An appendix from
Explorable Flexagons

Flexagon Playground

General tools for exploring flexagons

These tools simulate flexagons and the flexes you can use to manipulate them. They also give you the ability to create working flexagon models by printing and folding generated templates.

In order to understand the tools, you need to understand flexagons, some of the variety of flexes, and terminology like generating sequences and pats. The easiest way to learn this is to go through Explorable Flexagons, which explains the concepts and introduces you to much of the UI used in this page. Alternately, the Flexagon Theory paper provides some of the theoretical background.

  • The Flexagon Simulator allows you to create lots of different types of triangle flexagons and experiment with how various flexes modify them
  • Flexagon from Script is the same, except that you create flexagons using a flexagonator script
  • The Unfolded Template section gives you the ability to generate flexagon templates to print and fold
  • The Explorer can be used to find every state accessible by using the given flexes

Each individual section contains additional help.


Flexagon Simulator

The polygons: These represent a folded flexagon, both front (the larger one) and back (the smaller one). The buttons allow you to apply a flex at a given corner. The ^ on the smaller polygon is used to turn over the flexagon.

Apply Settings: Once you have entered in your settings in the various text boxes, click Apply Settings to use them to create a new flexagon. The Generating sequence is a series of flexes used to create the structure of the flexagon. Search flexes is used to determine which flex buttons should appear on the flexagon. Pats per face indicates the number of pats (stacks of triangles) to use for the flexagon. Center angle is the angle of each triangular leaf that starts out in the center of the flexagon. Angle 2 is the first angle clockwise from the center on the first leaf. Note: in order to get the default isosceles triangle, clear Center angle.

Undo, redo, reset: Use to undo or redo flexes, or to start over again from the given generating sequence.

Flex sequence: Allows you to type in a sequence of flexes. Click Apply Sequence to have it applied to the current flexagon if it can.

Generator: Shows the generating sequence for the current flexagon.

History: Shows a history of all flexes that have been applied either by clicking buttons on the flexagon or through applying a flex sequence.

Show Current Script: Fill the text box with a flexagonator script describing the state of the current flexagon. Can be used to supply input to other tools on this page.


Generating sequence:

Search flexes:

Pats per face:

Center angle:

Angle 2:

Flex sequence:

Generator: St*<<St*
History:


Flexagon from Script

The polygons: These represent a folded flexagon, both front (the larger one) and back (the smaller one). The buttons allow you to apply a flex at a given corner. The ^ on the smaller polygon is used to turn over the flexagon.

Undo, redo: Use to undo or redo flexes.

History: Shows a history of all flexes that have been applied by clicking buttons on the flexagon.

Run Script: Run the flexagonator script listed in the text box in order to create a new flexagon or modify an existing one.

Show Current Script: Fill the text box with a flexagonator script describing the state of the current flexagon. Can be used to supply input to other tools on this page.

History: P*P*P*


Unfolded Template:

Labels and colors: Gives you several options for how it displays labels and colors for the unfolded template. You can have it use values from the leafProperties portion of a script, display the leaf ids, display everything on one face, or display just the properties on the front or back.

Unfolded template: To make a real flexagon, print the unfolded template, cut it out, then fold like numbers on like numbers, and finish by taping together the first and last triangles.

Run Script: Create a new unfolded template by running the script in the text box, grabbing the one used by the Flexagon Simulator, or the one used by the Flexagon from Script.

Labels and colors:

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Explorer

Explore: Start by entering a flexagonator script that creates a flexagon. Click Explore all to have it use all the specified search flexes to find every accessible state (including <, >, and ^). Click Explore without ^ to have it skip turning the flexagon over during the search. Click Cancel to abort the current search.

States that support flexes: Once an exploration has been completed, this shows how many of the resulting states support each of the flexes it used during the search.

Show: Shows information about all the states it found after an exploration has been completed. You can have it show the internal structure of each state by clicking pat notation or all the flexes available from each state by clicking all available flexes. Or you can have it output information in DOT format, which can be used by some graph visualization tools, e.g. http://viz-js.com/.

States that support flexes:

- -
Explored 0 states, found 0 states

Show for each state:
Show DOT:


What next?

Go to the Table of Contents or the Flexagon Pages.

Copyright © 2018-2020 Scott Sherman