Explorable Flexagons

Learn to create and flex flexagons

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

pre: [[[1,2],3], 4, 5, [6,7], 8, 9]
post: [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