Learn to Create and Flex Flexagons

Chapter 1: Flexagon Intro

Exploring the hidden sides of hexaflexagons

• Introduces the hexaflexagon , with6equilateral triangles per side, and the pinch flex , a way of folding a flexagon to reveal previously hidden sides
• Let’s you explore how to visit every side of a flexagon
• Describes flex notation , which can be used to precisely describe a sequence of flexes
• Introduces the Tuckerman Traverse as a technique for visiting every side of a flexagon using the pinch flex
• Provides both a flexagon simulation and the unfolded strip for all hexaflexagons with3,4,5,6,7, and8sides so you can make your own
• Shows flexagons made up of different shapes of triangles, like using angles of30-60-90or54-54-72, and different numbers of triangles per side, like4,8,10, or12

Chapter 2: Flexing Flexagons

A short tour of different ways to flex a flexagon

• Describes a naming convention for a wide variety of triangle flexagons, like pentaflexagon and silver octaflexagon
• Defines a minimal flexagon as the simplest flexagon that supports a given flex
• Demonstrates a sampling of interesting flexes on triangle flexagons: the pyramid shuffle, flip flex, tuck flex, v-flex, and silver tetra flex
• Shows that once you generalize the types of flexes you use, the concept of “sides” of a flexagon is no longer sufficient for understanding all the states you can explore

Chapter 3: Generating Sequences

Creating flexagons with flex sequences

• Summarizes the flex notation that was previously introduced for describing sequences of flexes
• Defines a generating sequence as a sequence of flexes used to create the structure necessary for performing them on a given flexagon
• Shows how different flexes move you around a pinch flex diagram
• Gives you an interactive tool for typing in generating sequences to see what pinch flex diagram is generated
• Allows you to type in generating sequences consisting of any of the flexes introduced so far on a wide variety of triangle flexagons, giving you the unfolded strip that can be used to construct a working flexagon

Chapter 4: Pinch Flex Variations

Relatives of the pinch flex on different flexagons

• Describes a class of flexes that are related to the standard pinch flex
• Shows the minimal flexagon for a variety of flexes (e.g. P334and P3333) on several flexagons (e.g. the enneaflexagon and decaflexagon)
• Demonstrates generating sequences using these flexes to create interesting flexagons

Chapter 5: Slot Flexes

Flexes that involve sliding leaves through slots

• Describes a class of flexes called slot flexes
• Demonstrates the related flexes called the slot half and slot pocket
• Shows four slot tuck flex variants
• Finishes with the slot triple pocket flex on a pentaflexagon

Chapter 6: New flexes

How to define flexes with pat notation

• Describes using pat notation for the internal structure of a flexagon
• Shows how to define what a flex does by using pat notation to enumerate a flexagon’s structure before and after a flex
• Demonstrates using a flex definition to predict where you can do a flex and to predict exactly how a flexagon will change if you do the flex
• Provides definitions of the flexes discussed so far for hexaflexagons
• Lists additional details you need to fully understand when a flex can be performed
• Walks through an example of how to define a new flex and how to add it to the flexagon simulator

• I ~= (Sh>>T'>^T<<^) 2
• I ~= (Ltb>P>P>) 12
• P ~= T>>>V>>> so T' ~= >>>V>>>^P^

Chapter 7: Flex Sequences

Cycles, traversals, primes, and equalities

• Tools for finding cycles, flex sequences that end where they begin
• Examples of traversals, cycles that visit every state of a flexagon
• Prime flexes that can’t be replaced by other flexes
• Sequences of flexes that are equal or almost equal to each other

Appendix: Flexagon Playground

General tools for exploring flexagons

• Tools for creating new flexagons from generating sequences or scripts and exploring them using a variety of flexes
• Get the unfolded strip for any of your creations so you can try out the physical version
• An explorer that can find all the states accessible with a specified set of flexes from any flexagon

Appendix: Flex Compendium

Details about flexes on triangle flexagons

• Reference guide to a wide variety of flexes on triangle flexagons
• Shows sample slides and sample strips to fold on various triangle flexagons
• Shows flex definitions in pat notation, etc.

Flexagon References

Sources for additional flexagon information

• Bibliography for papers and publications
• Additional references

For additional information, seeloki3.com.