Symmetry Anyone? Willy Hereman. After Dinner Talk SANUM 2008 Conference Thursday, March 27, 2007, 9:00p.m.
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1 Symmetry Anyone? Willy Hereman After Dinner Talk SANUM 2008 Conference Thursday, March 27, 2007, 9:00pm
2 Tribute to Organizers
3 Karin Hunter
4 Karin Hunter Andre Weideman
5 Karin Hunter Andre Weideman Ben Herbst
6 Karin Hunter Andre Weideman Ben Herbst Dirk Laurie
7 Karin Hunter Andre Weideman Ben Herbst Dirk Laurie Stéfan van der Walt
8 Karin Hunter Andre Weideman Ben Herbst Dirk Laurie Stéfan van der Walt Neil Muller
9 Karin Hunter Andre Weideman Ben Herbst Dirk Laurie Stéfan van der Walt Neil Muller and the Support Staff Behind the Scenes
10 A Big THANK YOU!
11 Outline Symmetry Surrounding Us What is Symmetry? Father Daughter Puzzle Steiner Trees Puzzle Solving Quadratic, Cubic, Quartic Equations The Quintic and the French Revolutionary The Seven-Eleven Puzzle Modern Applications
12 Symmetry Surrounding Us Ask people on US campus
13 Symmetry Surrounding Us Ask people on US campus Ask an architect or artist
14 Symmetry Surrounding Us Ask people on US campus Ask an architect or artist Madam, I m Adam
15 Symmetry Surrounding Us Ask people on US campus Ask an architect or artist Madam, I m Adam Ask a mathematician
16 What is Symmetry? It is all about transformations
17 What is Symmetry? It is all about transformations Wigner: the unreasonable effectiveness of mathematics in the natural sciences the unreasonable effectiveness of using symmetries in mathematics A simple example: father-daughter puzzle
18 Father Daughter Puzzle Today, the ages of a father and his daughter add up to 40 years Five years from now, the father will be 4 times older than his daughter How old is each today?
19 Father Daughter Puzzle Today, the ages of a father and his daughter add up to 40 years Five years from now, the father will be 4 times older than his daughter How old is each today? Solution: F : age of the father (today) D : age of the daughter (today)
20 Father Daughter Puzzle Today, the ages of a father and his daughter add up to 40 years Five years from now, the father will be 4 times older than his daughter How old is each today? Solution: F : age of the father (today) D : age of the daughter (today) Then, F + D = 40 F + 5 = 4 (D + 5)
21 Father Daughter Puzzle Today, the ages of a father and his daughter add up to 40 years Five years from now, the father will be 4 times older than his daughter How old is each today? Solution: F : age of the father (today) D : age of the daughter (today) Then, F + D = 40 F + 5 = 4 (D + 5) Eliminate F = 40 D and solve 45 D = 4 (D + 5) or 5D = 25 Hence, D = 5 and F = 40 D = 35
22 Symmetry Reduces Complexity
23 Symmetry Reduces Complexity Solution: 20 + x : age of the father (today) 20 x : age of the daughter (today)
24 Symmetry Reduces Complexity Solution: 20 + x : age of the father (today) 20 x : age of the daughter (today) Then, 25 + x = 4 (25 x) or 5x = 75
25 Symmetry Reduces Complexity Solution: 20 + x : age of the father (today) 20 x : age of the daughter (today) Then, Hence, x = x = 4 (25 x) or 5x = 75
26 Symmetry Reduces Complexity Solution: 20 + x : age of the father (today) 20 x : age of the daughter (today) Then, Hence, x = x = 4 (25 x) or 5x = 75 So, father is 20 + x = 35, daughter is 20 x = 5
27 Steiner Trees Connecting Cities with Shortest Road System
28 D Three cities in equilateral triangle
29 L = 187 D One choice of a road system
30 L = 173 D Shortest road system connecting three cities
31 D Four cities in a square
32 L = 283 D One choice of a road system
33 L = 273 D Shortest road system connecting 4 cities
34 Solving Quadratic, Cubic, Quartic Equations Quadratic: ax 2 + bx + c = 0 Babylonians (400 BC)
35 Solving Quadratic, Cubic, Quartic Equations Quadratic: ax 2 + bx + c = 0 Babylonians (400 BC) Cubic: ax 3 + bx 2 + cx + d = 0 Italians ( ): dal Ferro & Fior, Tartaglia & Cardano
36 Solving Quadratic, Cubic, Quartic Equations Quadratic: ax 2 + bx + c = 0 Babylonians (400 BC) Cubic: ax 3 + bx 2 + cx + d = 0 Italians ( ): dal Ferro & Fior, Tartaglia & Cardano Quartic: ax 4 + bx 3 + cx 2 + dx + e = 0 Cardano & Ferrari
37 Solving Quadratic, Cubic, Quartic Equations Quadratic: ax 2 + bx + c = 0 Babylonians (400 BC) Cubic: ax 3 + bx 2 + cx + d = 0 Italians ( ): dal Ferro & Fior, Tartaglia & Cardano Quartic: ax 4 + bx 3 + cx 2 + dx + e = 0 Cardano & Ferrari Quintic: ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 The equation that couldn t be solved!
