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4 – Duplication, Trisection, Quadrature and .
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1 π 264 1509433962 14 . 3 106 333 4 1462643699 . 3 3 2 1416 . 3 ) 7 . 1 ( ) 4 . 1 ( ) 2 . 1 ( ) 1 . 1 ( 141592573 . 3 69999961 . 1 39999931 . 1 19999911 . 1 09999901 . 1 14159259 . 3 1 30 20 47 3 3 3
description
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. - PowerPoint PPT Presentation
Transcript of 4 – Duplication, Trisection, Quadrature and .
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π264150943396214.3
106333
41462643699.332
1416.3)7.1()4.1()2.1()1.1(
141592573.369999961.139999931.119999911.109999901.1
141592593.3130
20473
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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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§4-1 Thales to Euclid
Student Discussion.
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§4-1 Thales to Euclid 600 B.C. Thales initial efforts at demonstrative math546 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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§4-2 Lines of Math Development
Student Discussion.
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§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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§4-3 Three Famous Problems
Student Discussion.
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§4-3 Three Famous Problems More on these in a later chapter.
2. To square a circle. s r
s2 = r2
1. To double a cube.
2x 3 = y 3
x y
3. And To trisect an angle. α
β
3 α = βBACK
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§4-4 Euclidean Tools
Student Discussion.
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§4-4 Euclidean Tools
Copy a segment AB to line l starting at point C using Euclidean tools.
AB
C
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§4-5 Duplication of the Cube
Student Discussion.
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§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.
s2y
yx
xs
s23
3 2
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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One can construct
and one can construct
§4-5 Duplication of the Cube 3
To construct one can do the following:
since
and 21/3 = 21/4 + 21/16 + 21/64 + 21/256 + . . .
3 2
3 2
2 2 4 2 8 2 16 2 etc.
...2561
641
161
41
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to the needed accuracy.
Σ = a 1 / (1 – r)
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§4-6 Angle Trisection
Student Discussion.
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§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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Bisection/Trisection Link?
A B
2. Bisection of angle.
3. Trisection of line segment.
A B
C
4. Trisection of an angle ?
1. Bisection of line segment.
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§4-7 Quadrature of a Circle
Student Discussion.
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§4-7 Quadrature of a Circle 2
The Problem has an aesthetic appeal.s
r
s2 = r2
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§4-8 Chronology of
Student Discussion.
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§4- 8 Chronology of page 1
? Rhine Papyrus 3.16044
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continued
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
150 A.D. Ptolemy 377 / 120 3.1416
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
150 A.D. Ptolemy 377 / 120 3.1416480 Tsu Chung-chih 355 / 113 3.1415929
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
150 A.D. Ptolemy 377 / 120 3.1416480 Tsu Chung-chih 355 / 113 3.1415929530 Aryabhata 62832 / 20000 3.1416
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
150 A.D. Ptolemy 377 / 120 3.1416480 Tsu Chung-chih 355 / 113 3.1415929530 Aryabhata 62832 / 20000 3.14161150 Bhaskara 3927 / 1250 3.1416
? Rhine Papyrus 3.1604
240 B.C. Archimedes n = 96 (22/7) The Classic Method.
3.1418
150 A.D. Ptolemy 377 / 120 3.1416480 Tsu Chung-chih 355 / 113 3.1415929530 Aryabhata 62832 / 20000 3.14161150 Bhaskara 3927 / 1250 3.14161429 Al-Kashi classical method 16 places
3.1415927…
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§4- 8 Chronology of page 2 1529 Viete classical - 6 · 216 sides 9 places Also by formula
continued
1529 Viete classical - 6 · 216 sides 9 places Also by formula 1589 Anthonisz 3.141588
1529 Viete classical - 6 · 216 sides 9 places Also by formula 1589 Anthonisz 3.1415881593 Van Roomen 6 · 230 sides 15 places
1529 Viete classical - 6 · 216 sides 9 places Also by formula 1589 Anthonisz 3.1415881593 Van Roomen 6 · 230 sides 15 places1610 Van Ceulen 6 · 262 sides 35 places
1529 Viete classical - 6 · 216 sides 9 places Also by formula 1589 Anthonisz 3.1415881593 Van Roomen 6 · 230 sides 15 places1610 Van Ceulen 6 · 262 sides 35 places1621 Snell improved classical
1529 Viete classical - 6 · 216 sides 9 places Also by formula 1589 Anthonisz 3.1415881593 Van Roomen 6 · 230 sides 15 places1610 Van Ceulen 6 · 262 sides 35 places1621 Snell improved classical 1630 Grienberger using Snell’s 39 places
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§4- 8 Chronology of page 3 1650 Wallis π /2 = 2/1 · 2/3 · 4/3
