Icons of mathematics: an exploration of twenty key images
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Washington, DC
Math. Assoc. of America
2011
|
| Edition: | 1. print. |
| Series: | The Dolcani mathematical expositions
45 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 309 - 319 |
| Physical Description: | XVII, 327 S. Ill., graph. Darst. |
| ISBN: | 9780883853528 9780883859865 0883853523 |
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| 245 | 1 | 0 | |a Icons of mathematics |b an exploration of twenty key images |c Claudi Alsina ; Roger B. Nelsen |
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Record in the Search Index
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| adam_text | Titel: Icons of mathematics
Autor: Alsina, Claudi
Jahr: 2011
Contents
Preface ix
Twenty Key Icons of Mathematics xi
1 The Bride s Chair 1
1.1 The Pythagorean theorem?Eucüd s proof and more .... 2
1.2 The Vecten configuration................... 4
1.3 The law of cosines...................... 7
1.4 Grebe s theorem and van Lamoen s extension........ 8
1.5 Pythagoras and Vecten in recreational mathematics..... 9
1.6 Challenges.......................... 11
2 Zhou Bi Suan Jing 15
2.1 The Pythagorean theorem?a proof from ancient China 16
2.2 Two classical inequalities.................. 17
2.3 Two trigonometric formulas................. 18
2.4 Challenges.......................... 19
3 Garfield s Trapezoid 21
3.1 The Pythagorean theorem?the Presidential proof..... 22
3.2 Inequalities and Garfield s trapezoid............. 22
3.3 Trigonometrie formulas and identities............ 23
3.4 Challenges.......................... 26
4 The Semicircle 29
4.1 Thaies triangle theorem................... 30
4.2 The right triangle altitude theorem and the geometric mean . 31
4.3 Queen Dido s semicircle................... 32
4.4 The semicircles of Archimedes............... 34
4.5 Pappus and the harmonic mean............... 37
4.6 More trigonometric identities................ 38
4.7 Areas and perimeters of regulär polygons.......... 39
Xlil
xiv Contents
4.8 Euclid s construction of the five Piatonic solids....... 40
4.9 Challenges.......................... 41
5 Similar Figures 4S
5.1 Thaies proportionality theorem............... 46
5.2 Menelaus s theorem..................... 52
5.3 Reptiles............................ 53
5.4 Homothetic functions .................... 56
5.5 Challenges.......................... 58
6 Cevians 61
6.1 The theorems of Ceva and Stewart.............. 62
6.2 Medians and the centroid................... 65
6.3 Altitudes and the orthocenter................. 66
6.4 Angle-bisectors and the incenter............... 68
6.5 Circumcirele and circumcenter................ 70
6.6 Non-concurrent cevians................... 72
6.7 Ceva s theorem for circles.................. 73
6.8 Challenges.......................... 74
7 The Right Triangle 77
7.1 Right triangles and inequalities ............... 78
7.2 The incircle, circumcirele, and excircles........... 79
7.3 Right triangle cevians .................... 84
7.4 A characterization of Pythagorean triples.......... 85
7.5 Some trigonometric identities and inequalities ....... 86
7.6 Challenges.......................... 87
8 Napoleon s Triangles 91
8.1 Napoleon s theorem..................... 92
8.2 Fermat s triangle problem.................. 93
8.3 Area relationships among Napoleon s triangles....... 95
8.4 Escher s theorem....................... 98
8.5 Challenges.......................... 99
9 Ares and Angles 103
9.1 Angles and angle measurement............... 104
9.2 Angles intersecting circles.................. 107
9.3 The power ofapoint..................... 109
9.4 Euler s triangle theorem................... 111
9.5 The Taylor circle....................... 112
9.6 The Monge circle of an ellipse................ 113
9.7 Challenges.......................... 114
Contents xv
10 Polygons with Circles 117
10.1 Cyclic quadrilaterals..................... 118
10.2 Sangaku and Carnot s theorem............... 121
10.3 Tangential and bicentric quadrilaterals........... 125
10.4 Fuss s theorem........................ 126
10.5 The butterfly theorem.................... 127
10.6 Challenges.......................... 128
11 Two Circles 131
11.1 The eyeball theorem..................... 132
11.2 Generating the conics with circles ............. 133
11.3 Common chords....................... 135
11.4 Vesicapiscis......................... 137
11.5 The vesica piscis and the golden ratio............ 138
11.6 Lunes ............................ 139
11.7 The crescent puzzle..................... 141
11.8 Mrs. Miniver s problem................... 141
11.9 Concentric circles...................... 143
11.10 Challenges.......................... 144
12 Venn Diagrams 149
12.1 Three-circle theorems.................... 150
12.2 Triangles and intersecting circles.............. 153
12.3 Reuleaux polygons..................... 155
12.4 Challenges.......................... 158
13 Overlapping Figures 163
13.1 The carpets theorem..................... 164
13.2 The irrationality of -Jl and /3............... 165
13.3 Another characterization of Pythagorean
triples............................ 166
13.4 Inequalities between means................. 167
13.5 Chebyshev s inequality................... 169
13.6 Sumsofcubes........................ 169
13.7 Challenges.......................... 170
14 Yin and Yang 173
14.1 The great monad....................... 174
