Serious Fun with Flexagons

1 Introduction; 1.1 General Features; 1.2 Terminology; 1.3 Outline of Book; 1.4 Making Flexagons; 1.4.1 General Assembly Instructions; 1.4.2 The Two Sector First Order Fundamental Square Even Edge Flexagon; 2 Polygon Rings; 2.1 Introduction; 2.1.1 Multiple Polygons; 2.1.2 Combinations; 2.2 Edge Rings of Regular Polygons; 2.2.1 General Properties; 2.2.2 Regular Even Edge Rings; 2.2.3 Regular Odd Edge Rings; 2.2.4 Compound Edge Rings; 2.2.5 Irregular Edge Rings; 2.3 Edge Rings of Irregular Polygons; 2.3.1 General Properties; 2.3.2 Even Edge Rings of Silver and Bronze Triangles; 2.4 Vertex Rings; 2.4.1 General Properties; 2.4.2 Vertex Rings of Squares; 3 Fundamental Nets; 3.1 Introduction; 3.2 First Order Fundamental Edge Nets; 3.3 Second Order Fundamental Edge Nets; 3.4 Fundamental Vertex Nets; 3.5 Fundamental Silver and Bronze Edge Nets; 4 Fundamental Edge Flexagons; 4.1 Introduction; 4.1.1 Standard Face Numbering Sequences; 4.1.2 Truncated Flexagons; 4.2 First Order Fundamental Even Edge Flexagons; 4.2.1 General Properties; 4.2.2 Ring Even Edge Flexagons; 4.2.3 First Order Fundamental Triangle Even Edge Flexagons; 4.2.4 First Order Fundamental Square Even Edge Flexagons; 4.2.5 Detailed Analysis of Flexagons; 4.2.6 First Order Fundamental Pentagon Even Edge Flexagons; 4.2.7 First Order Fundamental Hexagon Even Edge Flexagons; 4.2.8 First Order Fundamental Octagon Even Edge Flexagons; 4.2.9 First Order Fundamental Dodecagon Even Edge Flexagons; 4.3 Second Order Fundamental Odd Edge Flexagons; 4.3.1 General Properties; 4.3.2 Second Order Fundamental Triangle Odd Edge Flexagons; 4.3.3 A Second Order Fundamental Square Odd Edge Flexagon; 4.3.4 A Second Order Fundamental 20-gon Odd Edge Flexagon; 5 Fundamental Skeletal and Point Flexagons; 5.1 Introduction; 5.2 First Order Fundamental Even Skeletal Flexagons; 5.2.1 General Properties; 5.2.2 First Order Fundamental Triangle Even Skeletal Flexagons; 5.2.3 A First Order Fundamental Square Even Skeletal Flexagon; 5.3 Fundamental Point Flexagons; 5.3.1 General Properties and Unagons; 5.3.2 The Fundamental Triangle Point Flexagon; 5.3.3 The Fundamental Square Point Flexagon; 5.3.4 Fundamental Pentagon Point Flexagons; 5.3.5 The Fundamental Hexagon Point Flexagon; 5.4 Interleaved Fundamental Point Flexagons; 5.4.1 General Properties; 5.4.2 The Interleaved Fundamental Pentagon Point Flexagon; 5.4.3 An Interleaved Fundamental Enneagon Point Flexagon.; 5.5 Augmented Fundamental Point Flexagons; 5.5.1 General Properties; 5.5.2 An Augmented Fundamental Triangle Point Flexagon; 5.5.3 An Augmented Fundamental Square Point Flexagon; 5.6 Augmented Interleaved Fundamental Point Flexagons; 5.6.1 General Properties; 5.6.2 Augmented Interleaved Fundamental Triangle Point Flexagons; 5.6.3 An Augmented Interleaved Fundamental Square Point Flexagon; 6 Fundamental Compound Edge Flexagons; 6.1 Introduction; 6.2 General Properties; 6.3 Triangular Fundamental Compound Edge Flexagons; 6.3.1 Some Properties; 6.3.2 A Fundamental Square Compound Edge Flexagon; 6.3.3 A Fundamental Pentagon Compound Edge Flexagon; 6.4 A Square-Like