· 4/5 · 6/5 · etc
continued
1650 Wallis π /2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · etc
1671 Gregory with trig series1 – 1/3 + 1/5 – 1/7 + . . .
1650 Wallis π /2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · etc
1671 Gregory with trig series1 – 1/3 + 1/5 – 1/7 + . . .
1699 Sharp using Gregory’s 71 places
1650 Wallis π /2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · etc
1671 Gregory with trig series1 – 1/3 + 1/5 – 1/7 + . . .
1699 Sharp using Gregory’s 71 places1706 Machin using Gregory’s 100 places
1650 Wallis π /2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · etc
1671 Gregory with trig series1 – 1/3 + 1/5 – 1/7 + . . .
1699 Sharp using Gregory’s 71 places1706 Machin using Gregory’s 100 places1719 DeLangny using Gregory’s 112 places
1650 Wallis π /2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · etc
1671 Gregory with trig series π /4 1 – 1/3 + 1/5 – 1/7 + . . .
1699 Sharp using Gregory’s 71 places1706 Machin using Gregory’s 100 places1719 DeLangny using Gregory’s 112 places1841 Rutherford 152 places And on and on and on
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§4- 8 Chronology of page 4 Gregory’s formula for calculating :
...91
71
51
311
4
n 2n+1 1/2n+1 sum avg previous two1 1 1 12 3 0.333333333 0.666666667 0.8333333333 5 0.2 0.866666667 0.7666666674 7 0.142857143 0.723809524 0.7952380955 9 0.111111111 0.834920635 0.7793650796 11 0.090909091 0.744011544 0.7894660897 13 0.076923077 0.820934621 0.782473082
continued
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§4- 8 Chronology of page 5 n 2n+1 1/2n+1 sum avg previous two1 1 1 1
13 25 0.04 0.804600691 0.78460069114 27 0.037037037 0.767563654 0.78608217315 29 0.034482759 0.802046413 0.78480503416 31 0.032258065 0.769788349 0.78591738117 33 0.03030303 0.800091379 0.78493986418 35 0.028571429 0.771519951 0.78580566519 37 0.027027027 0.798546977 0.78503346420 39 0.025641026 0.772905952 0.785726464
Note : 785398163.04
It converges rather slowly.continued
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§4- 8 Chronology of page 6
Polish Jesuit Adams Kochansky (1865) had a rather clever
method to approximate .
continued
1
1
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3
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π 2 = 2 2 + (3 - 3/3) 2
= 4 + 9 – 2 3 + 1/3
...1415333.3and3
13639
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§4-8 Chronology of page 7 Computer approximations.
1949 ENIAC 70 hr 2037 places
continued
1949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001974 CDC 7600 23+ hr 1,000,000
1949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001974 CDC 7600 23+ hr 1,000,0001983 HITAC M-280H 6 ¾ 10 mil
1949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001974 CDC 7600 23+ hr 1,000,0001983 HITAC M-280H 6 ¾ 10 mil1987 NEC SX-2 36 hr 134 mil
1949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001974 CDC 7600 23+ hr 1,000,0001983 HITAC M-280H 6 ¾ 10 mil1987 NEC SX-2 36 hr 134 mil1989 IBM 3090 1 billion
1949 ENIAC 70 hr 2037 places1961 IBM 7090 8 ¾ hr 100,0001974 CDC 7600 23+ hr 1,000,0001983 HITAC M-280H 6 ¾ 10 mil1987 NEC SX-2 36 hr 134 mil1989 IBM 3090 1 billion1997 Hitachi SR 2201 79 hr 51 billion
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§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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§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
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§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.
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§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.
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Assignment
Read chapter 5.
Paper 1 draft due Wednesday.
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World Rank
Name Country Continent Digits Date Notes
1 Lu, Chao China Asia 67890 20 Nov 2005 world record
2 Chahal, Krishan India Asia 43000 19 Jun 2006 details
3Goto, Hiroyuki
JapanAsia 42195 18 Feb 1995
World Record 1995-2006
4 Tomoyori, Hideaki JapanAsia
40000 10 Mar 1987
World Record 1987 - 1995
5Mahadevan, Rajan India Asia 31811 05 Jul 1981
World Record 1981 – 1987
6Tammet, Daniel Great Britain Europe 22514 14 Mar 2004
European/British Record
7Thomas, David Great Britain Europe 22500 01 May 1998
8Robinson, William Great Britain Europe 20220 05 May 1991
Europ./Brit. Record 1991-1998
9 Carvello, Creigthon Great Britain Europe 20013 27 Jun 1980 World Record 1980
10 Umile, Marc USA North America 15314 21 Jul 2007
