When I left my first teaching school (the one I’m back working at again now) I decided to make a piece of art as a leaving gift for my colleagues in the maths team. I’d seen an image on Pinterest of a mural made from different coloured paper tetrahedra, and wanted to try it for myself. However, being a maths teacher, I was intrigued by the possibilities of tessellating smaller tetrahedra within the gaps left by bigger tetrahedra....

Now I know that A-sized paper has some pretty cool properties, so, using the photocopier I reduced an A4 tetrahedron net down through the various A-sizes - and, yes, I made some lovely discoveries (which made perfect sense once I started thinking about it)!

Each iteration of A-sized paper decreases in area by half, (so the side lengths decrease by a factor of 1/(sqrt2)): A5 is half the area of A4, A6 half the area of A5, and so on. So the base of an A6 tetrahedron is a quarter the area of an A4 base, meaning that four A6 tetrahedron bases can be packaged in the gap left by an A4 tetrahedron. I played around and found some other combinations (see some examples below).

I printed and cut out all my nets, assembled my different sized tetrahedra (step-by-step instructions to follow below) and played around with lots of different arrangements. When finally happy, I stuck them down onto a ready-made canvas.

A week ago my ex-colleague Ali got in touch because she wanted to try the activity with her class. I talked her through the instructions and her students went on to produce the wonderful artwork below :)

Here are the step-by-step instructions if you would like to try this activity with your students.

When we think of paper snowflakes, we usually imagine the 'kirigami' type that involve first folding a sheet of paper into sixths, and then cutting bits out to create a papercut snowflake exhibiting perfect six-fold symmetry. (If you fancy trying some, there are some beautiful templates from First Palette here and some fun Star Wars and Harry Potter themed templates by Anthony Herrera, here). However, you can also make beautiful origami snowflakes from a single sheet of folded paper and no cutting, and they are wonderfully mathematical!

I have been planning an origami snowflake window display as part of my Christmas decorations this year, and this weekend got down to some folding. I chose to use tracing paper so that the internal structure and symmetry of the snowflakes would be visible when held up to the light - any translucent paper will do, including baking parchment or tissue paper.

In doing my research I came across several different styles, all starting from a paper hexagon. This is where the maths begins! The best method of cutting a paper hexagon for this type of snowflake is to start from an A4 (or letter size) rectangular sheet, as all the fold lines that remain form useful pre-creases for your final model.  The method itself is ingenious, and would make a great little geometric proof challenge for your students. Here is a video of me performing the method. Why does it produce a perfect hexagon?

Now to fold your snowflakes! Here are links to the instructions for the three most effective models that I found online. I have also included photos of the interim stages of the folding of my first attempts.

Diagram by Dennis Walker: http://www.oriwiki.com/origamidennis/diagrams/oridiag.htm

Video by Sara Adams, @happyfolding: https://www.youtube.com/watch?v=7m72m8L0xuA

Video by Riccardo Foschi: https://www.youtube.com/watch?v=x6UjDVLSqOk

Video by @senbazurueurope (in French, but clearly demonstrated): https://www.youtube.com/watch?v=EYMxVAlnnS0

I hope you enjoy folding these wonderful designs as much as I have. The layers upon layers of symmetry are incredibly pleasing to create, and there are lots of 'ooh' and 'aah' moments to enjoy as the cleverness of each design reveals itself. 

Let it snow! ❄️❄️❄️