Charming proofs a journey into elegant mathematics
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| Sprache: | English |
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Math. Assoc. of America
2010
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| Schriftenreihe: | The Dolciani mathematical expositions
42 |
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| 245 | 1 | 0 | |a Charming proofs |b a journey into elegant mathematics |c Claudi Alsina ; Roger B. Nelsen |
| 264 | 1 | |a Washington, DC |b Math. Assoc. of America |c 2010 | |
| 300 | |a XXIV, 295 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1819629806001586176 |
|---|---|
| adam_text | Titel: Charming proofs
Autor: Alsina, Claudi
Jahr: 2010
Contents
Preface ix
Introduction xix
A Garden of Integers 1
1.1 Figurate numbers....................... 1
2 Sums of Squares, triangulär numbers, and cubes....... 6
3 There are infinitely many primes............... 9
4 Fibonacci numbers...................... 12
5 Fermat s theorem....................... 15
6 Wilson s theorem....................... 16
7 Perfect numbers....................... 16
8 Challenges.......................... 17
1
1
1
1
1
1
1
2 Distinguished Numbers 19
2.1 The irrationality of V2.................... 20
2.2 The irrationality of Vk for non-square k .......... 21
2.3 The golden ratio....................... 22
2.4 n and the circle........................ 25
2.5 The irrationality of7r..................... 27
2.6 The Comte de Buffon and bis needle ............ 28
2.7 e as a limit.......................... 29
2.8 An infinite series for e.................... 32
2.9 The irrationality of ?..................... 32
2.10 Steiner s problem on the number e.............. 33
2.11 The Euler-Mascheroni constant............... 34
2.12 Exponents, rational and irrational.............. 35
2.13 Challenges.......................... 36
3 Points in the Plane 39
3.1 Pick s theorem........................ 39
3.2 Circles and sums of two Squares .............. 41
3.3 The Sylvester-Gallai theorem................ 43
Xlli
xiv Contents
3.4 Bisecting a set of 100,000 points .............. 44
3.5 Pigeons and pigeonholes................... 45
3.6 Assigning numbers to points in the plane.......... 47
3.7 Challenges.......................... 48
4 The Polygonal Playground 51
4.1 Polygonal combinatorics................... 51
4.2 Drawing an n-gon with given side lengths ......... 54
4.3 The theorems of Maekawa and Kawasaki.......... 55
4.4 Squaring polygons...................... 58
4.5 The stars of the polygonal playground............ 59
4.6 Guards in art galleries.................... 62
4.7 Triangulation of convex polygons.............. 63
4.8 Cycloids, cyclogons, and polygonal cycloids........ 66
4.9 Challenges.......................... 69
5 A Treasury of Triangle Theorems 71
5.1 The Pythagorean theorem.................. 71
5.2 Pythagorean relatives..................... 73
5.3 The inradius of a right triangle................ 75
5.4 Pappus generalization of the Pythagorean theorem..... 77
5.5 The incircle and Heron s formula.............. 78
5.6 The circumcircle and Euler s triangle inequality....... 80
5.7 The orthic triangle...................... 81
5.8 The Erdös-Mordell inequality................ 82
5.9 The Steiner-Lehmus theorem ................ 84
5.10 The medians of a triangle .................. 85
5.11 Are most triangles obtuse?.................. 87
5.12 Challenges.......................... 88
6 The Enchantment of the Equilateral Triangle 91
6.1 Pythagorean-like theorems.................. 91
6.2 The Fermat point of a triangle................ 94
6.3 Viviani s theorem....................... 96
6.4 A triangulär tiling of the plane and Weitzenböck s inequality 96
6.5 Napoleon s theorem..................... 99
6.6 Morley s miracle....................... 100
6.7 Van Schooten s theorem................... 102
6.8 The equilateral triangle and the golden ratio......... 103
6.9 Challenges.......................... 104
Contents xv
7 The Quadrilaterals Corner 107
7.1 Midpoints in quadrilaterals.................. 107
7.2 Cyclic quadrilaterals..................... 109
7.3 Quadrilateral equalities and inequalities........... 111
7.4 Tangential and bicentric quadrilaterals............ 115
7.5 Anne s and Newton s theorems ............... 116
7.6 Pythagoras with a parallelogram and equilateral triangles . . 118
7.7 Challenges.......................... 119
8 Squares Everywhere 121
8.1 One-square theorems..................... 121
8.2 Two-square theorems .................... 123
8.3 Three-square theorems ................... 128
8.4 Four and more Squares.................... 131
8.5 Squares in recreational mathematics ............ 133
8.6 Challenges.......................... 135
9 Curves Ahead 137
9.1 Squarable lunes........................ 137
