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A Mathematical Tapestry

Book Description

This easy-to-read book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgments
  9. 1. Flexagons – A beginning thread
    1. 1.1 Four scientists at play
    2. 1.2 What are flexagons?
    3. 1.3 Hexaflexagons
    4. 1.4 Octaflexagons
  10. 2. Another thread – 1-period paper-folding
    1. 2.1 Should you always follow instructions?
    2. 2.2 Some ancient threads
    3. 2.3 Folding triangles and hexagons
    4. 2.4 Does this idea generalize?
    5. 2.5 Some bonuses
  11. 3. More paper-folding threads – 2-period paper-folding
    1. 3.1 Some basic ideas about polygons
    2. 3.2 Why does the FAT algorithm work?
    3. 3.3 Constructing a 7-gon
    4. *3.4 Some general proofs of convergence
  12. 4. A number-theory thread – Folding numbers, a number trick, and some tidbits
    1. 4.1 Folding numbers
    2. *4.2 Recognizing rational numbers of the form
    3. *4.3 Numerical examples and why 3 × 7 = 21 is a very special number fact
    4. 4.4 A number trick and two mathematical tidbits
  13. 5. The polyhedron thread – Building some polyhedra and defining a regular polyhedron
    1. 5.1 An intuitive approach to polyhedra
    2. 5.2 Constructing polyhedra from nets
    3. 5.3 What is a regular polyhedron?
  14. 6. Constructing dipyramids and rotating rings from straight strips of triangles
    1. 6.1 Preparing the pattern piece for a pentagonal dipyramid
    2. 6.2 Assembling the pentagonal dipyramid
    3. 6.3 Refinements for dipyramids
    4. 6.4 Constructing braided rotating rings of tetrahedra
    5. 6.5 Variations for rotating rings
    6. 6.6 More fun with rotating rings
  15. 7. Continuing the paper-folding and number-theory threads
    1. 7.1 Constructing an 11-gon
    2. *7.2 The quasi-order theorem
    3. *7.3 The quasi-order theorem when t = 3
    4. 7.4 Paper-folding connections with various famous number sequences
    5. 7.5 Finding the complementary factor and reconstructing the symbol
  16. 8. A geometry and algebra thread – Constructing, and using, Jennifer’s puzzle
    1. 8.1 Facts of life
    2. 8.2 Description of the puzzle
    3. 8.3 How to make the puzzle pieces
    4. 8.4 Assembling the braided tetrahedron
    5. 8.5 Assembling the braided octahedron
    6. 8.6 Assembling the braided cube
    7. 8.7 Some mathematical applications of Jennifer’s puzzle
  17. 9. A polyhedral geometry thread – Constructing braided Platonic solids and other woven polyhedra
    1. 9.1 A curious fact
    2. 9.2 Preparing the strips
    3. 9.3 Braiding the diagonal cube
    4. 9.4 Braiding the golden dodecahedron
    5. 9.5 Braiding the dodecahedron
    6. 9.6 Braiding the icosahedron
    7. 9.7 Constructing more symmetric tetrahedra, octahedra, and icosahedra
    8. 9.8 Weaving straight strips on other polyhedral surfaces
  18. 10. Combinatorial and symmetry threads
    1. 10.1 Symmetries of the cube
    2. 10.2 Symmetries of the regular octahedron and regular tetrahedron
    3. 10.3 Euler’s formula and Descartes’ angular deficiency
    4. 10.4 Some combinatorial properties of polyhedra
  19. 11. Some golden threads – Constructing more dodecahedra
    1. 11.1 How can there be more dodecahedra?
    2. 11.2 The small stellated dodecahedron
    3. 11.3 The great stellated dodecahedron
    4. 11.4 The great dodecahedron
    5. 11.5 Magical relationships between special dodecahedra
  20. 12. More combinatorial threads – Collapsoids
    1. 12.1 What is a collapsoid?
    2. 12.2 Preparing the cells, tabs, and flaps
    3. 12.3 Constructing a 12-celled polar collapsoid
    4. 12.4 Constructing a 20-celled polar collapsoid
    5. 12.5 Constructing a 30-celled polar collapsoid
    6. 12.6 Constructing a 12-celled equatorial collapsoid
    7. 12.7 Other collapsoids (for the experts)
    8. 12.8 How do we find other collapsoids?
  21. 13. Group theory – The faces of the trihexaflexagon
    1. 13.1 Group theory and hexaflexagons
    2. 13.2 How to build the special trihexaflexagon
    3. 13.3 The happy group
    4. 13.4 The entire group
    5. 13.5 A normal subgroup
    6. 13.6 What next?
  22. 14. Combinatorial and group-theoretical threads – Extended face planes of the Platonic solids
    1. 14.1 The question
    2. 14.2 Divisions of the plane
    3. 14.3 Some facts about the Platonic solids
    4. 14.4 Answering the main question
    5. 14.5 More general questions
  23. 15. A historical thread – Involving the Euler characteristic, Descartes’ total angular defect, and Pólya’s dream
    1. 15.1 Pólya’s speculation
    2. 15.2 Pólya’s dream
    3. *15.3 . . . The dream comes true
    4. *15.4 Further generalizations
  24. 16. Tying some loose ends together – Symmetry, group theory, homologues, and the Pólya enumeration theorem
    1. 16.1 Symmetry: A really big idea
    2. *16.2 Symmetry in geometry
    3. *16.3 Homologues
    4. *16.4 The Pólya enumeration theorem
    5. *16.5 Even and odd permutations
    6. 16.6 Epilogue: Pólya and ourselves – Mathematics, tea, and cakes
  25. 17. Returning to the number-theory thread – Generalized quasi-order and coach theorems
    1. 17.1 Setting the stage
    2. 17.2 The coach theorem
    3. 17.3 The generalized quasi-order theorem
    4. *17.4 The generalized coach theorem
    5. 17.5 Parlor tricks
    6. 17.6 A little linear algebra
    7. 17.7 Some open questions
  26. References
  27. Index