Explorations in Topology: Map Coloring, Surfaces and Knots
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Table of contents
- Title pagei
- Copyright pageii
- Table of Contentsv
- Prefacevii
- Format of the Chaptersviii
- Overview of the Chaptersix
- Success in the Course this Book is based onx
- Instructor’s Manualx
- Acknowledgmentsxi
- Chapter 1: Acme Does Maps and Considers Coloring Them1
- Scene 11
- 1. Your Turn5
- 2. Your Turn5
- Scene 25
- 3. Your Turn6
- 4. Your Turn9
- Scene 39
- 5. Your Turn10
- 6. Your Turn11
- Scene 411
- Investigations, Questions, Puzzles, and More15
- 1. Investigation15
- 2. Investigation16
- 3. Investigation16
- 4. Investigation17
- 5. Question18
- 6. Question18
- 7. Question18
- 8. Investigation18
- 9. Investigation20
- 10. Question20
- 11. Puzzle20
- 12. Question21
- 13. Gathering Evidence22
- 14. Puzzle23
- 15. Summarizing23
- Notes23
- References25
- Chapter 2: Acme Adds Tours27
- Scene 127
- Scene 233
- 1. Your Turn34
- Scene 335
- 2. Your Turn36
- Scene 437
- 3. Your Turn38
- 4. Your Turn39
- Investigations, Questions, Puzzles, and More40
- 1. Investigation40
- 2. Investigation40
- 3. Investigation40
- 4. Puzzle40
- 5. Investigation41
- 6. Investigation41
- 7. Question42
- 8. Investigation42
- 9. Investigation42
- 10. Investigation43
- 11. Puzzle43
- 12. Puzzle44
- 13. Question44
- 14. Summarizing44
- 15. Connecting45
- Notes45
- References47
- Chapter 3: Acme Collects Data from Maps49
- Scene 149
- 1. Your Turn50
- Scene 251
- 2. Your Turn53
- Scene 353
- 3. Your Turn60
- Scene 460
- 4. Your Turn61
- 5. Your Turn63
- 6. Your Turn64
- 7. Your Turn65
- Scene 565
- Investigations, Questions, Puzzles, and More67
- 1. Investigation67
- 2. Gathering Data67
- 3. Investigation68
- 4. Investigation68
- 5. Gathering Data68
- 6. Question69
- 7. Question69
- 8. Question69
- 9. Question69
- 10. Question69
- 11. Investigation70
- 12. Investigation70
- 13. Investigation70
- 14. Summarizing70
- Notes70
- References71
- Chapter 4: Acme Collects More Data, Proves a Theorem, and Returns to Coloring Maps73
- Scene 173
- 1. Your Turn76
- Scene 276
- 2. Your Turn80
- Scene 380
- Investigations, Questions, Puzzles, and More83
- 1. Investigation83
- 2. Investigation84
- 3. Investigation84
- 4. Investigation84
- 5. Investigation84
- 6. Question85
- 7. Question85
- 8. Question85
- 9. Puzzle86
- 10. Summarizing86
- Notes87
- References87
- Chapter 5: Acme’s Solicitor Proves a Theorem: the Four-Color Conjecture89
- Scene 189
- Investigations, Questions, Puzzles, and More94
- 1. Gathering Evidence94
- 2. Gathering Evidence95
- 3. Game95
- 4. Game96
- 5. Investigation96
- 6. Investigation98
- 7. Looking Back98
- Notes98
- References102
- Chapter 6: Acme Adds Doughnuts to Its Repertoire103
- Scene 1103
- 1. Your Turn107
- 2. Your Turn107
- 3. Your Turn109
- Scene 2109
- 4. Your Turn110
- Scene 3111
- 5. Your Turn112
- Scene 4114
- Investigations, Questions, Puzzles, and More116
- 1. Investigation116
- 2. Investigation116
- 3. Question116
- 4. Investigation116
- 5. Investigation116
- 6. Investigation116
- 7. Question117
- 8. Question117
- 9. Question117
- 10. Question117
- 11. Question118
- 12. Game118
- 13. Puzzle118
- 14. Something to Make119
- 15. Investigation121
- 16. Summarizing121
- 17. Connecting121
- Notes121
- References124
- Chapter 7: Acme Considers the Möbius Strip125
- Scene 1125
- 1. Your Turn127
- Scene 2127
- 2. Your Turn129
- Scene 3129
- 3. Your Turn130
- 4. Your Turn131
- 5. Your Turn131
- 6. Your Turn132
- 7. Your Turn132
- Scene 4132
- Investigations, Questions, Puzzles, and More138
- 1. Gathering Evidence138
- 2. Investigation138
- 3. Generalizing139
- 4. Question139
- 5. Gathering Evidence139
- 6. Investigation140
- 7. Investigation140
- Edges of Twisted Strips141
- 8. Investigation142
- Cutting Twisted Strips Down the Middle142
- 9. Investigation143
- 10. Things To Make143
- 11. Something To Make: Möbius Shorts143
- 12. Game144
- 13. Investigation144
- 14. Question144
- 15. Question145
- 16. Something To Make145
- 17. Summarizing145
- Notes145
- References147
