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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4 11 15) Kolam (13 4 17 21) Design Since 21 is a **Fibonacci** **Number**, **the** canonical quartet code for 21 2 is (5 8 13 21), **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 four rectangles 4/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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