Explorable Flexagons

Learn to create and flex flexagons

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

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

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

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

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

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

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

For additional information, see loki3.com.