9.2 The amazing Archimedean Spiral.............. 144
9.3 The quadratrix of Hippias.................. 146
9.4 The shoemaker s knife and the salt cellar.......... 147
9.5 The Quetelet and Dandelin approach to conics....... 149
9.6 Archimedes triangles..................... 151
9.7 Helices............................ 154
9.8 Challenges.......................... 156
10 Adventures in Tiling and Coloring 159
10.1 Plane tilings and tessellations................ 160
10.2 Tiling with triangles and quadrilaterals........... 164
10.3 Infinitely many proofs of the Pythagorean theorem..... 167
10.4 The leaping frog....................... 170
10.5 The seven friezes....................... 172
10.6 Colorful proofs........................ 175
10.7 Dodecahedra and Hamiltonian circuits ........... 185
10.8 Challenges.......................... 186
11 Geometry in Three Dimensions 191
11.1 The Pythagorean theorem in three dimensions ....... 192
11.2 Partitioning space with planes................ 193
xvi Contents
11.3 Corresponding triangles on three lines........... 195
11.4 An angle-trisecting cone .................. 196
11.5 The intersection of three spheres.............. 197
11.6 The fourth circle....................... 199
11.7 The area of a spherical triangle............... 199
11.8 Euler s polyhedral formula................. 200
11.9 Faces and vertices in convex polyhedra........... 202
11.10 Whysometypes of faces repeat in polyhedra ....... 204
11.11 Euler and Descartes ä la Pölya............... 205
11.12 Squaring Squares and cubingcubes............. 206
11.13 Challenges.......................... 208
12 Additional Theorems, Problems, and Proofs 209
12.1 Denumerable and nondenumerble sets........... 209
12.2 The Cantor-Schröder-Bernstein theorem.......... 211
12.3 The Cauchy-Schwarz inequality.............. 212
12.4 The arithmetic mean-geometric mean inequality...... 214
12.5 Two pearls of origami.................... 216
12.6 How to draw a straight line................. 218
12.7 Some gems in functional equations............. 220
12.8 Functional inequalities................... 227
12.9 Euler s series for 7r2/6................... 230
12.10 The Wallis product..................... 233
12.11 Stirling s approximation of n!................ 234
12.12 Challenges.......................... 236
Solutions to the Challenges 239
Chapter 1.............................. 239
Chapter 2.............................. 241
Chapter 3.............................. 245
Chapter 4.............................. 247
Chapter 5.............................. 250
Chapter 6.............................. 255
Chapter 7.............................. 258
Chapter 8.............................. 261
Chapter 9.............................. 263
Chapter 10............................. 265
Contents xvii
Chapter 11.............................270
Chapter 12.............................272
References 275
Index 289
About the Authors 295
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| id | DE-604.BV036780242 |
| illustrated | Illustrated |
| indexdate | 2024-12-24T00:13:19Z |
| institution | BVB |
| isbn | 9780883853481 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020696913 |
| oclc_num | 705973823 |
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| owner_facet | DE-20 DE-188 |
| physical | XXIV, 295 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Math. Assoc. of America |
| record_format | marc |
| series | The Dolciani mathematical expositions |
| series2 | The Dolciani mathematical expositions |
| spellingShingle | Alsina, Claudi 1952- Nelsen, Roger B. 1942- Charming proofs a journey into elegant mathematics The Dolciani mathematical expositions Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd Beweis (DE-588)4132532-1 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4145177-6 (DE-588)4132532-1 |
| title | Charming proofs a journey into elegant mathematics |
| title_auth | Charming proofs a journey into elegant mathematics |
| title_exact_search | Charming proofs a journey into elegant mathematics |
| title_full | Charming proofs a journey into elegant mathematics Claudi Alsina ; Roger B. Nelsen |
| title_fullStr | Charming proofs a journey into elegant mathematics Claudi Alsina ; Roger B. Nelsen |
| title_full_unstemmed | Charming proofs a journey into elegant mathematics Claudi Alsina ; Roger B. Nelsen |
| title_short | Charming proofs |
| title_sort | charming proofs a journey into elegant mathematics |
| title_sub | a journey into elegant mathematics |
| topic | Mathematik (DE-588)4037944-9 gnd Beweistheorie (DE-588)4145177-6 gnd Beweis (DE-588)4132532-1 gnd |
| topic_facet | Mathematik Beweistheorie Beweis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020696913&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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