Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- 1 Flexagons - A beginning thread -- 1.1 Four scientists at play -- 1.2 What are flexagons? -- 1.3 Hexaflexagons -- Basic instructions -- 1.4 Octaflexagons -- 2 Another thread - 1-period paper-folding -- 2.1 Should you always follow instructions? -- A word to the wise -- Material error -- Geometrical error -- 2.2 Some ancient threads -- 2.3 Folding triangles and hexagons -- Why do we get equilateral triangles? -- Constructing bigger triangles -- Constructing hexagons -- 2.4 Does this idea generalize? -- Is there a general pattern to all this? -- 2.5 Some bonuses -- Uses for our folded tape -- 3 More paper-folding threads - 2-period paper-folding -- 3.1 Some basic ideas about polygons -- 3.2 Why does the FAT algorithm work? -- 3.3 Constructing a 7-gon -- 3.4 Some general proofs of convergence -- 4 A number-theory thread - Folding numbers, a number trick, and some tidbits -- 4.1 Folding numbers -- 4.2 Recognizing rational numbers of the form ta - 1/tb - 1 -- Case 1 -- Case 2 -- 4.3 Numerical examples -- 4.4 A number trick and two mathematical tidbits -- A number trick -- Tidbit 1 -- Tidbit 2 -- 5 The polyhedron thread - Building some polyhedra and defining a regular polyhedron -- 5.1 An intuitive approach to polyhedra -- 5.2 Constructing polyhedra from nets -- General instructions -- General instructions (alternative) -- Alternative construction -- Practical hints -- 5.3 What is a regular polyhedron? -- 6 Constructing dipyramids and rotating rings from straight strips of triangles -- 6.1 Preparing the pattern piece for a pentagonal dipyramid -- 6.2 Assembling the pentagonal dipyramid -- 6.3 Refinements for dipyramids -- 6.4 Constructing braided rotating rings of tetrahedra -- Practical hints -- 6.5 Variations for rotating rings.

Why do you suppose it was possible to make this model? -- 6.6 More fun with rotating rings -- Some history about rotating rings -- 7 Continuing the paper-folding and number-theory threads -- 7.1 Constructing an 11-gon -- 7.2 The quasi-order theorem -- 7.3 The quasi-order theorem when t = 3 -- 7.4 Paper-folding connections with various famous number sequences -- 7.5 Finding the complementary factor and reconstructing the symbol -- is a reduced and contracted 2-symbol -- 8 A geometry and algebra thread - Constructing, and using, Jennifer's puzzle -- 8.1 Facts of life -- 8.2 Description of the puzzle -- 8.3 How to make the puzzle pieces -- 8.4 Assembling the braided tetrahedron -- 8.5 Assembling the braided octahedron -- 8.6 Assembling the braided cube -- A Variation -- 8.7 Some mathematical applications of Jennifer's puzzle -- Volumes of some related polyhedra -- 9 A polyhedral geometry thread - Constructing braided Platonic solids and other woven polyhedra -- 9.1 A curious fact -- 9.2 Preparing the strips -- 9.3 Braiding the diagonal cube -- 9.4 Braiding the golden dodecahedron -- 9.5 Braiding the dodecahedron -- 9.6 Braiding the icosahedron -- 9.7 Constructing more symmetric tetrahedra, octahedra, and icosahedra -- 9.8 Weaving straight strips on other polyhedral surfaces -- 10 Combinatorial and symmetry threads -- 10.1 Symmetries of the cube -- 10.2 Symmetries of the regular octahedron and regular tetrahedron -- The diagonal cube has some combinatorial surprises -- Remark on orientation and symmetry -- 10.3 Euler's formula and Descartes' angular deficiency -- 10.4 Some combinatorial properties of polyhedra -- 11 Some golden threads - Constructing more dodecahedra -- 11.1 How can there be more dodecahedra? -- 11.2 The small stellated dodecahedron -- Instructions -- 11.3 The great stellated dodecahedron -- Instructions -- 11.4 The great dodecahedron.

Instructions -- 11.5 Magical relationships between special dodecahedra -- 12 More combinatorial threads - Collapsoids -- 12.1 What is a collapsoid? -- 12.2 Preparing the cells, tabs, and flaps -- Cells -- Tabs -- Flaps -- 12.3 Constructing a 12-celled polar collapsoid -- Practical hints -- 12.4 Constructing a 20-celled polar collapsoid -- 12.5 Constructing a 30-celled polar collapsoid -- 12.6 Constructing a 12-celled equatorial collapsoid -- Challenges -- For any collapsoid -- For the 12-celled collapsoid (polar or equatorial) -- For the 30-celled collapsoid -- 12.7 Other collapsoids (for the experts) -- 12.8 How do we find other collapsoids? -- 13 Group theory - The faces of the trihexaflexagon -- 13.1 Group theory and hexaflexagons -- 13.2 How to build the special trihexaflexagon -- 13.3 The happy group -- 13.4 The entire group -- 13.5 A normal subgroup -- 13.6 What next? -- 14 Combinatorial and group-theoretical threads - Extended face planes of the Platonic solids -- 14.1 The question -- 14.2 Divisions of the plane -- Case 1 -- Case 2 -- 14.3 Some facts about the Platonic solids -- 14.4 Answering the main question -- Tetrahedron -- Hexahedron (or cube) -- Octahedron -- Dodecahedron -- Icosahedron -- 14.5 More general questions -- 15 A historical thread - Involving the Euler characteristic, Descartes' total angular defect, and Polya's dream -- 15.1 Polya's speculation -- 15.2 Polya's dream -- Remarks -- 15.3 …The dream comes true -- 15.4 Further generalizations -- 16 Tying some loose ends together - Symmetry, group theory, homologues, and the Polya enumeration theorem -- 16.1 Symmetry: A really big idea -- 16.2 Symmetry in geometry -- Definition of a group -- Remarks -- 16.3 Homologues -- 16.4 The Polya enumeration theorem -- 16.5 Even and odd permutations -- Definition of even and odd permutations -- Step 1 -- Step 2 -- Step 3.

16.6 Epilogue: Polya and ourselves - Mathematics, tea, and cakes -- 17 Returning to the number-theory thread - Generalized quasi-order and coach theorems -- 17.1 Setting the stage -- 17.2 The coach theorem -- 17.3 The generalized quasi-order theorem -- 17.4 The generalized coach theorem -- 17.5 Parlor tricks -- 17.6 A little linear algebra -- 17.7 Some open questions -- References -- Index.