Fundamental Compound Edge Flexagon; 6.4.1 Some Properties; 6.4.2 A fundamental hexagon compound edge flexagon; 6.5 Pentagonal Fundamental Compound Edge Flexagons; 6.5.1 Some Properties; 6.5.2 A Fundamental Square Compound Edge Flexagon; 6.5.3 A Fundamental Hexagon Compound Edge Flexagon; 6.6 A Hexagonal Fundamental Compound Edge Flexagon; 6.6.1 Some Properties; 6.6.2 A Fundamental Octagon Compound Edge Flexagon; 6.7 Heptagonal Fundamental Compound Edge Flexagons; 6.7.1 Some Properties; 6.7.2 A Fundamental Hexagon Compound Edge Flexagon; 6.7.3 A Fundamental Decagon Compound Edge Flexagon; 7 Irregular Cycle Flexagons; 7.1 Introduction; 7.2 Irregular Cycle Even Edge Flexagons; 7.2.1 General Properties; 7.2.2 Derivation of Nets; 7.2.3 The Irregular Cycle Square Even Edge Flexagon; 7.2.4 An Irregular Cycle Pentagon Even Edge Flexagon; 7.2.5 Irregular Cycle Hexagon Even Edge Flexagons; 7.3 Irregular Cycle Interleaved Point Flexagons; 7.3.1 General Properties; 7.3.2 Interleaf Flexes; 7.3.3 The Irregular Cycle Interleaved Square Point; Flexagon; 7.3.4 Irregular Cycle Interleaved Pentagon Point Flexagons; 7.3.5 Irregular Cycle Interleaved Hexagon Point Flexagons; 7.3.6 Augmented Irregular Cycle Interleaved Triangle Point Flexagons; 7.4 Distinct Face Numbering Sequences; 7.5 Irregular Cycle Non Interleaved Point Flexagons; 7.5.1 General Properties; 7.5.2 The Irregular Cycle Non Interleaved Square; Point Flexagon; 7.5.3 An Irregular Cycle Non Interleaved Pentagon Point Flexagon; 8 Degenerate flexagons; 8.1 Introduction; 8.2 Degenerate Even Edge Flexagons; 8.2.1 General Properties; 8.2.2 A Degenerate Square Even Edge Flexagon; 8.2.3 Degenerate Pentagon Even Edge Flexagons; 8.2.4 Degenerate Hexagon Even Edge Flexagons; 8.2.5 Degenerate Octagon Even Edge Flexagons; 8.2.6 A Degenerate Dodecagon Even Edge Flexagon; 8.3 Degenerate Non Interleaved Point Flexagons; 8.3.1 General Properties; 8.3.2 The Degenerate Non Interleaved Square Point Flexagon; 8.3.3 Degenerate Non Interleaved Pentagon Point Flexagons; 8.4 Degenerate Irregular Cycle Interleaved Point Flexagons; 8.4.1 General Properties; 8.4.2 Degenerate Interleaved Triangle Point Flexagons; 8.5 Degenerate Compound Edge Flexagons; 8.5.1 General Properties; 8.5.2 The Degenerate Square-Like Hexagon Compound Edge Flexagon; 8.5.3 Degenerate Pentagonal Compound Edge Flexagons; 9 Irregular Ring Even Edge Flexagons; 9.1 Introduction; 9.1.1 Fundamental Irregular Ring Even Edge Flexagons; 9.2 Irregular Ring Triangle Even Edge Flexagons; 9.2.1 General Properties; 9.2.2 An Irregular Ring 12 Triangle Even Edge Flexagon; 9.2.3 An Irregular Ring 8 Triangle Even Edge Flexagon; 9.2.4 An Irregular Ring 16 Triangle Even Edge Flexagon; 9.3 Irregular Ring Square Even Edge Flexagons; 9.3.1 General Properties; 9.3.2 Irregular Ring 6 Square Even Edge Flexagons; 9.3.3 Irregular Ring 8 Square Even Edge Flexagons; 9.4 Irregular Ring Pentagon Even Edge Flexagons; 9.4.1 General Properties; 9.4.2 An Irregular Ring 6 Pentagon Even Edge Flexagon; 9.4.3 An Irregular Ring 8 Pentagon Even Edge Flexagon; 9.5 Irregular Ring Hexagon Even Edge Flexagons; 9.5.1 General