LESSON-1-NATURE-OF-MATHEMATICS.pptx patterns
- 1. The Nature of Mathematics 1.1 Mathematics in the World 1.2 Fibonacci Numbers 1.3 The Golden Ratio
- 2. Learning Objectives At the end of the lesson the student should be able to: Identify patterns in nature and regularities in the world. Explain the importance of mathematics in one’s life Express appreciation for mathematics as a human endeavor
- 3. As rational creatures we tend to identify and follow patterns whether consciously or subconsciously. Recognizing patterns feels natural, like our brain is hardwired to recognize them.
- 4. What are Patterns in Nature? These are repeated designs or behaviors found naturally. They show consistency and structure and often follow mathematical principles.
- 5. EXAMPLES
- 6. Is a curved that emanates from a central point, winding around it with increasing distance. Found in snail shells, galaxies, Sunflowers, and hurricanes. 1. SPIRALS
- 7. Patterns and numbers in nature and the world
- 8. 2. FRACTALS Seen in fern leaves, snowflakes, and coastlines. They look similar at different scales.
- 9. 3. SYMMETRY Present in butterflies, flowers, and human faces.
- 10. 4. WAVES In water, sound, and light.
- 11. 5. TESSELLATIONS Patterns of shapes that fit perfectly without gaps, like honeycombs made by bees.
- 14. What is the next figure in the given pattern?
- 19. FIBONACCI A GREAT EUROPEAN MATHEMATICIAN OF THE MIDDLE AGES. HIS FULL NAME AN ITALIAN IS LEONARDO PISANO, WHICH MEANS LEONARDO OF PISA, BECAUSE HE WAS BORN IN PISA, ITALY IN (1170-1240). FIBONACCI is the shortened word for the Latin term. “filius Bonacci,” which stands for “son of Bonacio” His father’s name was GUGLIELMO BONACCIO. FIBONACCI. Discovered the pattern of numbers form the set {1,1,2,3,5,8,13,….} 70
- 20. FIBONACCI. Observed numbers in nature. His most popular contribution perhaps is the number that is seen in the petals of flowers. A calla lily flower has only 1 petal, trillium has 3, hibiscus has 5, cosmos flower has 8, corn marigold has 13, some asters have 21, and a daisy can have 34,55, or 89 petals, surprisingly, these petal counts represent the first eleven numbers of the FIBONACCI sequence. Not all petal numbers of flowers, however follow this pattern discovered by Fibonacci.
- 21. Corn Marigold, Calla Lily, Trillium, Hibiscus, Cosmos
- 25. The principle behind the Fibonacci numbers is as follows: o Letbe the nth integer in the Fibonacci sequence, the next (n+1)th term o Consider the first few terms below: Let =1 be the second term, the third term Is found by += 1+1 = 2 o The fourth term o To find the new nth Fibonacci number, simply add the two numbers immediately preceding this nth number.
- 26. n= 3: 1+1 = 2 n= 4: = 1+2 = 3 n= 5: = 2+3 = 5 n= 6: = 3+5 = 8 n= 7: n= 8: = 8 + 13 = 21 N= 9: = 13 + 21 = 34
- 27. These numbers arranged in increasing order can be written as the sequence {1,1,2,3,5,8,13,21,34,55-89,…..) Similarly when you count the clockwise and counter clockwise spiral in the sun flower seed head. It is interesting to note that the numbers 34, and 55 occur which are consecutive Fibonacci numbers. Pineapple also have spiral formed by their hexagonal nubs.
- 31. The Golden Ratio
- 32. The Golden Ratio The ratio of two consecutive FIBONACCI NUMBERS as n becomes large, approaches the golden ratio; that is: Lim The golden Ratio denoted by φ is sometimes called the Golden mean or golden section.
- 37. JOHANNES KEPLER (Known for his laws of planetary motion) he observed that dividing a Fibonacci number by the number immediately before it in the ordered sequence yields a quotient approximately equal to 1.618. this amazing ratio is denoted by the symbol φ called the Golden Ratio.
- 38. It is interesting to note that the ratio of two adjacent Fibonacci Numbers approaches the golden ratio. That is: As an n becomes large. This is indeed a mystery. What does the golden ratio have to do with a rabbit population model?
- 39. The following table gives values of the ratio n n 3 = 2 = 1.617647.59 4 = 1.5 = 1.618181818 5 = 1.617977528 6 =1.6 = 1.61805556 7 =1.625 = 1.618025751 8 = 1.615384615 = 1.618037135 9 =1.619047619 = 1.618032787
- 40. KEPLER once claimed that GEOMETRY has two great treasures; 1.Theorem of Pythagoras. This treasure we may compare to measure of gold. ( 2.The division of a line into extreme and mean ratio. This may we may name a precious jewel.
- 44. The division of a line into EXTREME and MEAN ratio.