- Chapter 8: Acme Creates New Worlds: Klein Bottles and Other Surfaces149
- Scene 1149
- 1. Your Turn155
- 2. Your Turn155
- Scene 2156
- 3. Your Turn158
- Scene 3158
- Investigations, Questions, Puzzles, and More162
- 1. Puzzle162
- 2. Puzzle162
- 3. Investigation163
- 4. Investigation165
- 5. Gathering Evidence: Möbius Shorts165
- 6. Gathering Evidence165
- 7. Pattern Recognition166
- 8. Investigation166
- 9. Something To Make167
- 10. Something To Make167
- 11. A Game169
- 12. Investigation170
- 13. Experiment170
- 14. Question171
- 15. Question171
- 16. Something To Make171
- 17. Summarizing173
- Notes173
- References175
- Chapter 9: Acme Makes Order Out of Chaos: Surface Sums and Euler Numbers177
- Scene 1177
- 1. Your Turn179
- 2. Your Turn182
- Scene 2184
- 4. Your Turn187
- Scene 3187
- 5. Your Turn193
- Scene 4193
- Investigations, Questions, Puzzles, and More197
- 1. Gathering Evidence197
- 2. Question197
- 3. Question198
- 4. Investigation198
- 5. Investigation198
- 6. Question198
- 7. Question198
- 8. Question199
- 9. Investigation199
- 10. Question199
- 11. Investigation199
- 12. Investigation200
- 13. Investigation200
- 14. Investigation201
- 15. Summarizing201
- Notes201
- References202
- Chapter 10: Acme Classifies Surfaces205
- Scene 1205
- 1. Your Turn208
- Scene 2209
- 2. Your Turn211
- 3. Your Turn212
- Scene 3212
- 4. Your Turn215
- Investigations, Questions, Puzzles, and More218
- 1. Investigation218
- 2. Investigation218
- 3. Investigation218
- 4. Investigation218
- 5. Investigation219
- 6. Question219
- 7. Question219
- 8. Question219
- 9. Question219
- 10. Question219
- 11. Question220
- 12. Investigation220
- 13. Investigation220
- 14. Investigation220
- 15. Investigation221
- 16. Summarizing221
- Notes222
- References223
- Chapter 11: Acme Encounters the Fourth Dimension225
- Scene 1225
- 1. Your Turn228
- 2. Your Turn228
- Scene 2228
- 4. Your Turn233
- 5. Your Turn233
- Scene 3233
- 6. Your Turn236
- 7. Your Turn236
- Scene 4236
- Investigations, Questions, Puzzles, and More244
- 1. Investigation244
- 2. Investigation244
- 3. Question244
- 4. Investigation245
- 5. Investigation245
- 6. Question246
- 7. Investigation247
- 8. Investigation248
- 9. Investigation248
- 10. Investigation248
- 11. Investigation248
- 12. Question249
- 13. Summarizing249
- Notes249
- References252
- Chapter 12: Acme Colors Maps on Surfaces: Heawood's Estimate253
- 1. Your Turn255
- 2. Your Turn256
- 3. Your Turn258
- 4. Your Turn261
- Investigations, Questions, Puzzles and More262
- 1. Investigation262
- 2. Investigation263
- 3. Question263
- 4. Investigation263
- 5. Investigation263
- 6. Investigation263
- 7. Investigation263
- 8. Investigation264
- 9. Investigation264
- 10. Question264
- 11. Question264
- 12. Investigation264
- 13. Investigation264
- 14. Construction264
- 15. Construction265
- 16. Construction265
- 17. Question266
- 18. Summarizing266
- Notes267
- References270
- Chapter 13: Acme Gets All Tied Up with Knots271
- Scene 1271
- Your Turn 1275
- Your Turn 2276
- Your Turn 3276
- Your Turn 4277
- Your Turn 5277
- Your Turn 6281
- Scene 2281
- Your Turn 7282
- Your Turn 8284
- Your Turn 9284
- Your Turn 10287
- Scene 3287
- Your Turn 11290
- Your Turn 12292
- Your Turn 13293
- Scene 4294
- Investigations, Questions, Puzzles, and More299
- Investigation 1299
- Investigation 2300
- Investigation 3300
- Investigation 4300
- Investigation 5300
- Investigation 6302
- Investigation 7304
- Investigation 8304
- Investigation 9306
- Question 10307
- Investigation 11307
- Question 12307
- Question 13307
- Investigation 14307
- Question 15308
- Summary 16308
- Notes308
- References312
- Chapter 14: Where To Go from Here: Projects313
- Project Ideas313
- I. Map Coloring314
- II. Surfaces315
- III. Surfaces and Maps318
- IV. Three-Space319
- V. Knots and Links320
- VI. Miscellaneous322
- Resources324
- Communicating Project Results326
- Nitty-Gritty on Putting Together an Effective Display328
- Project Evaluation329
- The bottom line330
- Index331
This book 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
* 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