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

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3 4-1 Thales to Euclid Student Discussion.

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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.

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5 4-2 Lines of Math Development Student Discussion.

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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.

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8 4-3 Three Famous Problems Student Discussion.

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9 4-3 Three Famous Problems More on these in a later chapter. 2. To square a circle. sr s 2 = r 2 1. To double a cube. 2x 3 = y 3 x y 3. And To trisect an angle. 3 = BACK

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10 4-4 Euclidean Tools Student Discussion.

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11 4-4 Euclidean Tools Copy a segment AB to line l starting at point C using Euclidean tools. A B C

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12 4-5 Duplication of the Cube Student Discussion.

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13 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.

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14 One can construct and one can construct 4-5 Duplication of the Cube 3 To construct one can do the following: since and 2 1/3 = 2 1/4 + 2 1/16 + 2 1/64 + 2 1/256 +... etc. to the needed accuracy. = a 1 / (1 r)

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15 4-6 Angle Trisection Student Discussion.

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16 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.

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17 Bisection/Trisection Link? AB 2. Bisection of angle. 3. Trisection of line segment. AB C 4. Trisection of an angle ? 1. Bisection of line segment.

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18 4-7 Quadrature of a Circle Student Discussion.

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19 4-7 Quadrature of a Circle 2 The Problem has an aesthetic appeal. s r s 2 = r 2

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20 4-8 Chronology of Student Discussion.

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21 4- 8 Chronology of page 1 ?Rhine Papyrus3.1604 continued ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 150 A.D.Ptolemy 377 / 1203.1416 ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 150 A.D.Ptolemy 377 / 1203.1416 480Tsu Chung-chih 355 / 1133.1415929 ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 150 A.D.Ptolemy 377 / 1203.1416 480Tsu Chung-chih 355 / 1133.1415929 530Aryabhata 62832 / 200003.1416 ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 150 A.D.Ptolemy 377 / 1203.1416 480Tsu Chung-chih 355 / 1133.1415929 530Aryabhata 62832 / 200003.1416 1150Bhaskara 3927 / 12503.1416 ?Rhine Papyrus3.1604 240 B.C.Archimedes n = 96 (22/7) The Classic Method. 3.1418 150 A.D.Ptolemy 377 / 1203.1416 480Tsu Chung-chih 355 / 1133.1415929 530Aryabhata 62832 / 200003.1416 1150Bhaskara 3927 / 12503.1416 1429Al-Kashi classical method16 places 3.1415927

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22 4- 8 Chronology of page 2 1529Viete classical - 6 2 16 sides9 places Also by formula continued 1529Viete classical - 6 2 16 sides9 places Also by formula 1589Anthonisz3.141588 1529Viete classical - 6 2 16 sides9 places Also by formula 1589Anthonisz3.141588 1593Van Roomen 6 2 30 sides15 places 1529Viete classical - 6 2 16 sides9 places Also by formula 1589Anthonisz3.141588 1593Van Roomen 6 2 30 sides15 places 1610Van Ceulen 6 2 62 sides35 places 1529Viete classical - 6 2 16 sides9 places Also by formula 1589Anthonisz3.141588 1593Van Roomen 6 2 30 sides15 places 1610Van Ceulen 6 2 62 sides35 places 1621Snell improved classical 1529Viete classical - 6 2 16 sides9 places Also by formula 1589Anthonisz3.141588 1593Van Roomen 6 2 30 sides15 places 1610Van Ceulen 6 2 62 sides35 places 1621Snell improved classical 1630Grienberger using Snells39 places

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23 4- 8 Chronology of page 3 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc continued 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc 1671Gregory with trig series 1 1/3 + 1/5 1/7 +... 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc 1671Gregory with trig series 1 1/3 + 1/5 1/7 +... 1699Sharp using Gregorys71 places 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc 1671Gregory with trig series 1 1/3 + 1/5 1/7 +... 1699Sharp using Gregorys71 places 1706Machin using Gregorys100 places 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc 1671Gregory with trig series 1 1/3 + 1/5 1/7 +... 1699Sharp using Gregorys71 places 1706Machin using Gregorys100 places 1719DeLangny using Gregorys112 places 1650Wallis /2 = 2/1 2/3 4/3 4/5 6/5 etc 1671Gregory with trig series /4 1 1/3 + 1/5 1/7 +... 1699Sharp using Gregorys71 places 1706Machin using Gregorys100 places 1719DeLangny using Gregorys112 places 1841Rutherford152 places And on and on and on

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24 4- 8 Chronology of page 4 Gregorys formula for calculating : continued

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25 4- 8 Chronology of page 5 Note : It converges rather slowly. continued

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26 4- 8 Chronology of page 6 Polish Jesuit Adams Kochansky (1865) had a rather clever method to approximate . continued 1 1 30 3 2 = 2 2 + (3 - 3/3) 2 = 4 + 9 2 3 + 1/3

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27 4-8 Chronology of page 7 Computer approximations. 1949ENIAC 70 hr2037 places continued 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1974CDC 7600 23+ hr1,000,000 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1974CDC 7600 23+ hr1,000,000 1983HITAC M-280H 6 10 mil 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1974CDC 7600 23+ hr1,000,000 1983HITAC M-280H 6 10 mil 1987NEC SX-2 36 hr134 mil 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1974CDC 7600 23+ hr1,000,000 1983HITAC M-280H 6 10 mil 1987NEC SX-2 36 hr134 mil 1989IBM 30901 billion 1949ENIAC 70 hr2037 places 1961IBM 7090 8 hr100,000 1974CDC 7600 23+ hr1,000,000 1983HITAC M-280H 6 10 mil 1987NEC SX-2 36 hr134 mil 1989IBM 30901 billion 1997Hitachi SR 2201 79 hr51 billion

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28 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

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29 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 Eulers constant and the transcendental number e. continued

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30 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 couldnt even program their VCR.

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31 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 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.

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

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33 World Rank NameCountryContinentDigitsDateNotes 1 Lu, ChaoChinaAsia6789020 Nov 2005world record 2 Chahal, KrishanIndiaAsia4300019 Jun 2006details 3 Goto, Hiroyuki Japan Asia4219518 Feb 1995 World Record 1995- 2006 4Tomoyori, HideakiJapan Asia 4000010 Mar 1987 World Record 1987 - 1995 5 Mahadevan, RajanIndiaAsia3181105 Jul 1981 World Record 1981 1987 6 Tammet, DanielGreat BritainEurope2251414 Mar 2004 European/British Record 7 Thomas, DavidGreat BritainEurope2250001 May 1998 8 Robinson, WilliamGreat BritainEurope2022005 May 1991 Europ./Brit. Record 1991-1998 9 Carvello, CreigthonGreat BritainEurope2001327 Jun 1980World Record 1980 10 Umile, MarcUSANorth America1531421 Jul 2007