The Fibonacci Sequence - MMW Lesson 1.2 PDF

The Fibonacci Sequence  Who is Fibonacci? Leonardo Pisano Bogollo (Leonardo of Pisa), better known as Fibonacci was an Italian mathematician who is most famous for his Fibonacci sequence and for popularizing the Hindu-Arabic numeral system in Europe. He advocated the use of the digits 0 to 9 and...

The Fibonacci Sequence  Who is Fibonacci? Leonardo Pisano Bogollo (Leonardo of Pisa), better known as Fibonacci was an Italian mathematician who is most famous for his Fibonacci sequence and for popularizing the Hindu-Arabic numeral system in Europe. He advocated the use of the digits 0 to 9 and of the place values.  Who is Fibonacci? He is popularly known as Fibonacci, a shortened word for the Latin term "fillius Bonacci," which means "son of Bonacci" because his father was Guglielmo Bonaccio. He discovered one of the famous formulas in mathematics, the FIBONACCI SEQUENCE. Fibonacci was able to find this number sequence while looking at how generations of rabbits breed.  The Fibonacci Sequence The breeding of rabbits led to his discovery of the numbers in the Fibonacci sequence.  The Fibonacci Sequence The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, typically starting with 0 and 1, or 1 and 1. The sequence commonly begins: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... The Fibonacci Numbers  Sequences in the Fibonacci Numbers n F(n) n F(n) n F(n) 1 1 11 89 21 10946 Every third 2 1 12 144 22 17711 element of the 3 2 13 233 23 28657 Fibonacci 4 3 14 377 24 46368 Sequence is divisible by 2. 5 5 15 610 25 75025 6 8 16 987 26 121393 2, 8, 34, 144, 7 13 17 1597 27 196418 610, … 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040  Sequences in the Fibonacci Numbers n F(n) n F(n) n F(n) 1 1 11 89 21 10946 Every fourth 2 1 12 144 22 17711 element of the 3 2 13 233 23 28657 Fibonacci 4 3 14 377 24 46368 Sequence is divisible by 3. 5 5 15 610 25 75025 6 8 16 987 26 121393 3, 21, 144, 987, 6765, … 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040  Sequences in the Fibonacci Numbers n F(n) n F(n) n F(n) 1 1 11 89 21 10946 Every fifth 2 1 12 144 22 17711 element of the 3 2 13 233 23 28657 Fibonacci 4 3 14 377 24 46368 Sequence is divisible by 5. 5 5 15 610 25 75025 6 8 16 987 26 121393 5, 55, 610, 7 13 17 1597 27 196418 6765, … 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040  Sequences in the Fibonacci Numbers n F(n) n F(n) n F(n) 1 1 11 89 21 10946 Every sixth 2 1 12 144 22 17711 element of the 3 2 13 233 23 28657 Fibonacci 4 3 14 377 24 46368 Sequence is divisible by 8. 5 5 15 610 25 75025 6 8 16 987 26 121393 8, 144, 2584, 7 13 17 1597 27 196418 46368, … 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040  Sequences in the Fibonacci Numbers n F(n) n F(n) n F(n) 1 1 11 89 21 10946 2 1 12 144 22 17711 3 2 13 233 23 28657 1. Every nth Fibonacci 4 3 14 377 24 46368 number is 5 5 15 610 25 75025 divisible by F(n). 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040  Sequences in the Fibonacci Numbers n F(n) (F(n))2 1 1 1 Let us look at the 2 1 1 squares of 3 2 4 Fibonacci 4 3 9 numbers. 5 5 25 6 8 64 7 13 169 8 21 441 9 34 1156 10 55 3025  Sequences in the Fibonacci Numbers n F(n) (F(n))2 Sequence 1 1 1 1 1 Let’s make 2 1 1 2 another 3 2 4 5 sequence from 4 3 9 13 the squares of 5 5 25 34 the Fibonacci 6 8 64 89 sequence using 7 13 169 233 the same 8 21 441 610 method. 9 34 1156 1597 10 55 3025 4181 Sequences in the Fibonacci Numbers n F(n) (F(n))2 Sequence 1 1 1 1 1 2. The sum of 2 1 1 2 squares of two 3 2 4 5 consecutive 4 3 9 13 Fibonacci numbers is also a 5 5 25 34 Fibonacci number. 6 8 64 89 7 13 169 233 8 21 441 610 9 34 1156 1597 10 55 3025 4181  Sequences in the Fibonacci Numbers n F(n) (F(n))2 Sequence 2 What if we 1 1 1 1 add 2 1 1 1+1=2 consecutive 3 2 4 1+1+4=6 Fibonacci 4 3 9 1 + 1 + 4 + 9 = 15 numbers? 5 5 25 1 + 1 + 4 + 9 + 25 = 40 6 8 64 1 + 1 + 4 + 9 + 25 + 64 = 104 7 13 169 1 + 1 + 4 + 9 + 25 + 64 + 169 = 273 8 21 441 1 + 1 + 4 + 9 + 25 + 64 + 169 + 441 = 714 9 34 1156 1 + 1 + 4 + 9 + 25 + 64 + 169 + 441 + 1156 = 1870 10 55 3025 1 + 1 + 4 + 9 + 25 + 64 + 169 + 441 + 1156 + 3025 = 4895 Sequences in the Fibonacci Numbers n F(n) (F(n))2 Sequence 2 Factors 1 1 1 1 1x1 2 1 1 2 1x2 3 2 4 6 2x3 4 3 9 15 3x5 5 5 25 40 5x8 6 8 64 104 8 x 13 7 13 169 273 13 x 21 8 21 441 714 21 x 34 9 34 1156 1870 34 x 55 10 55 3025 4895 55 x 89  Sequences in the Fibonacci Numbers n F(n) (F(n))2 Sequence 2 Factors 1 1 1 1 1x1 2 1 1 2 1x2 3. The sum of the squares of 3 2 4 6 2x3 consecutive 4 3 9 15 3x5 Fibonacci numbers is 5 5 25 40 5x8 a product of two consecutive 6 8 64 104 8 x 13 Fibonacci numbers. 