Skip main navigation

What is a Tuckerman diagram?

Bryant Tuckerman, a prominent mathematician, designed a graph to explain what happens when you flex a flexagon
Tri-hexa-flexagon Tuckerman diagram in the shape of a triangle with a blue, a red and a yellow vertex. The blue vertex points to the red vertex which points to the yellow vertex which points back to the blue vertex.
© Davidson Institute of Science Education, Weizmann Institute of Science

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, a prominent mathematician, designed a graph to 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.

In a tri-hexa-flexagon, which 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 its 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.

The series of flexes that traverses a 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 loose in complexity you gain in detail. 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.

Flexagon facts

  • Every flexagon at a certain time has a structure that we will refer to as states. At any given time we see the upper face of the state.
  • Every face of the flexagon is divided into a certain number of stacks of identical geometrical shapes called leaves.
  • Each leaf has an upper side and a lower side.
  • Each stack is called a pat.
  • A flex changes the state of the flexagon.
© Davidson Institute of Science Education, Weizmann Institute of Science
This article is from the free online

Flexagons Galore: Advanced Flexagon Fun

Created by
FutureLearn - Learning For Life

Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education