Properties; 9.5.2 An Irregular Ring 6 Hexagon Even Edge Flexagon; 9.5.3 Irregular Ring 8 Hexagon Even Edge Flexagons; 9.6 Irregular Ring Dodecagon Even Edge Flexagons; 9.6.1 General Properties; 9.6.2 Irregular Ring 8 Dodecagon Even Edge Flexagons; 10 Irregular Polygon Edge Flexagons; 10.1 Introduction; 10.1.1 Transformation of Polygons; 10.1.2 Stretch Flexagons and Stretch Polygon Rings; 10.2 Irregular Triangle Edge Flexagons; 10.2.1 Irregular Triangles; 10.2.2 Isosceles Triangle Edge Rings; 10.2.3 Bronze Even Edge Rings; 10.2.4 Isosceles Triangle Even Edge Flexagons; 10.2.5 Isosceles Triangle Odd Edge Flexagons; 10.2.6 Scalene Triangle Even Edge Flexagons; 10.2.7 A Partial Overlap Bronze Even Edge Flexagon; 10.2.8 A Scalene Triangle Even Edge Flexagon; 10.3 Irregular Quadrilateral Even Edge Flexagons; 10.3.1 Irregular Quadrilaterals; 10.3.2 Irregular Quadrilateral Even Edge Rings; 10.3.3 A Rectangle Even Edge Flexagon; 10.3.4 Rhombus Even Edge Flexagons; 10.3.5 A Trapezium Even Edge Flexagon; 10.4 Irregular Pentagon Even Edge Flexagons; 10.4.1 Irregular Pentagons; 10.4.2 Irregular Pentagon Even Edge Rings; 10.4.3 An Equiangular Irregular Pentagon Even Edge Flexagon; 10.4.4 An Irregular Ring 8 Irregular Pentagon Even Edge Flexagon; 11 Complex Flexagons; 11.1 Introduction; 11.2 Linked Even Edge Flexagons; 11.2.1 Methods of Linking; 11.2.2 Linked Hexaflexagons; 11.2.3 Linked Square Even Edge Flexagons; 11.2.4 Linked Pentagon Even Edge Flexagons; 11.2.5 Linked Silver Even Edge Flexagons; 11.2.6 Linked Bronze Even Edge Flexagons; 11.3 Linked Point Flexagons; 11.3.1 Methods of Linking; 11.3.2 Linked Triangle Point Flexagons; 11.3.3 Linked Square Point Flexagons; 11.4 Conjoined Point Flexagons; 11.4.1 General Properties; 11.4.2 Conjoined Triangle Point Flexagons; 11.4.3 A Conjoined Pentagon Point Flexagon; 11.5 Bundled Odd Edge Flexagons; 11.5.1 General Properties; 11.5.2 A Five Sector Bundled Triangle Odd Edge Flexagon; 11.5.3 The Seven and Six Flexagon; 11.6 Slipagons; 11.6.1 General Properties; 11.6.2 A Trihexaflexagon Slipagon; 11.6.3 A Partial Overlap Silver Even Edge Slipagon; 11.7 Coupled Point Flexagons; 11.7.1 General Properties; 11.7.2 A Coupled Triangle Point Flexagon; 11.7.3 A Coupled Square Point Flexagon; 12 Miscellaneous Flexagons; 12.1 Introduction; 12.2 Three Sector Odd Flexagons; 12.2.1 General Properties; 12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon; 12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon; 12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon; 12.3 Degenerate Odd Edge Flexagons; 12.3.1 General Properties; 12.3.2 A Degenerate Square Odd Edge Flexagon; 12.4 Alternating Odd Edge Flexagons; 12.4.1 General Properties; 12.4.2 A Square Alternating Odd Edge Flexagon; 12.5 Flapagons; 12.5.1 General Properties; 12.5.2 The Fundamental Square Duplex Flapagon; 12.5.3 A Square Flapagon-Flexagon Hybrid; 12.5.4 The Fundamental Isosceles Triangle Triplex Flapagon; 12.6 Multiplex Edge Flexagons; 12.6.1 General Properties; 12.6.2 A Square Duplex Edge Flexagon; 12.6.3 The Thrice Three-Fold Flexagon; 12.7 A Hooke's Joint Flexagon; Index