Preface 1. Flexagons - a beginning thread 2. Another thread - 1-period paper folding 3. More paper folding threads - 2-period paper-folding 4. A number-theory thread - folding numbers, a number trick, and some titbits 5. The polyhedron thread - building some polyhedra and defining a regular polyhedron 6. Constructing dipyramids and rotating rings from straight strips of triangles 7. Continuing the paper-folding and number theory threads 8. A geometry and algebra thread - constructing, and using, Jennifer's puzzle 9. A polyhedral geometry thread - constructing braided platonic solids and other woven polyhedra 10. Combinatorial and symmetry threads 11. Some golden threads - constructing more dodecahedra 12. More combinatorial threads - collapsoids 13. Group theory - the faces of the tri-hexaflexagon 14. Combinatorial and group theory threads - extended face planes of the platonic solids 15. A historical thread - involving the Euler characteristic, Descartes' total angular defect, and Polya's dream 16. Tying some loose ends together - symmetry, group theory, homologues, and the Polya enumeration theorem 17. Returning to the number theory thread - generalized quasi-order and coach theorems References Index.