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What is a trihexaflexagon?

Trihexaflexagons are part of the flexagon family. Eighteen equilateral triangles are folded into a hexagon which reveals three different faces as it is ‘flexed’.

How hexaflexagons came to be 

What would you do if you discovered your school note paper did not fit into your folder? 

This is the problem Arthur Stone faced when he moved from England to America in 1939, due to the different paper sizes. Arthur Stone cut his school note paper to fit into his folder and was left with long strips of excess paper. 

Arthur Stone started folding the paper in different ways. He ended up folding the paper into triangles then he twisted them to form a hexagon. Once he had formed a hexagon, he continued playing with the paper by twisting and flexing it. He then noticed a small hole in the middle of the shape. As he played a bit more with the shape he was able to open it to form a new face (side) to discover the hexaflexagon! 

Arthur Stone could not wait to share this discovery with his friends and together, they started a committee to investigate the hexaflexagon further. The committee wanted to understand how the hexaflexagon worked and whether it could be replicated with other shapes.  After playing around and folding it in various ways, they found they could recreate it with other shapes.

The trihexaflexagon activity

 This activity is designed for students to gain an appreciation of the many wonderful properties of shapes and symmetry. All the instructions and templates are provided on this page.

Students can enjoy the trihexaflexagon templates provided or you may like to download blank templates so students can experiment with their own design and try and work out what is happening to the triangles. They can also explore what happens if they fold the flexagon in a different direction and then how to get the faces to return to their correct placement. As an extension, students may like to investigate how to find all six faces of a hexahexaflexagon – there are lots of templates for these available on the internet.

We love trihexaflexagons

Our Numeracy Ambassador, Simon Pampena has created some wonderful templates that demonstrate how the symmetry changes when you flex the trihexaflexagon as he explains below:

The symmetry within the trihexaflexagon, with Simon Pampena

How to make a trihexaflexagon

If you have not made one before we recommend you watch Simon Pampena make one:

How to make a trihexaflexagon, with Simon Pampena

Simon has also developed step by step instructions:

Trihexaflexagon folding instructions (DOCX 6.35MB)

Trihexaflexagon folding instructions (PDF 460.47KB)

There is also an activity guide and detailed illustrations of how the triangles rotate:

Trihexaflexagon Activity Guide (DOCX 507KB)

Trihexaflexagon Activity Guide (PDF 261KB)

The symmetry of the Trihexaflexagon illustrated (DOCX 199KB)

The symmetry of the Trihexaflexagon illustrated (PDF 245KB)

Trihexaflexagon templates

Simon’s trihexaflexagons

Thumbnail of the black and white trihexaflexagon template
A4 black and white (PDF 464.38KB)
A3 black and white (PDF 784.02KB)

Thumbnail of the colour trihexaflexagon template
A4 colour (PDF 1.28MB)
A3 colour (PDF 2.18MB)

thumbnail of the blank trihexaflexagon template
Blank template (PDF 10.2KB)

2015 National Literacy and Numeracy Week design

thumbnail of the black and white National Literacy and Numeracy Week trihexaflexagon template
A4/A3 Outline (PDF 2.57MB)

thumbnail of the colour National Literacy and Numeracy Week design trihexaflexagon template
A4/A3 Colour (PDF 2.51MB)

Lower primary school students may prefer to create their own tessellation pattern or create different pictures using the Changing Shape activity.

Investigating Patterns: Create your own tessellation
Investigating Patterns: Create your own tessellation (PDF 982.73KB)
Investigating Patterns: Create your own tessellation (DOCX 385.85KB)

Did you know that mathematicians have made a tessellation discovery? You can read all about it on the Helix @ CSIRO website

Double Helix also has a tessellation explorer activity you can access.


Changing Shapes
Changing Shape (PDF 249.86KB)
Changing Shape (DOCX 190.96KB)