12 Oct 2013•0 j'aime•14,565 vues

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Technologie

Rachit BhallaSuivre

- 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 What is Arithmetic progression? 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, ....
- 3. Arithmetic progression in nature Old Faithful is a popular attraction at Yellowstone National Park, because the geyser produces long eruptions that are fairly predictable. When tourists visit Old Faithful, they will see a sign that indicates an estimated time that the geyser will next erupt. No one controls the geyser like an amusement park ride. Its patterns over time have caused park rangers to develop predictable eruption times using an arithmetic sequence.
- 4. Old Faithful The time between eruptions is based on the length of the previous eruption If an eruption lasts one minute, then the next eruption will occur in approximately forty-six minutes (plus or minus ten minutes). If an eruption lasts two minutes, then the next eruption will occur in approximately fifty-eight minutes. This pattern continues based on a constant difference of twelve minutes, forming an arithmetic sequence of 46, 58, 70, 82, 94, .. An eruption of n minutes will indicate that the next eruption, an, will occur in an = a1 + (n î 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. In this particular situation, the next eruption will occur in an = 46+(n î 1)12 minutes, if the previous eruption was n minutes long.
- 5. Trigonometry in daily life What is Trigonometr y?And how is it used? Trigonometry is the branch of mathematics that studies triangles and their relationships.
- 6. The Pyramids of Giza Primitive forms of trigonometry were used in the construction of these wonders of the world.
- 7. Architecture In architecture, trigonometry plays a massive role in the compilation of building plans. For example, architects would have to calculate exact angles of intersection for components of their structure to ensure stability and safety. Some instances of trigonometric use in architecture include arches, domes, support beams, and suspension bridges. 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.
- 8. Jantar Mantar observatory For millenia, trigonometry has played a major role in calculating distances between stellar objects and their paths.
- 9. Astronomy Astronomy has been studied for millennia by civilizations in all regions of the world. In our modern age, being able to apply Astronomy helps us to calculate distances between stars and learn more about the universe. 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. Menelaus’ Theorem helps astronomers gather information by providing a backdrop in spherical triangle calculation.
- 10. Grand Canyon Skywalk Geologists had to measure the amount of pressure that surrounding rocks could withstand before constructing the skywalk.
- 11. Geology Trigonometry is used in geology to estimate the true dip of bedding angles. Calculating the true dip allows geologists to determine the slope stability. Although not often regarded as an integral profession, geologists contribute to the safety of many building foundations. Any adverse bedding conditions can result in slope failure and the entire collapse of a structure.
- 12. Fibonacci series around us What are fibonacci series? Were introduced in The Book of Calculating Series begins with 0 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 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … Fn + 2 = Fn + 1 + Fn
- 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. Fibonacci Numbers ~ Were introduced in The Book of Calculating ~ Series begins with 0 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 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … Fn + 2 = Fn + 1 + Fn
- 15. Fibonacci’s 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. Fibonacci’s Rabbits Continued End of the first month = 1 pair End of the second month = 2 pair End of the third month = 3 pair End of the fourth month = 5 pair 5 pairs of rabbits produced in one year 1, 1, 2, 3, 5, 8, 13, 21, 34, …
- 17. Fibonacci Numbers in Nature Fibonacci spiral is found in both snail and sea shells
- 18. Fibonacci Numbers in Nature Sneezewort (Achillea ptarmica) shows the Fibonacci numbers
- 19. Fibonacci Numbers in Nature Plants show the Fibonacci numbers in the arrangements of their leaves Three clockwise rotations, passing five leaves Two counter-clockwise rotations
- 20. Fibonacci Numbers in Nature Pinecones clearly show the Fibonacci spiral
- 21. Fibonacci Numbers in Nature A Lilies and irises = 3 petals Buttercups and wild roses = 5 petals Corn marigolds = 13 petals Black-eyed Susan’s = 21 petals
- 22. Fibonacci Numbers in Nature The Fibonacci numbers are found in the arrangement of seeds on flower heads 55 spirals spiraling outwards and 34 spirals spiraling inwards
- 23. Fibonacci Numbers in Nature Fibonacci spiral can be found in cauliflower
- 24. Fibonacci Numbers in Nature The Fibonacci numbers can be found in pineapples and bananas Bananas have 3 or 5 flat sides Pineapple scales have Fibonacci spirals in sets of 8, 13, 21
- 25. Golden Section What is golden section ? Represented by the Greek letter Phi Phi equals … and 0.6180339887 … Ratio of Phi is 1 : 1.618 or 0.618 : 1 Mathematical definition is Phi2 = Phi + 1 Euclid showed how to find the golden section of a line <-------1-------> A G g GB = AG or 1 – g = g B 1-g so that g2 = 1 – g
- 26. Golden Section and Fibonacci Numbers The Fibonacci numbers arise from the golden section The graph shows a line whose gradient is Phi First point close to the line is (0, 1) Second point close to the line is (1, 2) Third point close to the line is (2, 3) Fourth point close to the line is (3, 5) The coordinates are successive Fibonacci numbers
- 27. Golden Section and Fibonacci Numbers The golden section 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 section
- 28. Golden Section and Geometry Is the ratio of the side of a regular pentagon to its diagonal The diagonals cut each other with the golden ratio Pentagram describes a star which forms parts of many flags European Union United States
- 29. Golden Section in Nature Arrangements of leaves are the same as for seeds and petals All are placed at 0.618 per turn Is 0.618 of 360o which is 222.5o One sees the smaller angle of 137.5o Plants seem to produce their leaves, petals, and seeds based upon the golden section
- 30. Golden Section in Architecture Golden section appears in many of the proportions of the Parthenon in Greece Front elevation is built on the golden section (0.618 times as wide as it is tall)
- 31. Golden Section in Architecture Golden section can be found in the Great pyramid in Egypt Perimeter of the pyramid, divided by twice its vertical height is the value of Phi
- 32. Golden Section in Architecture Golden section can be found in the design of Notre Dame in Paris Golden section continues to be used today in modern architecture United Nations Headquarters Secretariat building
- 33. Golden Section in Art Golden section can be found in Leonardo da Vinci’s artwork The Annunciation
- 34. Golden Section in Art a The Last Supper Madonna with Child and Saints
- 35. Golden Section in Art Golden section can be seen in Albrecht Durer’s paintings Trento Nurnberg
- 36. Golden Section in Music Stradivari used the golden section to place the f-holes in his famous violins Baginsky used the golden section to construct the contour and arch of violins
- 37. Golden Section in Music Mozart used the golden section when composing music Divided sonatas according to the golden section Exposition consisted of 38 measures Development and recapitulation consisted of 62 measures Is a perfect division according to the golden section
- 38. Golden Section in Music Beethoven used the golden section in his famous Fifth Symphony Opening of the piece appears at the golden section point (0.618) Also appears at the recapitulation, which is Phi of the way through the piece
- 39. Fibonacci Numbers and Golden Section at Towson
- 40. Fibonacci Numbers and Golden Section at Towson