38 Challenge Mathematica to Solve the Quadratic, Cubic, Quartic, Quintic Equations
39 The Quintic and the French Revolutionary
40 The Quintic and the French Revolutionary Evariste Galois (1830): inventor of group theory, which quintic equations can (cannot) be solved
41 The Quintic and the French Revolutionary Evariste Galois (1830): inventor of group theory, which quintic equations can (cannot) be solved Niels Hendrik Abel (1821): general quintic equation can not be solved analytically
42 The Quintic and the French Revolutionary Evariste Galois (1830): inventor of group theory, which quintic equations can (cannot) be solved Niels Hendrik Abel (1821): general quintic equation can not be solved analytically Joseph Liouville, Camille Jordan, Felix Klein, Sophus Lie,
43 The Seven Eleven Puzzle The sum and product of the prices of four items is R 711
44 The Seven Eleven Puzzle The sum and product of the prices of four items is R 711 Hence, x + y + z + w = 711 x y z w = 711
45 The Seven Eleven Puzzle The sum and product of the prices of four items is R 711 Hence, x + y + z + w = 711 x y z w = 711 Solution:
46 The Seven Eleven Puzzle The sum and product of the prices of four items is R 711 Hence, x + y + z + w = 711 x y z w = 711 Solution: Prices:
47 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w =
48 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w = Integer factors: =
49 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w = Integer factors: = Thus, x = 79 2 a 1 3 b 1 5 c 1 y = 2 a 2 5 c 2 z = 2 a 3 3 b 3 5 c 3 w = 2 a 4 3 b 4 5 c 4
50 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w = Integer factors: = Thus, x = 79 2 a 1 3 b 1 5 c 1 y = 2 a 2 5 c 2 z = 2 a 3 3 b 3 5 c 3 w = 2 a 4 3 b 4 5 c 4 With a 1 + a 2 + a 3 + a 4 = 6, b 1 + b 3 + b 4 = 2, c 1 + c 2 + c 3 + c 4 = 6
51 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w = Integer factors: = Thus, x = 79 2 a 1 3 b 1 5 c 1 y = 2 a 2 5 c 2 z = 2 a 3 3 b 3 5 c 3 w = 2 a 4 3 b 4 5 c 4 With a 1 + a 2 + a 3 + a 4 = 6, b 1 + b 3 + b 4 = 2, c 1 + c 2 + c 3 + c 4 = 6 Actually, x = n 79 with n = 1, 2, 3, 4, 5, 6, 8
52 One Solution Strategy : Convert to integer problem x + y + z + w = 711 x y z w = Integer factors: = Thus, x = 79 2 a 1 3 b 1 5 c 1 y = 2 a 2 5 c 2 z = 2 a 3 3 b 3 5 c 3 w = 2 a 4 3 b 4 5 c 4 With a 1 + a 2 + a 3 + a 4 = 6, b 1 + b 3 + b 4 = 2, c 1 + c 2 + c 3 + c 4 = 6 Actually, x = n 79 with n = 1, 2, 3, 4, 5, 6, 8 Using y z w (y+z+w)3 27 eliminates n = 5, 6, 8
53 Reject x = 79, x = 2 79 = 158, and x = 3 79 = 237 because = 632, = 553, and = 474 are not five-folds (+ argument)
54 Reject x = 79, x = 2 79 = 158, and x = 3 79 = 237 because = 632, = 553, and = 474 are not five-folds (+ argument) Bingo! x = 4*79 = 316 = and y + z + w = 395
55 Reject x = 79, x = 2 79 = 158, and x = 3 79 = 237 because = 632, = 553, and = 474 are not five-folds (+ argument) Bingo! x = 4*79 = 316 = and y + z + w = 395 So, at least one number is a five-fold: either only one number is (excluded), or all three are
