix to Power We already have a generic function to raise something to a power And it’s O(log /+ z) = xy + xz(y + z)x = yx + zx 191 Canonical Semiring: Natural Numbers Natural numbers do not have additive inverses Matrix multiplication on matrices with natural number coefficients makes perfect sense 192 Sample Graph Problem: Social Network If you’re friends with X, X /

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

is 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 under/

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

, a second black, a third spotted, and a fourth brown. The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55/

##### CS503: Fifth Lecture, Fall 2008 Recursion and Linked Lists. Michael Barnathan.

, 8, 13, 21, 34, … And here’s its function: – F(n) = F(n-1) + F(n-2) – F(1) = F(2) = 1 Fibonacci: two-piece recursion. In order to find the nth Fibonacci number, we need to simply add the n-1th and n-2th Fibonacci numbers. Ok, so here’s a Java function/ an element:O(1) Search:O(N). Merge:O(1). Dynamically sized by nature. – Just stick a new node at the end. Modifications are fast, but node access is the killer. – And you need to access the nodes before performing other operations on them. Three main uses/

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 reserved/ 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

##### CS503: Fifth Lecture, Fall 2008 Recursion and Linked Lists. Michael Barnathan.

, 8, 13, 21, 34, … And here’s its function: – F(n) = F(n-1) + F(n-2) – F(1) = F(2) = 1 Fibonacci: two-piece recursion. In order to find the nth Fibonacci number, we need to simply add the n-1th and n-2th Fibonacci numbers. Ok, so here’s a Java function/ an element:O(1) Search:O(N). Merge:O(1). Dynamically sized by nature. – Just stick a new node at the end. Modifications are fast, but node access is the killer. – And you need to access the nodes before performing other operations on them. Three main uses/

##### 1 Recursion Overview l Introduction to recursion and recursive methods l Simple popular recursive algorithms l Writing recursive methods l Preview: 1-D.

the increase in computation time is almost unnoticeable to the user. The choice between recursion and iteration more often depends on the the nature of the problem being solved 3 Steps to solve a recursive Problem l 1- Try /th Fibonacci number is: "+ answer); } 13 Simple Recursive Algorithms (cont.) l Exercises: Write complete recursive programs for the following algorithms 1 power(x,y) that implements x^y using repeated additions and without using multiplication. Assume x to be a floating point value and /

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

, 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. Proof. (by induction on k) n Base cases: – F 0 = 1, F 1/

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

to test convergence??? The RATIO TEST! Applications of Phi Phi in Nature There is no other number that recurs throughout life more so than does phi. When looking at nature, we see Phi, often times without realizing it. Phi in Nature The golden spiral is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with/

##### 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.

Fibonacci number sequence is If we take the ratio （比例） of two successive numbers in Fibonacci series and we divide each by the number before it, we will get the following series of numbers. The ratio seems to approach to a particular number,which we call the golden number( 黄金数 ).It is often represented by a Greek letter phi(φ).It is also a very useful thing in our life and in the nature/

##### Developing Mathematics Patterns and Ideas Presented By Sekender & Shahjehan Khan February 27, 2005.

Camels- Middle East Traffic Jam Flight Patterns Crop Circles Numerals Arabic Numbers Bengali Numbers Hindi Numbers Chinese Numbers Alphabet Bengali Arabic Hindi Greek Our Nature, objects in Nature and Biological symmetry Common Snail (Helix) Ovulate Cone (Pinus) Muscadine / 1234567*9+8=? 11111111 (8) Fibonacci Leonardo Pisano ( 1170- 1250? ) our Bigolllo is known better by his nickname Fibonacci. He is best remembered for the introduction of Fibonacci numbers and the Fibonacci sequence. The sequence is 1,1,2/

##### 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 golden/ 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

##### Programming for Engineers in Python Sawa 2015 Lecture 2: Lists and Loops 1.

prev = curr curr = new print "The nth Fibonacci number is", curr 30 Fibonacci – Another Code n = input(“Insert a non-negative number") fibs = [0, 1] for i in range/sum_dist / num_points 32 range(num_points) sum_dist += end_points[i] – start_points[i] For Loop and Strings Iterate over strings: name = "Kobe" for letter in name: print "Give me/natural to use for In some cases it is better to use while for: Predefined number of iterations No need to initialize or advance the loop variable while: Unknown number/

##### This presentation will be demonstrating my understanding of the similarities and differences between mathematics and numeracy and how these concepts can.

sort of mathematical understanding. The Fibonacci sequence is known to be Natures numbering system because of the Fibonacci number patterns that recurrently occur in nature. They appear everywhere in Nature, from the leaf arrangement in / Number and Algebra, Measurement and Geometry, and Statistics and Probability (Australian Curriculum, 2013). Australian Curriculum Learning Outcomes: Number and Algebra Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten/

##### Note to teachers I’m a member of MAA, and “Found Math” pictures is one of their regular features. Here are a few. And this summer, there was an article.

is an avid nature photographer and author of A Mathematical Nature Walk. He recently captured the Fibonacci sequence in a daisy. See more of his photos here.Old Dominion Universityalk. He recently captured FOUND MATH: Fibonacci spirals on the ceiling/ Fibonacci sequence. Photo by William Turner. FOUND MATH: Although many flowers have three, five, or eight petals (Fibonacci numbers), some have six petals (not a Fibonacci number). Photo by Julian Fleron, Westfield State College. FOUND MATH: A Fibonacci spiral/

##### 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,... Proof/

##### 1 CSC 222: Computer Programming II Spring 2004 Sorting and recursion  insertion sort, O(N 2 )  selection sort  recursion  merge sort, O(N log N)

naturally defined as recursive algorithms a recursive algorithm is one that refers to itself when solving a problem  to solve a problem, break into smaller instances of problem, solve & combine Fibonacci numbers: 1 st Fibonacci number = 1 2 nd Fibonacci number = 1 Nth Fibonacci number = (N-1)th Fibonacci number + (N-2)th Fibonacci number/ non-linear data structures (CSC427)  recursion is essential to understanding and implementing fast sorting algorithms 17 Merge sort merge sort is defined recursively /

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

 Born in Italy and traveled excessively  Arabic Numeral enthusiast  Liber Abaci  Lived 1170-1250  Born in Italy and traveled excessively  Arabic Numeral enthusiast  Liber Abaci What is the Fibonacci sequence?  How do we get these numbers?  1,1,2/= 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/

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

discuss and comment. -Students will 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/