14.2 Combinatorial yin and yang................. 176
14.3 Integration via the symmerryof yin and yang........ 178
14.4 Recreational yin and yang.................. 179
14.5 Challenges.......................... 181
xvi Contents
15 Polygonal Lines 183
15.1 Lines and line segments................... 184
15.2 Polygonal numbers...................... 186
15.3 Polygonal lines in calculus.................. 188
15.4 Convex polygons....................... 189
15.5 Polygonal cycloids...................... 192
15.6 Polygonal cardioids ..................... 196
15.7 Challenges.......................... 198
16 Star Polygons 201
16.1 The geometry of star polygons................ 202
16.2 Thepentagram........................ 206
16.3 The star of David....................... 208
16.4 The star of Lakshmi and the octagram............ 211
16.5 Star polygons in recreational mathematics.......... 214
16.5 Challenges.......................... 217
17 Self-similar Figures 221
17.1 Geometrie series....................... 222
17.2 Growing figures iteratively.................. 224
17.3 Folding paper in half twelve times.............. 227
17.4 The spira mirabilis...................... 228
17.5 The Menger sponge and the Sierpinski carpet........ 230
17.6 Challenges.......................... 231
18 Tatami 233
18.1 The Pythagorean theorem?Bhäskara s proof........ 234
18.2 Tatami mats and Fibonacci numbers............. 235
18.3 Tatami mats and representations of Squares......... 237
18.4 Tatami inequalities...................... 238
18.5 Generalized tatami mats................... 239
18.6 Challenges.......................... 240
19 The Rectangolar Hyperhola 243
19.1 One curve, many definitions.................245
19.2 The reetangular hyperhola and its tangent lines.......245
19.3 Inequalities for natural logarithms..............247
19.4 The hyperbolic sine and cosine...............249
19.5 The series of reeiproeals of triangulär numbers.......250
19.6 Challenges.......................... 251
Contents xvii
20 Tiüng 253
20.1 Lattice multiplication..................... 254
20.2 Tiling as a proof technique.................. 255
20.3 Tiling a rectangle with rectangles.............. 256
20.4 The Pythagorean theorem?infinitely many proofs..... 257
20.5 Challenges.......................... 258
Solutions to the Challenges 261
Chapter 1.............................. 261
Chapter 2.............................. 264
Chapter 3.............................. 265
Chapter 4.............................. 267
Chapter 5.............................. 270
Chapter 6.............................. 272
Chapter 7.............................. 276
Chapter 8.............................. 280
Chapter 9.............................. 283
Chapter 10............................. 285
Chapter 11............................. 287
Chapter 12............................. 290
Chapter 13............................. 293
Chapter 14............................. 295
Chapter 15............................. 297
Chapter 16............................. 298
Chapter 17............................. 302
Chapter 18............................. 303
Chapter 19............................. 305
Chapter 20............................. 306
References 309
Index 321
About the Authors 327
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| discipline | Mathematik |
| edition | 1. print. |
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| spelling | Alsina, Claudi 1952- Verfasser (DE-588)143990934 aut Icons of mathematics an exploration of twenty key images Claudi Alsina ; Roger B. Nelsen 1. print. Washington, DC Math. Assoc. of America 2011 XVII, 327 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Dolcani mathematical expositions 45 Literaturverz. S. 309 - 319 Beweis (DE-588)4132532-1 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Diagramm (DE-588)4012044-2 gnd rswk-swf Diagramm (DE-588)4012044-2 s Geometrie (DE-588)4020236-7 s Beweis (DE-588)4132532-1 s DE-604 Nelsen, Roger B. 1942- Verfasser (DE-588)12076945X aut The Dolcani mathematical expositions 45 (DE-604)BV001900740 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024975405&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Alsina, Claudi 1952- Nelsen, Roger B. 1942- Icons of mathematics an exploration of twenty key images The Dolcani mathematical expositions Beweis (DE-588)4132532-1 gnd Geometrie (DE-588)4020236-7 gnd Diagramm (DE-588)4012044-2 gnd |
| subject_GND | (DE-588)4132532-1 (DE-588)4020236-7 (DE-588)4012044-2 |
| title | Icons of mathematics an exploration of twenty key images |
| title_auth | Icons of mathematics an exploration of twenty key images |
| title_exact_search | Icons of mathematics an exploration of twenty key images |
| title_full | Icons of mathematics an exploration of twenty key images Claudi Alsina ; Roger B. Nelsen |
| title_fullStr | Icons of mathematics an exploration of twenty key images Claudi Alsina ; Roger B. Nelsen |
| title_full_unstemmed | Icons of mathematics an exploration of twenty key images Claudi Alsina ; Roger B. Nelsen |
| title_short | Icons of mathematics |
| title_sort | icons of mathematics an exploration of twenty key images |
| title_sub | an exploration of twenty key images |
| topic | Beweis (DE-588)4132532-1 gnd Geometrie (DE-588)4020236-7 gnd Diagramm (DE-588)4012044-2 gnd |
| topic_facet | Beweis Geometrie Diagramm |
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