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By Julian F. Fleron and Volker Ecke Discovering the Art of Mathematics Geometry by Julian F. Fleron and Volker Ecke with Philip K. Hotchkiss and Christine von Renesse c 2010{2015 (Rev.: 2015-09-03) Working Draft: May be copied and distributed for educational purposes only. Such use requires notification of the authors. (Send email to jfl[email protected] .) Not for any other quotation or distribution without written consent of the authors. For more information about this manuscript or the larger project to develop a library of 10 inquiry-based guides, please see http://www. artofmathematics.org/. DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse ii Acknowledgements This work has been supported by the National Science Foundation under awards NSF0836943 and NSF1229515. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. These materials are also based on work supported by Project PRIME which was made possible by a generous gift from Mr. Harry Lucas. These materials were originally developed for use in the course Mathematical Explorations at Westfield State. They are now part of the curriculum library for Discovering the Art of Mathe- matics, all of whose volumes are freely available at http://artofmathematics.org . The Department of Mathematics at Westfield State has provided fertile ground and enthusias- tic support for pedagogical innovation, scholarship on teaching and learning in mathematics, and classroom experimentation. Under the direction of Chairs Karin Vorwerk and previously John Judge, and the organization of Deb Samwell, the department continues to be deeply committed to student learning - including in its many required core courses. We are thankful to all depart- mental faculty, current and past, for their support of Discovering the Art of Mathematics. We are grateful for the support of many at our institution over many years. In particular we thank those in Academic Affairs, our fine library faculty, Robert Martin, William Lopes, Louann D'Angelo, Elizabeth Preston, and Andrew Bonacci. We are grateful for the support our our Advisory Board: Jere Confrey, David Farmer, Sandra Laursen, Carmel Schettino, Deborah Schifter and Dorothy Wallace. We thank Lee Zia for his guidance and encouragement. Student interns Jillian Bates, Chelsea Baker, Joey Grabowski, Jen Roy, Rachel Cloutier, Zach Lancto, Alyssa Danilow, and Barbara Mulvaney helped with many tasks that supported the project - Jillian, Chelsea and Joey trying out many chapters in this book. Thanks to Sage Fox Consulting, Ken Rath and Emily DeHaro-Otero in particular, for their work on assessment and evaluation. A great debt of gratitude to Tom Hoogendyk who designed and keeps running our Discovering the Art of Mathematics website. In different incarnations, this work was reviewed and/or beta tested by Robert Stein and iii DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse Fumiko Futamura. We are indebted to Analisa Crannell, Marc Frantz, Jenny Livingstone, and Robert Bosch for inspiration and many wonderful conversations about geometry. The authors are grateful for their many insights and detailed work. Any and all imperfections remain with the authors. c 2010{2015 by Julian Fleron, Phil Hotchkiss, Volker Ecke, Christine von Renesse. iv Contents Acknowledgements iii Preface 1 0.1 Notes to the Explorer . 1 0.2 Navigating This Book . 3 0.3 Directions for the Guides . 4 0.3.1 Notes for Guides on the Geometry Exploration . 5 0.3.2 Chapter Dependencies . 7 1 Introduction 9 1.1 The Many Faces of Geometry . 9 1.2 Dimension . 10 1.3 Geometry Around Us . 12 1.4 Geometric Construction . 16 1.5 Conclusion . 18 1.6 Further Investigations . 19 1.6.1 Kepler's Laws of Planetary Motion . 19 1.6.2 Platonic Solids and Kepler's Solar System Model . 19 1.6.3 The Six-Cornered Snow Flake as an Archetype for Learning . 21 2 Navigating Between Dimensions 23 2.1 Flatland { A Romance of Many Dimensions . 23 2.2 Cross Sections . 27 2.3 The Flatland Game . 31 2.4 Making Your Own Flatland Movie . 33 2.5 Sliceforms . 33 2.6 More General Cross Sections . 37 2.7 Alternate Approach: Projections . 38 2.8 Connections . 38 3 Dimensional Interplay in Other Fields 43 3.1 Blueprints, building, and architecture . 43 3.1.1 Computer Aided Design (CAD) . 43 3.2 CAT scans, MRI, PET scans, and other medical imaging. 46 v DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 3.3 Geography, maps, and topography . 49 3.3.1 Topographic Maps: . 49 3.3.2 Maps and Sliceforms: . 50 3.4 3D Laser Scanning and 3D Printing . 53 3.4.1 Three-dimensional laser scanning and stereolithographic building . 53 3.4.2 3D Laser Scanning: . 53 3.4.3 3D Laser Building: . 55 3.5 CAVES and Virtual Reality Theaters . 56 3.5.1 CAVE Game Activity . 58 3.6 Blueprints, building, and architecture . 58 4 Visualizing Between Dimensions 59 4.1 Perspectives on Art . 59 4.2 Perspective Drawing . 60 4.2.1 One-Point Perspective . 60 4.2.2 Two-Point Perspective . 63 4.3 The Necker Cube . 65 4.4 Projection . 66 4.4.1 Parallel Projections . 66 4.4.2 Point Projections . 69 4.5 3D Drawing . 70 5 Climbing a Ladder to the Higher Dimensions 77 5.1 The Dimensional Ladder . 77 5.2 The Fourth Dimension in Art . 79 5.3 Drawing Shadows of the Higher Dimensions . 80 5.4 Patterns: Gear to Help Climb the Dimensional Ladder . 83 5.4.1 Point, Line, Square, Cube and Hypercube Components . 83 5.4.2 Euler's Formula for Polyhedra . 86 5.5 Coordinate Geometry . 88 5.6 Defining Dimension . 89 5.7 Implications of the Fourth Dimension . 93 5.8 The Fourth Dimension in Literature . 95 5.9 Further Investigations . 96 5.9.1 Different Types of Hypercube Projections . 96 5.9.2 Extending the Pattern of Hypercube Components . 96 5.9.3 Euler's Formula . 97 6 Inbetween the Dimensions - Fractals 101 6.1 Proportion and Scale . 101 6.2 Scaling Perimeters, Areas and Volumes . 104 6.3 Power Laws and Familiar Formulas . 108 6.4 Fractals . 110 6.4.1 The Sierpinski Triangle . 113 6.4.2 Koch Curve . 115 6.4.3 Cantor Set . 117 vi DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 6.5 Fractal Dimensions . 118 6.6 A Few More Fractals . 121 6.7 Conclusion . 122 6.8 Further Investigation . 123 6.8.1 Zipf's Law and the Pareto Principle . 123 6.8.2 Kleiber's Law . 124 6.8.3 More on Scaling Perimeters, Areas and Volumes . 124 6.8.4 Menger Sponge Surface Area . 125 6.9 Teacher Notes . 126 6.10 JF Notes . 127 6.10.1 Things still to get in there. 127 6.11 Scaling . 128 7 So, What is Geometry? [Working Draft] 133 7.1 Introduction . 133 7.2 Ancient Geometry . 134 7.3 Euclid and the Platonic Ideal . 140 7.4 Analytic Geometry . 142 7.5 Non-Euclidean Geometry . 142 7.6 Klein's Erlanger Program . 142 7.7 The Shape of the Universe . 144 7.8 So What is Geometry? . ..
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