Patterns in Nature and Regularities in the World Module 1 PDF

Patterns in Nature and Regularities in the World MODULE 1 August 12, 2024 Module Overview As we continue to examine in what ways our world works, we discover configurations explaining several “how's” of nature and human creations, like art. In this module, you will be introduced...

Patterns in Nature and Regularities in the World MODULE 1 August 12, 2024 Module Overview As we continue to examine in what ways our world works, we discover configurations explaining several “how's” of nature and human creations, like art. In this module, you will be introduced to two of the mathematical concepts embedded in the environment and have inspired some of our inventions – Fibonacci sequence and golden ratio. We begin by examining these concepts and then enrich our understanding by looking into their presence in nature and various art forms. Lecture’s Objective/s At the end of this lecture, we should be able to: Describe the Fibonacci Sequence and Golden Ratio Find the nth term of the Fibonacci Sequence Explain the relationship between Fibonacci Numbers and the Golden Ratio Identify and appreciate the Fibonacci Sequence and Golden Ratio in nature and arts  Sequence It is a function whose domain is the set of positive integer. We use the notation an to denote the image of the integer n. For illustration: 3, 6, 9, 12, 15, ⋯ , 3𝑛, ⋯ Example 1 Given the sequence 𝑎𝑛 = 8𝑛 − 3 Answer the following: 1. Write the first 5 terms of the sequence. 2. What is the 27th, 28th, and 29th term of the sequence?  Example 1 Given the sequence 𝑎𝑛 = 8𝑛 − 3 Answer the following: 1. Write the first 5 terms of the sequence. Solution: The first five terms of the sequence given by 𝑎𝑛 = 8𝑛 − 3 are 𝑎1 = 8 1 −3=5 𝑎2 = 8 2 − 3 = 13 𝑎3 = 8 3 − 3 = 21 𝑎4 = 8 4 − 3 = 29 𝑎5 = 8 5 − 3 = 37 Example 1 Given the sequence 𝑎𝑛 = 8𝑛 − 3 Answer the following: 2. What is the 27th, 28th, and 29th term of the sequence? Solution The 27th, 28th, and 29th term of the sequence are 𝑎27 = 8 27 − 3 = 216 𝑎28 = 8 28 − 3 = 221 𝑎29 = 8 29 − 3 = 229 Example 2 Given the sequence 𝑎𝑛 = −𝑛2 − 1 Answer the following: 1. Write the first 5 terms of the sequence. 2. What is the 27th, 28th, and 29th term of the sequence? Example 2 Given the sequence 𝑎𝑛 = −𝑛2 − 1 Answer the following: 1. Write the first 5 terms of the sequence. Solution: The first five terms of the sequence given by 𝑎𝑛 = −𝑛2 − 1 are 𝑎1 = − 1 2 − 1 = −2 𝑎2 = − 2 2 − 1 = −5 𝑎3 = − 3 2 − 1 = −10 2 𝑎4 = − 4 − 1 = −17 𝑎5 = − 5 2 − 1 = −26 Example 2 Given the sequence 𝑎𝑛 = −𝑛2 − 1 Answer the following: 2. What is the 27th, 28th, and 29th term of the sequence? Solution The 27th, 28th, and 29th term of the sequence are 𝑎27 = − 27 2 − 1 = −730 𝑎28 = − 28 2 − 1 = −785 𝑎29 = − 29 − 1 = −842  Recurrence Relation A recurence relation for the sequence {𝑎𝑛 } is an equation that expresses 𝑎𝑛 in terms of one or more of the previous terms of the sequence, namely, 𝑎0 , 𝑎1 , 𝑎2 , ⋯ , 𝑎𝑛−1 for all integer n with 𝑛 ≥ 𝑛0 , where 𝑛0 is a nonnegative integer. The initial conditions for sequence specify the terms before 𝑛0 (before the recurrence relation takes effect). Example 3 Find the next 5 terms of the sequence with 𝑎1 = 5 and 𝑎𝑛 = 2𝑎𝑛−1 + 1 for 𝑛 ≥ 2. Solution: Start with 𝑛 = 2. That is, Followed by 𝑛 = 3. That is, Followed by 𝑛 = 4. That is, 𝑎2 = 2𝑎2−1 + 1 𝑎3 = 2𝑎3−1 + 1 𝑎4 = 2𝑎4−1 + 1 = 2𝑎1 + 1 = 2𝑎2 + 1 = 2𝑎3 + 1 =2 5 +1 = 2 11 + 1 = 2 23 + 1 𝑎2 = 11 𝑎3 = 23 𝑎4 = 47 Followed by 𝑛 = 5. That is, Followed by 𝑛 = 6. That is, 𝑎5 = 2𝑎5−1 + 1 𝑎6 = 2𝑎6−1 + 1 = 2𝑎4 + 1 = 2𝑎5 + 1 = 2 47 + 1 = 2 95 + 1 𝑎5 = 95 𝑎6 = 191 Example 4 What