**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/

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/

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))/

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/

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/

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 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/

= 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/

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/

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/

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/

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/

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/

– 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/

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/

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/

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/

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/

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/

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/

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/

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**/

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 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/

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**/

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 >>>/

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 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/

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/

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/

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 /

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/

, 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/

, 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

, 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/

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** /

, 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/

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/

**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**/

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/

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

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**/

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/

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 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/

**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 /

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/

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/

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