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4 – Duplication, Trisection, Quadrature and  .

π. 4 – Duplication, Trisection, Quadrature and  . The student will learn about. Some of the famous problems from antiquity and the search for  . §4-1 Thales to Euclid . Student Discussion. §4-1 Thales to Euclid . 600 B.C. Thales initial efforts at demonstrative math.

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4 – Duplication, Trisection, Quadrature and  .

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  1. π

  2. 4 – Duplication, Trisection, Quadrature and . The student will learn about Some of the famous problems from antiquity and the search for .

  3. §4-1 Thales to Euclid Student Discussion.

  4. §4-1 Thales to Euclid 600 B.C. Thales initial efforts at demonstrative math 546 B.C. Persia conquered Ionian cities. Pythagoras and others left for southern Italy. 492 B.C. Darius of Persia tried to punish Athens and failed. 480 B.C. Xerxes, son of Darius, tried again. Athens persevered. Peace and growth. 431 B.C. Peloponnesian war between Athens and Sparta with Athens losing.

  5. §4-2 Lines of Math Development Student Discussion.

  6. §4-2 Lines of Math Development Line 1 The Elements - Pythagoreans, Hippocrates, Eudoxus, Theodorus, and Theaetetus Line 2 Development of infinitesimals, limits, summations, paradoxes of Zeno, method of exhaustion of Antiphon and Eudoxus. Line 3 Higher geometry, curves other than circles and straight lines, surfaces other than sphere and plane.

  7. §4-3 Three Famous Problems Student Discussion.

  8. §4-3 Three Famous Problems More on these in a later chapter. 1. To double a cube. y x 2x 3 = y 3 2. To square a circle. s r 3. And To trisect an angle. s2 = r2 β α 3 α = β BACK

  9. §4-4 Euclidean Tools Student Discussion.

  10. §4-4 Euclidean Tools Copy a segment AB to line l starting at point C using Euclidean tools. B A C

  11. §4-5 Duplication of the Cube Student Discussion.

  12. §4-5 Duplication of the Cube 2 Hippocrates reduced the problem to Which reduces to x = The construction of the is necessary. The text has a clever mechanical construction. Archytas used the intersection of a right circular cylinder, right circular cone and torus. Cissoid of Diocles in study problem 4.4.

  13. §4-5 Duplication of the Cube 3 To construct one can do the following: since and one can construct etc. and 21/3 = 21/4 + 21/16 + 21/64 + 21/256 + . . . One can construct to the needed accuracy. Σ = a 1 / (1 – r)

  14. §4-6 Angle Trisection Student Discussion.

  15. §4-6 Angle Trisection 2 This has been the most popular of the three problems. This is easy to understand since one can both bisect and trisect a segment easily and further one can bisect an angle easily. (Next Slide.) Some mechanical devices – “the tomahawk”.

  16. Bisection/Trisection Link? 2. Bisection of angle. C 3. Trisection of line segment. A B 1. Bisection of line segment. A B 4. Trisection of an angle?

  17. §4-7 Quadrature of a Circle Student Discussion.

  18. §4-7 Quadrature of a Circle 2 s r s2 = r2 The Problem has an aesthetic appeal.

  19. §4-8 Chronology of  Student Discussion.

  20. §4- 8 Chronology of  page 1 3.1415927… continued

  21. §4- 8 Chronology of  page 2 continued

  22. §4- 8 Chronology of  page 3 continued

  23. §4- 8 Chronology of  page 4 Gregory’s formula for calculating : continued

  24. §4- 8 Chronology of  page 5 Note : It converges rather slowly. continued

  25. §4- 8 Chronology of  page 6 30 1  1 3 Polish Jesuit Adams Kochansky (1865) had a rather clever method to approximate . π 2 = 2 2 + (3 - 3/3) 2 = 4 + 9 – 2 3 + 1/3 continued

  26. §4-8 Chronology of  page 7 Computer approximations. continued

  27. §4-8 Chronology of  page 8 Now I, even I, would celebrate In rhymes unapt, the great Immortal Syracusian, rivaled nevermore, Who in his wondrous lore, Passed on before, Left men his guidance How to circles mensurate. A. C. Orr 1906 continued

  28. §4-8 Chronology of  page 9 In 1966 Martin Gardner predicted through imaginary Dr. Matrix that the millionth digit of  would be 5: It will not be long until pi is known to a million decimals. In anticipation, Dr. Matrix, the famous numerologist, has sent a letter asking that I put his prediction on record that the millionth digit of pi will be found to be 5. This calculation is based on the third book of the King James Bible, chapter 14, verse 16 (It mentions the number 7, and the seventh word has five letters), combined with some obscure calculations involving Euler’s constant and the transcendental number e. continued

  29. §4-8 Chronology of  page 10 Memorizing a piece of cake. 1975 – Osan Saito of Tokyo set a world record of memorizing  to 15,151 decimal places. Osan called out the digits to 3 reporters. It took her 3 hours and 10 minutes including a rest after every 1000 places. 1983 – Rajan Mahadenan memorizing  to 31,811. He was a student at Kansas State University. His roommate complained that he couldn’t even program their VCR.

  30. §4-8 Chronology of  page 11 Memorizing a piece of cake. SYDNEY, AUSTRALIA - January, 2006 - For Chris Lyons, reciting a 4,400 digit number was as easy as Pi. Lyons, 36, recited the first 4,400 digits of Pi without a single error at the 2006 Mindsports Australian Festival. It took 2 ½ hours to complete the feat. Lyons said he spent just one week memorizing the digits. In July 2006, a Japanese psychiatric counselor recited Pi to 83,431 decimal places from memory, breaking his own personal best of 54,000 digits and setting an unofficial world record, according to media reports.

  31. Assignment Read chapter 5. Paper 1 draft due Wednesday.

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