are 𝑎2 and 𝑎3 given the sequence to satisfies the recurrence relation for 𝑛 ≥ 2, 𝑎𝑛 = 𝑎𝑛−1 − 𝑎𝑛−2 with 𝑎0 = 3 and 𝑎1 = 5 Solution Start with 𝑛 = 2. That is, Followed by 𝑛 = 3. That is, 𝑎𝑛 = 𝑎𝑛−1 − 𝑎𝑛−2 𝑎𝑛 = 𝑎𝑛−1 − 𝑎𝑛−2 𝑎2 = 𝑎2−1 − 𝑎2−2 𝑎3 = 𝑎3−1 − 𝑎3−2 = 𝑎1 − 𝑎0 = 𝑎2 − 𝑎1 =5−3 =2−5 𝑎2 = 2 𝑎3 = −3 Fibonacci Numbers and the Golden Ratio Who was Fibonacci? Born in Pisa, Italy in 1175 AD Full name was Leonard Pisano Grew up with a North African education under the Moors Traveled extensively around Mediaterranean coast Met with many merchants and learned their systems of arithmetic Realized the advantages of the Hindu Arabic system Contribution to Mathematics Introduced the Hindu-Arabic number system into Europe. Based on ten digits and a decimal point. Europe previously used the Roman number system. Consisted of Roman numerals. Persuaded mathematicians to use the Hindu-Arabic number system Works in Mathematics Wrote five mathematical works, four books and one preserved letter. Liber Abbaci (The Book of Calculating) written in 1202. Practica geometriae (Practical Geometry) written in 1220. Flos written in 1225 Liber quadratorum (The Book of Squares) written in 1225. A letter to Master Theodorus written around 1225. The Fibonacci Rabbit Problem  The Fibonacci Rabbit Problem Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year? Generalize Pairs this Pairs last + Pairs 2 month = month months ago  Change Fibonacci’s problem slightly so that each pair of adult rabbits produces 2 pairs of baby rabbit. Which recursion formula best describes the rabbit population? A. This month = Last month + (Two months ago) B. This month = Last month + 2*(Two months ago) C. This month = 2*Last month + (Two months ago)  The Fibonacci Numbers  Were introduced in The Book of Calculating (Abacus)  Originally the series begins with 0 and 1  Later on Fibonacci omitted the 0 and starts and 1  Next number is found by adding the last two numbers together  Number obtained is the next number in the series  Pattern is repeated over and over 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … F(n + 2) = F(n + 1) + F(n)  The 16th term of the Fibonacci sequence is 987 and the 17th term is 1597. What is the 19th term? A. 2584 B. 4181 C. 6765 What is the 25th Fibonacci number? 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, … What is the 25th Fibonacci number? 𝐹25 = 75025 Binet’s Formula (when F1 & F2 = 1): 𝑛 𝑛 1+ 5 1− 5 − 𝐹𝑛 = 2 2 5 25 25 1+ 5 1− 5 − 𝐹25 = 2 2 5 = 75025 What do Fibonacci numbers have to do with the Golden Ratio?  Fibonacci Numbers 1 1 2 3 5 8 21 34 55 … 13 1 1 4 9 25 64 169 441 1156 3025 Adding up the squares 1+1+4=6 =3x2 1 + 1 + 4 + 9 = 15 =5x3 1 + 1 + 4 + 9 +25 = 40 =8x5 1 + 1 + 4 + 9 + 25 + 64 = 104 = 13 x 8 12 + 12 + 22 + 32 + 52 + 82 = 13 x 8 Constructing a Golden Rectangle What is the area of a rectangle? - Base x Height - Sum of all the area of the squares in the rectangle Golden Spiral  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. Let's look at some of the ratios of these numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…  2 =2 Bigger 1 3 = 1.5 Smaller 2 5 Notice that as we continue 3 = 1.67 Bigger 8 = 1.6 Smaller down the sequence, the 5 13 = 1.625 Bigger ratios seem to be 8 21 = 1.615 Smaller 13 converging upon one 34 21 = 1.619 Bigger 89 number (from both sides of 55 = 1.618 Smaller 55 = 1.618 the number). 