56 Reject x = 79, x = 2 79 = 158, and x = 3 79 = 237 because = 632, = 553, and = 474 are not five-folds (+ argument) Bingo! x = 4*79 = 316 = and y + z + w = 395 So, at least one number is a five-fold: either only one number is (excluded), or all three are Then, y = 5y, z = 5z, w = 5w and y + z + w = 79 y z w = 18000
57 Reject x = 79, x = 2 79 = 158, and x = 3 79 = 237 because = 632, = 553, and = 474 are not five-folds (+ argument) Bingo! x = 4*79 = 316 = and y + z + w = 395 So, at least one number is a five-fold: either only one number is (excluded), or all three are Then, y = 5y, z = 5z, w = 5w and y + z + w = 79 y z w = Not all three are five folds A single one cannot be a five fold (125 > 79)
58 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144
59 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2
60 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2 Test either case conclude that y = 2 is impossible
61 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2 Test either case conclude that y = 2 is impossible Then, y = 1 Bingo! y = 125
62 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2 Test either case conclude that y = 2 is impossible Then, y = 1 Bingo! y = 125 Finally, we must solve 5z + w = 54 and z w = 144
63 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2 Test either case conclude that y = 2 is impossible Then, y = 1 Bingo! y = 125 Finally, we must solve 5z + w = 54 and z w = 144 Solve a quadratic equation: z = 6, w = 24
64 So, one must be a multiple of 25; the other multiple of 5: y = 25y, z = 5z Then, 25y + 5z + w = 79 and y z w = 144 So, y = 1 or 2 Test either case conclude that y = 2 is impossible Then, y = 1 Bingo! y = 125 Finally, we must solve 5z + w = 54 and z w = 144 Solve a quadratic equation: z = 6, w = 24 Summary: x = 316, y = 125, z = 25 6 = 150, w = 5 24 = 120
65 Solution: x = 316 = y = 125 = 5 3 z = 150 = w = 120 =
66 Solution: x = 316 = y = 125 = 5 3 z = 150 = w = 120 = Prices:
67 Solution: x = 316 = y = 125 = 5 3 z = 150 = w = 120 = Prices: Challenge: Can the 7-11 puzzle be solved using symmetries (group theory)?
68 Modern Applications Maxwell s equations: merging electricity with magnetism
69 Modern Applications Maxwell s equations: merging electricity with magnetism Einstein: general relativity
70 Modern Applications Maxwell s equations: merging electricity with magnetism Einstein: general relativity Merging general relativity and quantum mechanics
71 Modern Applications Maxwell s equations: merging electricity with magnetism Einstein: general relativity Merging general relativity and quantum mechanics String theory, super string theory
72 Modern Applications Maxwell s equations: merging electricity with magnetism Einstein: general relativity Merging general relativity and quantum mechanics String theory, super string theory A theory for everything
73 Literature Ian Stewart, Why Beauty is Truth: The History of Symmetry, Basic Books, The Perseus Books Group, April 2007, 290 pages Podcast series, University of Warwick, 2007 (7 episodes, 90 minutes total) Mario Livio, The Equation That Couldn t Be Solved, Simon & Schuster, 2005, 368 pages
74 Thank You!
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