Transcript 1. Mathematics around us Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing our understanding of all that we observe. This presentation explores the many wonders and uses of mathematics in daily lives. This exhibition is divided into nine areas focusing on different aspects of mathematics. 2. Arithmetic Progression is useful in predicting an event if the pattern of the event is known. For Example, If an asset costs v when new, and is depreciated by d per year, its value each year can be represented by an arithmetic progression v, v-d, v-2d, ....Arithmetic progression What is Arithmetic progression? 3. Its patterns over time have caused park rangers to develop predictable eruption times using an arithmetic sequence. No one controls the geyser like an amusement park ride. When tourists visit Old Faithful, they will see a sign that indicates an estimated time that the geyser will next erupt. Old Faithful is a popular attraction at Yellowstone National Park, because the geyser produces long eruptions that are fairly predictable. Arithmetic progression in nature 4. In this particular situation, the next eruption will occur in an = 46+(n 1)12 minutes, if the previous eruption was n minutes long. An eruption of n minutes will indicate that the next eruption, an, will occur in an = a1 + (n 1)d minutes, where a1is the length after a one-minute eruption, and d is the constant difference of waiting time among eruptions that are a one-minute difference in time. This pattern continues based on a constant difference of twelve minutes, forming an arithmetic sequence of 46, 58, 70, 82, 94, .. If an eruption lasts two minutes, then the next eruption will occur in approximately fifty-eight minutes. If an eruption lasts one minute, then the next eruption will occur in approximately forty-six minutes (plus or minus ten minutes). The time between eruptions is based on the length of the previous eruption Old Faithful 5. Trigonometry is the branch of mathematics that studies triangles and their relationships.Trigonometry in daily life What is Trigonometr y?And how is it used? 6. The Pyramids of Giza Primitive forms of trigonometry were used in the construction of these wonders of the world. 7. Architecture remains one of the most important sectors of our society as they plan the design of buildings and ensure that they are able to withstand pressures from inside. Some instances of trigonometric use in architecture include arches, domes, support beams, and suspension bridges. For example, architects would have to calculate exact angles of intersection for components of their structure to ensure stability and safety. In architecture, trigonometry plays a massive role in the compilation of building plans. Architecture 8. Jantar Mantar observatory For millenia, trigonometry has played a major role in calculating distances between stellar objects and their paths. 9. Menelaus Theorem helps astronomers gather information by providing a backdrop in spherical triangle calculation. Astronomers use the method of parallax, or the movement of the star against the background as we orbit the sun, to discover new information about galaxies. In our modern age, being able to apply Astronomy helps us to calculate distances between stars and learn more about the universe. Astronomy has been studied for millennia by civilizations in all regions of the world. Astronomy 10. Grand Canyon Skywalk Geologists had to measure the amount of pressure that surrounding rocks could withstand before constructing the skywalk. 11. Any adverse bedding conditions can result in slope failure and the entire collapse of a structure. Although not often regarded as an integral profession, geologists contribute to the safety of many building foundations. Trigonometry is used in geology to estimate the true dip of bedding angles. Calculating the true dip allows geologists to determine the slope stability. Geology 12. Pattern is repeated over and over 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, Fn + 2 = Fn + 1 + Fn Number obtained is the next number in the series Next number is found by adding the last two numbers together Series begins with 0 and 1 Were introduced in The Book of Calculating Fibonacci series around us What are fibonacci series? 13. Who Was Fibonacci? ~ Born in Pisa, Italy in 1175 AD ~ Full name was Leonardo Pisano ~ Grew up with a North African education under the Moors ~Traveled extensively around the Mediterranean coast ~ Met with many merchants and learned their systems of arithmetic ~Realized the advantages of the Hindu-Arabic system 14. ~ Pattern is repeated over and over 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, Fn + 2 = Fn + 1 + Fn ~ Number obtained is the next number in the series ~ Next number is found by adding the last two numbers together ~ Series begins with 0 and 1 ~ Were introduced in The Book of Calculating Fibonacci Numbers 15. Fibonaccis Rabbits 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 the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year? 16. 