7 13 169 273 13 x 21 8 21 441 714 21 x 34 9 34 1156 1870 34 x 55 10 55 3025 4895 55 x 89 The Fibonacci Sequence in Nature  The Fibonacci Sequence in Nature Apple, Banana and Star Fruit  The Fibonacci Sequence in Nature Pineapples and Pinecone The Fibonacci Spiral 4. When arranged in a certain way, the Fibonacci sequence creates a special spiral pattern known as the Fibonacci spiral.  The Fibonacci Spiral in Nature Nautilus shell, Weather forecast, Sunflower  The Fibonacci Spiral in Nature Human Body The Fibonacci Sequence and the Golden Ratio 5. As the sequence progress, the ratio of consecutive Fibonacci numbers approximates to 1.618, which is approximately the value of the golden ratio.  Summary (Part 3) The Fibonacci Sequence is a number series in which each number is obtained by adding its two preceding numbers and was named after Leonardo Pisano. Interesting Patterns about the Fibonacci Sequence 1. Every nth Fibonacci number is divisible by F(n). 2. The sum of squares of two consecutive Fibonacci numbers is also a Fibonacci number. 3. The sum of the squares of consecutive Fibonacci numbers is a product of two consecutive Fibonacci numbers. 4. When arranged in a certain way, the Fibonacci sequence creates a special spiral pattern known as the Fibonacci spiral. 5. As the sequence progresses, the ratio of consecutive Fibonacci numbers approximates to 1.618 (golden ratio). The Golden Ratio  What is the Golden Ratio? The Golden ratio is also known as the golden section, golden mean or divine proportion. It is an irrational number approximately equal to 1.618 and is named after the Greek sculptor Phidias. The symbol of the golden ratio is the Greek letter "phi" – Ф (uppercase letter) or φ (lowercase letter).  What is the Golden Ratio? The ratio itself has earned names like Divine and Golden because it has very unique properties in mathematics and geometry and appears surprising numbers of places in nature. The golden ratio is called "golden" due to its historical and aesthetic significance, dating back to ancient times when it was associated with beauty, harmony, and perfection. Many artists and architects apply the Golden Ratio in their artworks and creative designs, believing that their works would be more pleasing and beautiful.  What are the values of Golden Ratio? 1.61803398874989484820... is an approximation of the actual golden ratio, which is an irrational number with an infinite number of digits that is equal to (√5+1)/2. The Golden Ratio is also equal to 2sin(54°).  What are the values of Golden Ratio? It can be expanded into this fraction that goes on forever (called a "continued fraction") which makes it “the most irrational number”.  What are the values of Golden Ratio? From this, it can be defined in terms of itself. Where did the Golden Ratio come from? The Golden Ratio and Fibonacci Sequence  The Golden Spiral The Fibonacci spiral, as mentioned in the previous lesson, is constructed using the numbers in the Fibonacci sequence. The Fibonacci spiral is also known as the Golden Spiral.  Application of the Golden Ratio The Fibonacci Spiral projected over the paintings of the Mona Lisa by Leonardo da Vinci (public domain, Creative Commons) and the Girl With a Pearl Earring by Johannes Vermeer (public domain, Creative Commons).  Application of the Golden Ratio The Golden Rectangle is a rectangle whose sides are in the proportion of the Golden Ratio. This may be observed in notable architectural structures dating back to ancient times as well as art. Temples like the Parthenon in Greece are believed to have the Golden Ratio in them.  Application of the Golden Ratio Proportions of the human body, such as the face, follows the Divine Proportion. The closer the body parts' proportion is to the Golden Ratio, the more aesthetic and beautiful the body is.  Application of the Golden Ratio Proportions of the human body, such as the face, follows the Divine Proportion. The closer the body parts' proportion is to the Golden Ratio, the more aesthetic and beautiful the body is.  Summary (Part 4) The Golden Ratio (φ) is an irrational number approximately equal to 1.618. Values of the Golden Ratio 1.61803398874989484820… or approximately 1.618 2sin(54°)