You are here

Serious Fun with Flexagons: A Compedium and Guide

Les Pook
Publication Date: 
Number of Pages: 
Solid Mechanics and Its Applications 164
[Reviewed by
Collin Carbno
, on

This book comes across as a sort of encyclopedia or a classification of flexagons. As such, the one real strength of the book is that numerous kinds of flexagons that are covered.

While mathematically I found the flexagon information exciting, the descriptions somehow failed to capture my imagination. For a start, I often found it difficult to grasp the significance of much of what the author was stating. For example, “The first two topological invariants for an ideal vertex ring of regular polygons are the same as those for an ideal edge ring of identical regular polygons but the third is different” is an interesting statement, but surrounding discussion wasn’t really enough to fully convince. Pook gives a lot of mathematical information on the flexagons in question, but he does not give the supporting arguments and proofs so that one can actually believe and understand. Thus, I was left wondering whether he determined these statements experimentally by building models, by finding proofs, or from the work of others.

The book claims to have more than 100 designs for constructions of various flexagons, many of them new to the world. Although I didn’t count them, it is true that the book is filled with various flexagon designs. It took me over an hour, however, to figure out how to get the first simple flexagon made, as the instructions and notation for creating the flexagons did not seem that well explained. No doubt, an experienced flexagon builder, more familiar with the notation, would have no problem following this sort of construction, and I found building the second and third ones easier. Thus, if you are a novice flexagon builder, looking for designs, you would be better off just grabbing flexagon information from the internet. The more serious researcher, however, will find this book an amazing resource for various kinds of flexagons.

I found the diagrams in the book adequate but too small and the 3-dimensional photographs of the finished flexagons were of surprisingly poor quality and shading resolution. If this was a $9.99 book this might be acceptable, but given the actual price, it is not.

In all fairness to the book, it has a wonderful index, and the author has put intense amount of thought into how to classify and organize the various categories and properties of the flexagons. I’m sure this book will remain a reference book of choice in the subject of flexagons for a long time. But if you looking for a book to have some fun with some interesting constructions, and hope to learn a little mathematics on the side, I’m think you find this book disappointing. If you are looking for some sort of “Encyclopedia of Flexagons” this flexagon book appears to be the best on the market.

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.

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