Gay Explorations in Topology

Map Coloring, Surfaces and Knots
1. Auflage 2007
ISBN: 978-0-08-049266-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 352 Seiten, Web PDF

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

Autoren/Hrsg.

Gay, David

Inhaltsverzeichnis

1;Cover;1
2;Title page;2
3;Copyright page;3
4;Table of Contents;6
5;Preface;8
5.1;Format of the Chapters;9
5.2;Overview of the Chapters;10
5.3;Success in the Course this Book is based on;11
5.4;Instructor’s Manual;11
6;Acknowledgments;12
7;Chapter 1: Acme Does Maps and Considers Coloring Them;14
7.1;Scene 1;14
7.2;1. Your Turn;18
7.3;2. Your Turn;18
7.4;Scene 2;18
7.5;3. Your Turn;19
7.6;4. Your Turn;22
7.7;Scene 3;22
7.8;5. Your Turn;23
7.9;6. Your Turn;24
7.10;Scene 4;24
7.11;Investigations, Questions, Puzzles, and More;28
7.12;1. Investigation;28
7.13;2. Investigation;29
7.14;3. Investigation;29
7.15;4. Investigation;30
7.16;5. Question;31
7.17;6. Question;31
7.18;7. Question;31
7.19;8. Investigation;31
7.20;9. Investigation;33
7.21;10. Question;33
7.22;11. Puzzle;33
7.23;12. Question;34
7.24;13. Gathering Evidence;35
7.25;14. Puzzle;36
7.26;15. Summarizing;36
7.27;Notes;36
7.28;References;38
8;Chapter 2: Acme Adds Tours;40
8.1;Scene 1;40
8.2;Scene 2;46
8.3;1. Your Turn;47
8.4;Scene 3;48
8.5;2. Your Turn;49
8.6;Scene 4;50
8.7;3. Your Turn;51
8.8;4. Your Turn;52
8.9;Investigations, Questions, Puzzles, and More;53
8.10;1. Investigation;53
8.11;2. Investigation;53
8.12;3. Investigation;53
8.13;4. Puzzle;53
8.14;5. Investigation;54
8.15;6. Investigation;54
8.16;7. Question;55
8.17;8. Investigation;55
8.18;9. Investigation;55
8.19;10. Investigation;56
8.20;11. Puzzle;56
8.21;12. Puzzle;57
8.22;13. Question;57
8.23;14. Summarizing;57
8.24;15. Connecting;58
8.25;Notes;58
8.26;References;60
9;Chapter 3: Acme Collects Data from Maps;62
9.1;Scene 1;62
9.2;1. Your Turn;63
9.3;Scene 2;64
9.4;2. Your Turn;66
9.5;Scene 3;66
9.6;3. Your Turn;73
9.7;Scene 4;73
9.8;4. Your Turn;74
9.9;5. Your Turn;76
9.10;6. Your Turn;77
9.11;7. Your Turn;78
9.12;Scene 5;78
9.13;Investigations, Questions, Puzzles, and More;80
9.14;1. Investigation;80
9.15;2. Gathering Data;80
9.16;3. Investigation;81
9.17;4. Investigation;81
9.18;5. Gathering Data;81
9.19;6. Question;82
9.20;7. Question;82
9.21;8. Question;82
9.22;9. Question;82
9.23;10. Question;82
9.24;11. Investigation;83
9.25;12. Investigation;83
9.26;13. Investigation;83
9.27;14. Summarizing;83
9.28;Notes;83
9.29;References;84
10;Chapter 4: Acme Collects More Data, Proves a Theorem, and Returns to Coloring Maps;86
10.1;Scene 1;86
10.2;1. Your Turn;89
10.3;Scene 2;89
10.4;2. Your Turn;93
10.5;Scene 3;93
10.6;Investigations, Questions, Puzzles, and More;96
10.7;1. Investigation;96
10.8;2. Investigation;97
10.9;3. Investigation;97
10.10;4. Investigation;97
10.11;5. Investigation;97
10.12;6. Question;98
10.13;7. Question;98
10.14;8. Question;98
10.15;9. Puzzle;99
10.16;10. Summarizing;99
10.17;Notes;100
10.18;References;100
11;Chapter 5: Acme’s Solicitor Proves a Theorem: the Four-Color Conjecture;102
11.1;Scene 1;102
11.2;Investigations, Questions, Puzzles, and More;107
11.3;1. Gathering Evidence;107
11.4;2. Gathering Evidence;108
11.5;3. Game;108
11.6;4. Game;109
11.7;5. Investigation;109
11.8;6. Investigation;111
11.9;7. Looking Back;111
