Explorations in Topology (eBook)

Map Coloring, Surfaces and Knots

David Gay (Autor)

eBook Download: PDF

2007 | 1. Auflage
352 Seiten
Elsevier Science (Verlag)
978-0-08-049266-7 (ISBN)

Infos

Explorations in Topology gives students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that would help them make sense of a future, more formal topology course.

The innovative story-line style of the text models the problems-solving process, presents the development of concepts in a natural way, and through its informality seduces the reader into engagement with the material. The end-of-chapter Investigations give the reader opportunities to work on a variety of open-ended, non-routine problems, and, through a modified 'Moore method', to make conjectures from which theorems emerge. The students themselves emerge from these experiences owning concepts and results. The end-of-chapter Notes provide historical background to the chapter's ideas, introduce standard terminology, and make connections with mainstream mathematics.

The final chapter of projects provides opportunities for continued involvement in 'research' beyond the topics of the book.

* Students begin to solve substantial problems right from the start
* Ideas unfold through the context of a storyline, and students become actively involved
* The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material

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Explorations in Topology gives students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that would help them make sense of a future, more formal topology course. The innovative story-line style of the text models the problems-solving process, presents the development of concepts in a natural way, and through its informality seduces the reader into engagement with the material. The end-of-chapter Investigations give the reader opportunities to work on a variety of open-ended, non-routine problems, and, through a modified "e;Moore method"e;, to make conjectures from which theorems emerge. The students themselves emerge from these experiences owning concepts and results. The end-of-chapter Notes provide historical background to the chapter's ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides opportunities for continued involvement in "e;research"e; beyond the topics of the book. * Students begin to solve substantial problems right from the start* Ideas unfold through the context of a storyline, and students become actively involved* The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material

