Flexagons

If children are interested in creating different shapes, they may find flexagons particularly interesting. Flexagons are a type of model that has hidden surfaces that can be revealed by folding the model. One example of this is the trihexaflexagon, this long name means that it is a model with 6 different sides and can show 3 different faces. These faces can also appear in different orientations; these faces and their orientations are shown in the image below. To change the visible surfaces of this model, you fold it in/out of itself. To do this, you pinch the edges of the model together until the next surface, which was hidden inside the model, reveals itself.

Six images of a hexagon-like shape, there are two images where the pattern on the surface is green and also two where it is purple and two where it is blue. For each colour, there is one image of the pattern being a circle and one image of the pattern being a split circle.

Making a Trihexaflexagon

Children can make a trihexaflexagon by folding equilateral triangles on a strip of paper as shown in the image below. This amount of triangles works when using sellotape, but if you want to use a glue stick, you can add an extra triangle.

A strip of paper folded so that it is made of equalateral triangles, one point up, one point down, etc. There are 9 triangles in total.

Once you have the above strip of paper, you can twist it around so that it looks like the image below. We need two flat faces and then a twist, then another two flat faces and a twist and so on. Once the strip is twisted in this way, you can use tape to secure the two end triangles together along the outside edge. Now you are ready to start experimenting!

An image of a hexagon shape labelled to show where the folds in the surface are. There are two triangles and then a fold, this pattern then repeats and all the folds are orientated in the same way.

Children might find it interesting to mark the model with different colours to discover how there are actually three surfaces. If they use a pattern like circles in the centre of the model, they will be able to see that the circles split up when they turn the model over.

Other Flexagons

There are many other types of flexagons with a variety of shapes, but another popular model is the hexahexaflexagon. This long name means that it is a model with 6 different sides and can show 6 different faces. See the picture below to see all the different faces this model can show.

Six images of a hexagon-like shape, in each figure the shape is a different colour

Children might find it an interesting challenge to find all the different faces of the model. They might also notice that some faces appear more than others.

How to Make a Hexahexaflexagon

To make a hexahexaflexagon, you again need to draw some equilateral triangles on a strip of paper and then fold along the triangular lines. As this model is a bit more complicated, it can help to colour-code the faces of the model before the folding begins. To do this, you can copy the patterns in the image below, where the triangle that is left blank is a different triangle on each side (i.e. no triangle is blank on both sides). The blank triangles will be glued together to make the final model. To fold the model together, start by folding the orange, blue and pink triangles together so that they are hidden. Next, twist the strip around so that you get a hexagonal shape where all the triangles on the front are the same colour and all the triangles on the back are the same colour. Finally, glue the blank triangles together, and your model is ready for experimenting.

Two images showing how to colour the faces of a hexahexaflexagon before it is folded.

To see a demonstration of how to use flexagons, you can view the video below.

Flexagons were of great interest to some mathematicians and scientists from the late 1930s to the early 1940s at Princeton University, and they even formed a committee to investigate flexagons. One member of the committee was the famous physicist Richard Feynman.