Some of the most important advances in the study of flexagons in recent years have been made by Scott Sherman. It all started when Sherman started to write “Flexagonator”, a computer program that can simulate flexagons. Before we go any further, it’s time to recap the definitions we already know.

Flexagon Terms

• Every flexagon has a number of faces that we will refer to also as states.
• Every state of the flexagon consists of a number of stacks of identical geometrical shapes.
• The stacks are called pats.
• The individual, 2-sided, identical, geometrical shapes in the stack are called leaves.
• A flex changes the appearance of the flexagon from one state to another.
• The Tuckerman diagram or state map is a graph that shows all the different states of the flexagon. The states are the nodes of the graph. The lines between two nodes represent the flexes that connect them.

Sherman’s Flexagon Notation

Scott Sherman realised that if you want to write a computer program that flexes flexagons, you need to create a language that represents both the objects – flexagon states, leaves, pats, etc – and the different operations that you can do on a flexagon (turn it over, flex it, etc).

Here are some of the terms that are used in his language. Don’t worry if there are some terms that you don’t understand. We will encounter them later on in the course.

• P – Pinch flex
• V – V flex
• T – Tuck flex
• F – Flip flex
• Tw – Twist flex
• Sh – Pyramid shuffle flex
• ^ – Turn the flexagon over
• > – Shift current vertex one place clockwise
• < – Shift current vertex one place counter-clockwise
• ‘ – Reverse the flex – do it backwards

Examples

• P – perform a pinch flex at a vertex
• P> – perform a pinch flex and then move the flexagon round clockwise to the next vertex
• (P>)3 – perform a pinch flex and a clockwise move, three times. Note the use of parentheses and the number outside the right bracket to signal that the same move is performed three times. In the tri-hexa-flexagon, this is the Tuckerman traverse. It leads to all possible states of the flexagon.
• (P<)2^V – perform a pinch flex, move to the next counter-clockwise vertex, do this again, then turn the flexagon over and perform a V-flex (from the same vertex – just the other side).

Discussion

Any questions? Feel free to ask.