5 pairs of rabbits produced in one year 1, 1, 2, 3, 5, 8, 13, 21, 34, End of the fourth month = 5 pair End of the third month = 3 pair End of the second month = 2 pair End of the first month = 1 pair Fibonaccis Rabbits Continued 17. Fibonacci spiral is found in both snail and sea shellsFibonacci Numbers in Nature 18. Sneezewort (Achillea ptarmica) shows the Fibonacci numbersFibonacci Numbers in Nature 19. Two counter-clockwise rotations Three clockwise rotations, passing five leaves Plants show the Fibonacci numbers in the arrangements of their leaves Fibonacci Numbers in Nature 20. Pinecones clearly show the Fibonacci spiralFibonacci Numbers in Nature 21. A Lilies and irises = 3 petals Buttercups and wild roses = 5 petals Corn marigolds = 13 petals Black-eyed Susans = 21 petals Fibonacci Numbers in Nature 22. 55 spirals spiraling outwards and 34 spirals spiraling inwards The Fibonacci numbers are found in the arrangement of seeds on flower heads Fibonacci Numbers in Nature 23. Fibonacci spiral can be found in cauliflowerFibonacci Numbers in Nature 24. Pineapple scales have Fibonacci spirals in sets of 8, 13, 21 Bananas have 3 or 5 flat sides The Fibonacci numbers can be found in pineapples and bananas Fibonacci Numbers in Nature 25. Euclid showed how to find the golden section of a line Mathematical definition is Phi2 = Phi + 1 Ratio of Phi is 1 : 1.618 or 0.618 : 1 Phi equals and 0.6180339887 Represented by the Greek letter Phi Golden Section What is golden section ? < -------1------->A G g GB = AG or 1 g = g B 1-g so that g2 = 1 g 26. The coordinates are successive Fibonacci numbers Fourth point close to the line is (3, 5) Third point close to the line is (2, 3) Second point close to the line is (1, 2) First point close to the line is (0, 1) The graph shows a line whose gradient is Phi The Fibonacci numbers arise from the golden section Golden Section and Fibonacci Numbers 27. Limit is the positive root of a quadratic equation and is called the golden section Obtained by taking the ratio of successive terms in the Fibonacci series The golden section arises from the Fibonacci numbers Golden Section and Fibonacci Numbers 28. Pentagram describes a star which forms parts of many flags European Union United States The diagonals cut each other with the golden ratio Is the ratio of the side of a regular pentagon to its diagonal Golden Section and Geometry 29. Plants seem to produce their leaves, petals, and seeds based upon the golden section One sees the smaller angle of 137.5o Is 0.618 of 360o which is 222.5o All are placed at 0.618 per turn Arrangements of leaves are the same as for seeds and petals Golden Section in Nature 30. Front elevation is built on the golden section (0.618 times as wide as it is tall) Golden section appears in many of the proportions of the Parthenon in Greece Golden Section in Architecture 31. Perimeter of the pyramid, divided by twice its vertical height is the value of Phi Golden section can be found in the Great pyramid in Egypt Golden Section in Architecture 32. Golden section continues to be used today in modern architecture United Nations Headquarters Secretariat building Golden section can be found in the design of Notre Dame in Paris Golden Section in Architecture 33. Golden section can be found in Leonardo da Vincis artwork The AnnunciationGolden Section in Art 34. a The Last Supper Madonna with Child and SaintsGolden Section in Art 35. Golden section can be seen in Albrecht Durers paintings Trento NurnbergGolden Section in Art 36. Baginsky used the golden section to construct the contour and arch of violins Stradivari used the golden section to place the f-holes in his famous violins Golden Section in Music 37. Is a perfect division according to the golden section Development and recapitulation consisted of 62 measures Exposition consisted of 38 measures Divided sonatas according to the golden section Mozart used the golden section when composing music Golden Section in Music 38. Also appears at the recapitulation, which is Phi of the way through the piece Opening of the piece appears at the golden section point (0.618) Beethoven used the golden section in his famous Fifth Symphony Golden Section in Music 39. Fibonacci Numbers and Golden Section at Towson 40. Fibonacci Numbers and Golden Section at Towson 41. Thank You -Rachit Bhalla