11.10;Notes;111
11.11;References;115
12;Chapter 6: Acme Adds Doughnuts to Its Repertoire;116
12.1;Scene 1;116
12.2;1. Your Turn;120
12.3;2. Your Turn;120
12.4;3. Your Turn;122
12.5;Scene 2;122
12.6;4. Your Turn;123
12.7;Scene 3;124
12.8;5. Your Turn;125
12.9;Scene 4;127
12.10;Investigations, Questions, Puzzles, and More;129
12.11;1. Investigation;129
12.12;2. Investigation;129
12.13;3. Question;129
12.14;4. Investigation;129
12.15;5. Investigation;129
12.16;6. Investigation;129
12.17;7. Question;130
12.18;8. Question;130
12.19;9. Question;130
12.20;10. Question;130
12.21;11. Question;131
12.22;12. Game;131
12.23;13. Puzzle;131
12.24;14. Something to Make;132
12.25;15. Investigation;134
12.26;16. Summarizing;134
12.27;17. Connecting;134
12.28;Notes;134
12.29;References;137
13;Chapter 7: Acme Considers the Möbius Strip;138
13.1;Scene 1;138
13.2;1. Your Turn;140
13.3;Scene 2;140
13.4;2. Your Turn;142
13.5;Scene 3;142
13.6;3. Your Turn;143
13.7;4. Your Turn;144
13.8;5. Your Turn;144
13.9;6. Your Turn;145
13.10;7. Your Turn;145
13.11;Scene 4;145
13.12;Investigations, Questions, Puzzles, and More;151
13.13;1. Gathering Evidence;151
13.14;2. Investigation;151
13.15;3. Generalizing;152
13.16;4. Question;152
13.17;5. Gathering Evidence;152
13.18;6. Investigation;153
13.19;7. Investigation;153
13.20;Edges of Twisted Strips;154
13.21;8. Investigation;155
13.22;Cutting Twisted Strips Down the Middle;155
13.23;9. Investigation;156
13.24;10. Things To Make;156
13.25;11. Something To Make: Möbius Shorts;156
13.26;12. Game;157
13.27;13. Investigation;157
13.28;14. Question;157
13.29;15. Question;158
13.30;16. Something To Make;158
13.31;17. Summarizing;158
13.32;Notes;158
13.33;References;160
14;Chapter 8: Acme Creates New Worlds: Klein Bottles and Other Surfaces;162
14.1;Scene 1;162
14.2;1. Your Turn;168
14.3;2. Your Turn;168
14.4;Scene 2;169
14.5;3. Your Turn;171
14.6;Scene 3;171
14.7;Investigations, Questions, Puzzles, and More;175
14.8;1. Puzzle;175
14.9;2. Puzzle;175
14.10;3. Investigation;176
14.11;4. Investigation;178
14.12;5. Gathering Evidence: Möbius Shorts;178
14.13;6. Gathering Evidence;178
14.14;7. Pattern Recognition;179
14.15;8. Investigation;179
14.16;9. Something To Make;180
14.17;10. Something To Make;180
14.18;11. A Game;182
14.19;12. Investigation;183
14.20;13. Experiment;183
14.21;14. Question;184
14.22;15. Question;184
14.23;16. Something To Make;184
14.24;17. Summarizing;186
14.25;Notes;186
14.26;References;188
15;Chapter 9: Acme Makes Order Out of Chaos: Surface Sums and Euler Numbers;190
15.1;Scene 1;190
15.2;1. Your Turn;192
15.3;2. Your Turn;195
15.4;Scene 2;197
15.5;4. Your Turn;200
15.6;Scene 3;200
15.7;5. Your Turn;206
15.8;Scene 4;206
15.9;Investigations, Questions, Puzzles, and More;210
15.10;1. Gathering Evidence;210
15.11;2. Question;210
15.12;3. Question;211
15.13;4. Investigation;211
15.14;5. Investigation;211
15.15;6. Question;211
15.16;7. Question;211
15.17;8. Question;212
15.18;9. Investigation;212
15.19;10. Question;212
15.20;11. Investigation;212
15.21;12. Investigation;213
15.22;13. Investigation;213
15.23;14. Investigation;214
15.24;15. Summarizing;214
15.25;Notes;214
15.26;References;215
16;Chapter 10: Acme Classifies Surfaces;218
16.1;Scene 1;218
16.2;1. Your Turn;221
16.3;Scene 2;222
