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Analysis of Mathematical Fiction with Geometric Themes

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Analysis of Mathematical Fiction with Geometric Themes ANALYSIS OF MATHEMATICAL FICTION WITH GEOMETRIC THEMES Jennifer Rebecca Shloming Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2012 © 2012 Jennifer Rebecca Shloming All rights reserved ABSTRACT Analysis of Mathematical Fiction with Geometric Themes Jennifer Rebecca Shloming Analysis of mathematical fiction with geometric themes is a study that connects the genre of mathematical fiction with informal learning. This study provides an analysis of 26 sources that include novels and short stories of mathematical fiction with regard to plot, geometric theme, cultural theme, and presentation. The authors’ mathematical backgrounds are presented as they relate to both geometric and cultural themes. These backgrounds range from having little mathematical training to advance graduate work culminating in a Ph.D. in mathematics. This thesis demonstrated that regardless of background, the authors could write a mathematical fiction novel or short story with a dominant geometric theme. The authors’ pedagogical approaches to delivering the geometric themes are also discussed. Applications from this study involve a pedagogical component that can be used in a classroom setting. All the sources analyzed in this study are fictional, but the geometric content is factual. Six categories of geometric topics were analyzed: plane geometry, solid geometry, projective geometry, axiomatics, topology, and the historical foundations of geometry. Geometry textbooks aligned with these categories were discussed with regard to mathematical fiction and formal learning. Cultural patterns were also analyzed for each source of mathematical fiction. There were also an analysis of the integration of cultural and geometric themes in the 26 sources of mathematical fiction; some of the cultural patterns discussed are gender bias, art, music, academia, mysticism, and social issues. On the basis of this discussion, recommendations for future studies involving the use of mathematical fiction were made. TABLE OF CONTENTS Chapter I INTRODUCTION ..................................................................................................... 1 Need for the Study ..................................................................................................... 1 Purpose of the Study .................................................................................................. 6 Procedures of the Study ............................................................................................. 7 II LITERATURE REVIEW .......................................................................................... 10 Overview.................................................................................................................... 10 Informal and Formal Learning................................................................................... 12 Informal and Formal Geometry ................................................................................. 17 Linking Literature and Mathematics.......................................................................... 19 Integrating Literature and Mathematics..................................................................... 20 Using Literature to Teach Mathematics..................................................................... 23 Mathematical Fiction: Research and Usage............................................................... 33 An Introduction to Mathematical Fiction ......................................................... 33 Current Usage and Research of Mathematical Fiction ..................................... 37 Summary.................................................................................................................... 41 III METHODOLOGY .................................................................................................... 43 Overview.................................................................................................................... 43 Sources of Data.......................................................................................................... 45 Grounded Theory....................................................................................................... 49 Summary.................................................................................................................... 51 i Chapter IV ANALYSIS OF MATHEMATICAL FICTION: TWENTY-SIX NOVELS AND SHORT STORIES WITH GEOMETRIC THEMES...................... 53 Overview.................................................................................................................... 53 The Three Categories of Mathematical Fiction with Geometric Themes.................. 53 Category I.......................................................................................................... 54 Category II ........................................................................................................ 66 Category III....................................................................................................... 74 Summary.................................................................................................................... 77 V THE AUTHORS’ BACKGROUNDS AND LINKS TO MATHEMATICS............ 79 Overview.................................................................................................................... 79 Group 1: Writers with Little Mathematical or Scientific Training............................ 80 Group 2: Writers with a Background in a Related Field to Mathematics.................. 83 Group 3: Writers with a Strong Background in Mathematics ................................... 85 Summary.................................................................................................................... 91 VI DISCUSSION............................................................................................................ 92 Mathematical Fiction and Formal Learning .............................................................. 92 Geometry Textbooks Aligned with Plane Geometry [P/G].............................. 94 Geometry Textbooks Aligned with Solid Geometry [S/G] .............................. 95 Geometry Textbooks Aligned with Historical Foundations of Geometry [HF/G]........................................................................................ 96 Geometry Textbooks Aligned with Topology [Tp].......................................... 99 Geometry Textbooks Aligned with Axiomatics [Ax].......................................100 ii Chapter VI Geometry Textbooks Aligned with Projective Geometry [Pj] .........................101 (cont’d) Organizing the Collected Data of the Study ..............................................................105 On the Topics in the Fictions............................................................................105 Dimensionality Connected to Mathematical Fiction ........................................106 Famous Theorems and Problems Connected to Mathematical Fiction ............107 Geometric Fallacies and Mysticism Connected to Mathematical Fiction ........108 Pythagoras and the Pythagoreans Connected to Mathematical Fiction............109 Famous Mathematicians Connected to Mathematical Fiction..........................111 Axiomatics Connected to Mathematical Fiction...............................................114 On Pedagogical Approaches......................................................................................116 Informal Presentations of the Geometric Themes ............................................116 Formal Presentations of the Geometric Themes...............................................119 What Are the Authors’ Mathematical Backgrounds?................................................122 Cultural Themes.........................................................................................................124 Gender Bias Connected to Mathematical Fiction .............................................125 Art and Music Connected to Mathematical Fiction..........................................127 Mysticism and Magic Connected to Mathematical Fiction ..............................132 Academia Connected to Mathematical Fiction.................................................134 Social Issues Connected to Mathematical Fiction ............................................135 Warfare Connected to Mathematical Fiction....................................................137 Poetry Connected to Mathematical Fiction.......................................................137 Philosophy Connected to Mathematical Fiction...............................................138 iii Chapter VII SUMMARY, FINDINGS, AND RECOMMENDATIONS......................................140 Limitations of the Study.............................................................................................146 Recommendations for Future Studies........................................................................148 Recommendations for Educators ...............................................................................150 REFERENCES ..........................................................................................................154 APPENDIX................................................................................................................173 iv LIST OF TABLES Table 1 Number of Works of Mathematical Fiction Published between 2000 and 2009.......148
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