Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 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 tidbits
- 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 trihexaflexagon
- 14 Combinatorial and group-theoretical threads – Extended face planes of the Platonic solids
- 15 A historical thread – Involving the Euler characteristic, Descartes' total angular defect, and Pólya's dream
- 16 Tying some loose ends together – Symmetry, group theory, homologues, and the Pólya enumeration theorem
- 17 Returning to the number-theory thread – Generalized quasi-order and coach theorems
- References
- Index
13 - Group theory – The faces of the trihexaflexagon
Published online by Cambridge University Press: 10 November 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 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 tidbits
- 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 trihexaflexagon
- 14 Combinatorial and group-theoretical threads – Extended face planes of the Platonic solids
- 15 A historical thread – Involving the Euler characteristic, Descartes' total angular defect, and Pólya's dream
- 16 Tying some loose ends together – Symmetry, group theory, homologues, and the Pólya enumeration theorem
- 17 Returning to the number-theory thread – Generalized quasi-order and coach theorems
- References
- Index
Summary
Group theory and hexaflexagons
We described how to build a variety of hexaflexagons in Chapter 1. Here we give another description of how to build the 3-faced flexagon which has designs on each of its faces that enable us to track the set of motions bringing the flexagon into coincidence with itself.
The particular hexaflexagon we will consider in this chapter is the trihexaflexagon (also called the 3-6-flexagon), so named because it has 3 faces; that is, in any given state of the flexagon, one face (consisting of 6 equilateral triangles) will be up, one face will be down and one face will be hidden. Although the orientation of the faces will vary from state to state, the same 6 triangles will always appear together on a face.
By drawing a human visage on each face of the flexagon, and using a different color for each face, we can keep track of all the possible positions of the flexagon as it lies in a plane. We are thereby able to discover that the set of motions of this flexagon which bring it into coincidence with itself constitutes the dihedral group D18.
How to build the special trihexaflexagon
The trihexaflexagon is constructed from a strip of paper containing 10 equilateral triangles as shown in Figure 13.1. In order that the final model will flex easily the fold lines between the triangles should be creased firmly in both directions.
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- Information
- A Mathematical TapestryDemonstrating the Beautiful Unity of Mathematics, pp. 195 - 205Publisher: Cambridge University PressPrint publication year: 2010