Chapter 1: Flexagon Intro

Exploring the hidden sides of hexaflexagons

- Introduces the
*hexaflexagon*, with 6 equilateral 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 with 3 , 4 , 5 , 6 , 7 , and 8 sides so you can make your own
- Shows flexagons made up of different shapes of triangles, like using angles of 3 0 - 6 0 - 9 0 or 5 4 - 5 4 - 7 2 , and different numbers of triangles per side, like 4 , 8 , 1 0 , or 1 2

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. P 3 3 4 and P 3 3 3 3 ) 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

`[[[1,2],3], 4, 5, [6,7], 8, 9]`

`[2, 4, 5, [6,7], 8, [-1,[9,-3]]]`

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, see loki3.com.

Copyright © 2018-2019 Scott Sherman