16.4;2. Your Turn;224
16.5;3. Your Turn;225
16.6;Scene 3;225
16.7;4. Your Turn;228
16.8;Investigations, Questions, Puzzles, and More;231
16.9;1. Investigation;231
16.10;2. Investigation;231
16.11;3. Investigation;231
16.12;4. Investigation;231
16.13;5. Investigation;232
16.14;6. Question;232
16.15;7. Question;232
16.16;8. Question;232
16.17;9. Question;232
16.18;10. Question;232
16.19;11. Question;233
16.20;12. Investigation;233
16.21;13. Investigation;233
16.22;14. Investigation;233
16.23;15. Investigation;234
16.24;16. Summarizing;234
16.25;Notes;235
16.26;References;236
17;Chapter 11: Acme Encounters the Fourth Dimension;238
17.1;Scene 1;238
17.2;1. Your Turn;241
17.3;2. Your Turn;241
17.4;Scene 2;241
17.5;4. Your Turn;246
17.6;5. Your Turn;246
17.7;Scene 3;246
17.8;6. Your Turn;249
17.9;7. Your Turn;249
17.10;Scene 4;249
17.11;Investigations, Questions, Puzzles, and More;257
17.12;1. Investigation;257
17.13;2. Investigation;257
17.14;3. Question;257
17.15;4. Investigation;258
17.16;5. Investigation;258
17.17;6. Question;259
17.18;7. Investigation;260
17.19;8. Investigation;261
17.20;9. Investigation;261
17.21;10. Investigation;261
17.22;11. Investigation;261
17.23;12. Question;262
17.24;13. Summarizing;262
17.25;Notes;262
17.26;References;265
18;Chapter 12: Acme Colors Maps on Surfaces: Heawood's Estimate;266
18.1;1. Your Turn;268
18.2;2. Your Turn;269
18.3;3. Your Turn;271
18.4;4. Your Turn;274
18.5;Investigations, Questions, Puzzles and More;275
18.6;1. Investigation;275
18.7;2. Investigation;276
18.8;3. Question;276
18.9;4. Investigation;276
18.10;5. Investigation;276
18.11;6. Investigation;276
18.12;7. Investigation;276
18.13;8. Investigation;277
18.14;9. Investigation;277
18.15;10. Question;277
18.16;11. Question;277
18.17;12. Investigation;277
18.18;13. Investigation;277
18.19;14. Construction;277
18.20;15. Construction;278
18.21;16. Construction;278
18.22;17. Question;279
18.23;18. Summarizing;279
18.24;Notes;280
18.25;References;283
19;Chapter 13: Acme Gets All Tied Up with Knots;284
19.1;Scene 1;284
19.2;Your Turn 1;288
19.3;Your Turn 2;289
19.4;Your Turn 3;289
19.5;Your Turn 4;290
19.6;Your Turn 5;290
19.7;Your Turn 6;294
19.8;Scene 2;294
19.9;Your Turn 7;295
19.10;Your Turn 8;297
19.11;Your Turn 9;297
19.12;Your Turn 10;300
19.13;Scene 3;300
19.14;Your Turn 11;303
19.15;Your Turn 12;305
19.16;Your Turn 13;306
19.17;Scene 4;307
19.18;Investigations, Questions, Puzzles, and More;312
19.19;Investigation 1;312
19.20;Investigation 2;313
19.21;Investigation 3;313
19.22;Investigation 4;313
19.23;Investigation 5;313
19.24;Investigation 6;315
19.25;Investigation 7;317
19.26;Investigation 8;317
19.27;Investigation 9;319
19.28;Question 10;320
19.29;Investigation 11;320
19.30;Question 12;320
19.31;Question 13;320
19.32;Investigation 14;320
19.33;Question 15;321
19.34;Summary 16;321
19.35;Notes;321
19.36;References;325
20;Chapter 14: Where To Go from Here: Projects;326
20.1;Project Ideas;326
20.2;I. Map Coloring;327
20.3;II. Surfaces;328
20.4;III. Surfaces and Maps;331
20.5;IV. Three-Space;332
20.6;V. Knots and Links;333
20.7;VI. Miscellaneous;335
20.8;Resources;337
20.9;Communicating Project Results;339
20.10;Nitty-Gritty on Putting Together an Effective Display;341
20.11;Project Evaluation;342
20.12;The bottom line;343
21;Index;344