Inhalt

Cover 1
Title page 2
Copyright page 3
Table of Contents 6
Preface 8
Format of the Chapters 9
Overview of the Chapters 10
Success in the Course this Book is based on 11
Instructor’s Manual 11
Acknowledgments 12
Chapter 1: Acme Does Maps and Considers Coloring Them 14
Scene 1 14
1. Your Turn 18
2. Your Turn 18
Scene 2 18
3. Your Turn 19
4. Your Turn 22
Scene 3 22
5. Your Turn 23
6. Your Turn 24
Scene 4 24
Investigations, Questions, Puzzles, and More 28
1. Investigation 28
2. Investigation 29
3. Investigation 29
4. Investigation 30
5. Question 31
6. Question 31
7. Question 31
8. Investigation 31
9. Investigation 33
10. Question 33
11. Puzzle 33
12. Question 34
13. Gathering Evidence 35
14. Puzzle 36
15. Summarizing 36
Notes 36
References 38
Chapter 2: Acme Adds Tours 40
Scene 1 40
Scene 2 46
1. Your Turn 47
Scene 3 48
2. Your Turn 49
Scene 4 50
3. Your Turn 51
4. Your Turn 52
Investigations, Questions, Puzzles, and More 53
1. Investigation 53
2. Investigation 53
3. Investigation 53
4. Puzzle 53
5. Investigation 54
6. Investigation 54
7. Question 55
8. Investigation 55
9. Investigation 55
10. Investigation 56
11. Puzzle 56
12. Puzzle 57
13. Question 57
14. Summarizing 57
15. Connecting 58
Notes 58
References 60
Chapter 3: Acme Collects Data from Maps 62
Scene 1 62
1. Your Turn 63
Scene 2 64
2. Your Turn 66
Scene 3 66
3. Your Turn 73
Scene 4 73
4. Your Turn 74
5. Your Turn 76
6. Your Turn 77
7. Your Turn 78
Scene 5 78
Investigations, Questions, Puzzles, and More 80
1. Investigation 80
2. Gathering Data 80
3. Investigation 81
4. Investigation 81
5. Gathering Data 81
6. Question 82
7. Question 82
8. Question 82
9. Question 82
10. Question 82
11. Investigation 83
12. Investigation 83
13. Investigation 83
14. Summarizing 83
Notes 83
References 84
Chapter 4: Acme Collects More Data, Proves a Theorem, and Returns to Coloring Maps 86
Scene 1 86
1. Your Turn 89
Scene 2 89
2. Your Turn 93
Scene 3 93
Investigations, Questions, Puzzles, and More 96
1. Investigation 96
2. Investigation 97
3. Investigation 97
4. Investigation 97
5. Investigation 97
6. Question 98
7. Question 98
8. Question 98
9. Puzzle 99
10. Summarizing 99
Notes 100
References 100
Chapter 5: Acme’s Solicitor Proves a Theorem: the Four-Color Conjecture 102
Scene 1 102
Investigations, Questions, Puzzles, and More 107
1. Gathering Evidence 107
2. Gathering Evidence 108
3. Game 108
4. Game 109
5. Investigation 109
6. Investigation 111
7. Looking Back 111
Notes 111
References 115
Chapter 6: Acme Adds Doughnuts to Its Repertoire 116
Scene 1 116
1. Your Turn 120
2. Your Turn 120
3. Your Turn 122
Scene 2 122
4. Your Turn 123
Scene 3 124
5. Your Turn 125
Scene 4 127
Investigations, Questions, Puzzles, and More 129
1. Investigation 129
2. Investigation 129
3. Question 129
4. Investigation 129
5. Investigation 129
6. Investigation 129
7. Question 130
8. Question 130
9. Question 130
10. Question 130
11. Question 131
12. Game 131
13. Puzzle 131
14. Something to Make 132
15. Investigation 134
16. Summarizing 134
17. Connecting 134
Notes 134
References 137
Chapter 7: Acme Considers the Möbius Strip 138
Scene 1 138
1. Your Turn 140
Scene 2 140
2. Your Turn 142
Scene 3 142
3. Your Turn 143
4. Your Turn 144
5. Your Turn 144
6. Your Turn 145
7. Your Turn 145
Scene 4 145
Investigations, Questions, Puzzles, and More 151
1. Gathering Evidence 151
2. Investigation 151
3. Generalizing 152
4. Question 152
5. Gathering Evidence 152
6. Investigation 153
7. Investigation 153
Edges of Twisted Strips 154
8. Investigation 155
Cutting Twisted Strips Down the Middle 155
9. Investigation 156
10. Things To Make 156
11. Something To Make: Möbius Shorts 156
12. Game 157
13. Investigation 157
14. Question 157
15. Question 158
16. Something To Make 158
17. Summarizing 158
Notes 158
References 160
Chapter 8: Acme Creates New Worlds: Klein Bottles and Other Surfaces 162
Scene 1 162
1. Your Turn 168
2. Your Turn 168
Scene 2 169
3. Your Turn 171
Scene 3 171
Investigations, Questions, Puzzles, and More 175
1. Puzzle 175
2. Puzzle 175
3. Investigation 176
4. Investigation 178
5. Gathering Evidence: Möbius Shorts 178
6. Gathering Evidence 178
7. Pattern Recognition 179
8. Investigation 179
9. Something To Make 180
10. Something To Make 180
11. A Game 182
12. Investigation 183
13. Experiment 183
14. Question 184
15. Question 184
16. Something To Make 184
17. Summarizing 186
Notes 186
References 188
Chapter 9: Acme Makes Order Out of Chaos: Surface Sums and Euler Numbers 190
Scene 1 190
1. Your Turn 192
2. Your Turn 195
Scene 2 197
4. Your Turn 200
Scene 3 200
5. Your Turn 206
Scene 4 206
Investigations, Questions, Puzzles, and More 210
1. Gathering Evidence 210
2. Question 210
3. Question 211
4. Investigation 211
5. Investigation 211
6. Question 211
7. Question 211
8. Question 212
9. Investigation 212
10. Question 212
11. Investigation 212
12. Investigation 213
13. Investigation 213
14. Investigation 214
15. Summarizing 214
Notes 214
References 215
Chapter 10: Acme Classifies Surfaces 218
Scene 1 218
1. Your Turn 221
Scene 2 222
2. Your Turn 224
3. Your Turn 225
Scene 3 225
4. Your Turn 228
Investigations, Questions, Puzzles, and More 231
1. Investigation 231
2. Investigation 231
3. Investigation 231
4. Investigation 231
5. Investigation 232
6. Question 232
7. Question 232
8. Question 232
9. Question 232
10. Question 232
11. Question 233
12. Investigation 233
13. Investigation 233
14. Investigation 233
15. Investigation 234
16. Summarizing 234
Notes 235
References 236
Chapter 11: Acme Encounters the Fourth Dimension 238
Scene 1 238
1. Your Turn 241
2. Your Turn 241
Scene 2 241
4. Your Turn 246
5. Your Turn 246
Scene 3 246
6. Your Turn 249
7. Your Turn 249
Scene 4 249
Investigations, Questions, Puzzles, and More 257
1. Investigation 257
2. Investigation 257
3. Question 257
4. Investigation 258
5. Investigation 258
6. Question 259
7. Investigation 260
8. Investigation 261
9. Investigation 261
10. Investigation 261
11. Investigation 261
12. Question 262
13. Summarizing 262
Notes 262
References 265
Chapter 12: Acme Colors Maps on Surfaces: Heawood's Estimate 266
1. Your Turn 268
2. Your Turn 269
3. Your Turn 271
4. Your Turn 274
Investigations, Questions, Puzzles and More 275
1. Investigation 275
2. Investigation 276
3. Question 276
4. Investigation 276
5. Investigation 276
6. Investigation 276
7. Investigation 276
8. Investigation 277
9. Investigation 277
10. Question 277
11. Question 277
12. Investigation 277
13. Investigation 277
14. Construction 277
15. Construction 278
16. Construction 278
17. Question 279
18. Summarizing 279
Notes 280
References 283
Chapter 13: Acme Gets All Tied Up with Knots 284
Scene 1 284
Your Turn 1 288
Your Turn 2 289
Your Turn 3 289
Your Turn 4 290
Your Turn 5 290
Your Turn 6 294
Scene 2 294
Your Turn 7 295
Your Turn 8 297
Your Turn 9 297
Your Turn 10 300
Scene 3 300
Your Turn 11 303
Your Turn 12 305
Your Turn 13 306
Scene 4 307
Investigations, Questions, Puzzles, and More 312
Investigation 1 312
Investigation 2 313
Investigation 3 313
Investigation 4 313
Investigation 5 313
Investigation 6 315
Investigation 7 317
Investigation 8 317
Investigation 9 319
Question 10 320
Investigation 11 320
Question 12 320
Question 13 320
Investigation 14 320
Question 15 321
Summary 16 321
Notes 321
References 325
Chapter 14: Where To Go from Here: Projects 326
Project Ideas 326
I. Map Coloring 327
II. Surfaces 328
III. Surfaces and Maps 331
IV. Three-Space 332
V. Knots and Links 333
VI. Miscellaneous 335
Resources 337
Communicating Project Results 339
Nitty-Gritty on Putting Together an Effective Display 341
Project Evaluation 342
The bottom line 343
Index 344

Produkt-Details

• Erscheint lt. Verlag: 7.8.2007
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-049266-5 / 0080492665
• ISBN-13: 978-0-08-049266-7 / 9780080492667