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This content is taken from the Davidson Institute of Science Education at the Weizmann Institute of Science's online course, Flexagons and the Math Behind Twisted Paper. Join the course to learn more.
Tri-hexa-flexagon Tuckerman diagram
Tri-hexa-flexagon Tuckerman diagram

Tuckerman Diagrams

The previous article may have left you confused, so let’s recap what we know about flexagons and be specific as to how we are going to refer to them for the rest of the course.

  • Every flexagon has a number of faces that we will refer to also as states.
  • Every face of the flexagon is divided into a certain number of identical geometrical shapes called leaves.
  • Each flexagon has a thickness. Each of the (visible) leaves on the face is stacked as a folded wad of leaves. These stacks are called pats and they can contain any number of leaves from 1 to n-1, n being the number of leaves on the face of the flexagon.
  • The number of visible leaves on each face is always equal to the number of pats.
  • A flex changes the state of the flexagon.
  • We will use the Hilton-Pederson and it’s equivalent Greek wording to name flexagons, according to the number of states (the first prefix), and the number of geometrical shapes that make up the initial state(the second prefix). For example, the basic cyclic flexagon that you just made will be referred to as the 3-6-flexagon or the tri-hexa-flexagon. We realise that this becomes ambiguous with some of the advanced flexagons, but we won’t worry about that for the moment.

Flexagon maps

Mathematicians aim to give a precise description of the objects that they are studying so that people can understand what they are doing. Bryant Tuckerman, one of the flexagon committee members and a prominent mathematician, designed a graph that could explain what happens when you flex a flexagon. These diagrams became known as Tuckerman diagrams. Interestingly, they later served as an inspiration for the Nobel physicist Richard Feynman’s own diagrams in particle physics.

You can see Tuckerman’s diagram for the tri-hexa-flexagon above. This is the simplest Tuckerman diagram. The three states of the flexagons are at the vertices of the diagrams. The colour of the circle at the vertices is the colour of a face-up state and it’s shadow colour is the state’s face-down colour. These colours are sometimes written using words with the face-down colour written in brackets after the main, face-up colour. Arrows between two states represent the flexes. Different types of flexes, such as ‘pinch flex’ or ‘book flex’ can be depicted using words along each arrow or by using different styles of lines for different flexes. It is customary that the absence of a label means pinch flexing.

Reading the tri-hexa-flexagon Tuckerman diagram

Let’s read together the Tuckerman diagram above. The diagram tells us that if we pinch flex the flexagon with red face up and blue face down we will get the flexagon with its yellow face up and red face down. Pinch flexing from this state will bring us to the blue face up, yellow face down position. A third pinch flex from this state brings us back to the original red up, blue down state.

The series of flexes that traverses the Tuckerman diagram completely by using the minimum number of flexes, is called the Tuckerman traverse.

It is possible to make Tuckerman diagrams that are much more complex. What you lose in complexity you gain in detail. Obviously, the Tuckerman diagram for a flexagon where initially each leaf has a non-symmetrical marking scheme will be much more detailed as it will have to show us the different orientations of the leaves within each face. In this course we will stick with the basic Tuckerman diagram.


Do you have any questions about the Tuckerman diagram?

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This article is from the free online course:

Flexagons and the Math Behind Twisted Paper

Davidson Institute of Science Education at the Weizmann Institute of Science