34 Fibonacci Number calculator  1 377 1.61802575 1 610 1.61803714 2 987 1.61803279 3 1.5 1,597 1.61803445 5 1.66666667 2,584 1.61803381 8 1.6 4,181 1.61803406 13 1.625 6,765 1.61803396 21 1.61538462 10,946 1.618034 34 1.61904762 17,711 1.61803399 55 1.61764706 28,657 1.61803399 89 1.61818182 46,368 1.61803399 144 1.61797753 75,025 1.61803399 233 1.61805556  The Golden Ratio is what we call an irrational number: it has an infinite number of decimal places and it never repeats itself! Generally, we round the Golden Ratio to 1.618. The Golden Ratio  Represented by the Greek letter phi ( or  ) 1+ 5  It is an irrational number with a value = 2  Ratio of phi is 1 : 1.618 or 0.618 : 1  Mathematical definition is 2 =  +1  The Golden Ratio and The Fibonacci Numbers  The golden ratio arises from the Fibonacci numbers  Obtained by taking the ratio of successive terms in the Fibonacci series  Limit is the positive root of a quadratic equation and is called the golden ratio If you take two successive terms of the series a, b, and a + b then b a+b a   +1 a b b b We define the golden ratio,  to be the limit of ,so a 1 = +1  2 − −1 = 0 1+ 5 =  1.618 2  The Golden Ratio in Music The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as: There are 13 notes in the span of any note through its octave. A scale is composed of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on a tone which are combination of 2 steps and 1 step from the root tone, that is the 1st note of the scale. Fibonacci piano scale ratios Fibonacci piano scale 8 notes Fibonacci piano scale 13 notes 8 notes of the octave scale 13 notes of the chomatic scale Click here to play video Click here to play video Click to visit site The Golden Ratio in Music Mozart wrote some of the most beautiful piano concertos. Within these pieces of music, Mozart implemented the Fibonacci sequence. In the margins of the score for different compositions, Mozart jotted down mathematical equations. He began composing piano sonatas at the age of 18 and wrote a total of 18 sonatas with three movements, each in sonata form. Click picture to watch video Mozart composed his famous sonatas so that the movement from the Exposition to the Development and Recapitulation was at the Golden Ratio. For example, the Mozart Sonata 279, No. 1 contained a total of 100 measures. The first movement was 38 measures for the Exposition. The second and third movements were 62 measures. The ratio of 62 to 100 is 0.618 which equals exactly the golden proportion  The Golden Section in Architecture The Parthenon in Athens, built by the ancient Greeks from 447 to 438 BC, is regarded by many to illustrate the application of the Golden Ratio in design. Click here to view video  The Great Pyramid of Giza (also known as the Pyramid of Khufu or the Pyramid of Cheops) is the oldest and largest of the three pyramids in the Giza pyramid complex bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact. The Golden Ratio in Art Annunciation is a painting by the Italian Renaissance artists Leonardo da Vinci and Andrea del Verrocchio, dating from circa 1472–1475. It is housed in the Uffizi gallery of Florence, Italy. The Mona Lisa is a half- length portrait painting by the Italian Renaissance artist Leonardo da Vinci that has been described as "the best known, the most visited, the most written about, the most sung about, the most parodied work of art in the world" The Last Supper is a late 15th-century mural painting by Leonardo da Vinci housed by the refectory of the Convent of Santa Maria delle Grazie in Milan. The Vitruvian Man, created around 1490. The official title of the drawing is “Le proporzioni del corpo umano secondo Vitruvio,” or “The proportions of the human body according to Vitruvius.”  The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. Golden Spiral/Ratio in Cinematography