Flexagon Analysis

Let’s return to the basic tri-hexa-flexagon and explore some of its properties. You made this flexagon from a pre-coloured strip of paper. Of course, Arthur Stone and his friends didn’t have this privilege. They started out with a strip that wasn’t coloured. How did they know to colour it with just three colours? What happens if you colour the strip differently?

Colouring the tri-hexa-flexagon

These are not simple questions to answer, except perhaps for the tri-hexa-flexagon and I’ll explain why. The easiest way to colour a flexagon is to first make the flexagon from a blank strip and then colour each face when it appears. Since the tri-hexa-flexagon strip has 10 triangles on each side, of which two are the ‘envelope’ triangles that are glued face to face, there are 18 triangles distributed evenly between the 3 faces, 6 on each face. Once we colour each of the faces, we can undo the strip and see the correct colouring. Simple enough.

Different flexagons from the same strip

Some paper strips can actually make different flexagons, depending on the way they are folded. This makes colouring a challenge and depends on the required outcome. Still, as long as we first make the flexagon, then colour the faces and then unroll the strip, we can learn the correct colouring for each specific flexagon.

Flexagons with a different number of triangles on different-shaped faces

Some advanced flexagons actually have a different number of triangles on some of the faces, that is, they change shape when you flex. Imagine a flexagon shaped like a 6-triangle hexagon that changes into a 4-triangle square when flexed. This makes the problem of correct colouring much more difficult. Should we use the same colours for the square face or for the hexagon? Either way, we’re going to end up with some of the faces multi-coloured.

Patterns on flexagon faces

Perhaps the most confusing (and interesting) aspect of flexagons is how to count the number of faces when a pattern is drawn on the triangles. Draw a different pattern on each of the triangles on all the faces of the tri-hexa-flexagon. That’s 18 different patterns in total. It is preferable to draw non-symmetric patterns. You can use letters, numbers and symbols if you like, Here are 18 possible patterns:

Q, E, R, P, F, G, J, K, C, e, r, t, y, a, f, g, h, c

When you flex the flexagon, you’ll notice that the patterns move around and even rotate within the triangles in many different ways. There seem to be many more different faces now! How should these faces be considered? It seems hopeless to try and keep track of all these patterns, so people usually stick to colouring the triangles on each face with the same colour. In advanced different-shaped flexagons, the faces are renamed ‘states’, so include the multicoloured faces. A systematic study, however, should really be done by studying flexagons created by strips with a different, non-symmetrical pattern on each triangle. This is beyond the scope of this course, but if you are looking for something to do – go ahead!

The pat structure of hexa-flexagons

We saw that the tri-hexa-flexagon has 6 pats, alternating between 1 or 2 leaves thick as we rotate the flexagon in our hand. Now take the hexa-hexa-flexagon and pinch flex it between different states. You can follow the Tuckerman diagram of the flexagon from the previous step if you like.

What is the pat/leave structure of the hexa-hexa-flexagon? Do the pats in every different state have the same number of leaves?

Discussion

Share with us any insights you have from exploring the tri- and hexa- hexaflexagons.