I loved my Spirograph set as a child and spent many happy hours exploring the patterns and symmetries I could create with it. And a few years ago, before teaching, I made my own Christmas cards decorated with Spirograph 'snowflakes'. This year I thought I'd give them a go again. I've spent such an enjoyable afternoon, warm and cosy indoors while the rain's been drizzling down, with my little Spirograph set, that I thought I'd share some quick instructions for making these cards so that you can too.

You'll need a Spirograph set, or, to use the correct mathematical name, a hypotrochoid set (small ones of these can be bought very cheaply from about £1.50), coloured or metallic pens (I used silver, which catches the light beautifully) and some card. And that's it!  (I bought ready-made cards but that's not at all necessary).

Have a quick practice with your set to find the combinations of 'cogs' and 'wheels' that create the shapes you like, in a range of sizes. I marked the holes that I wanted to use with a dot, so that I didn't forget which positions I'd chosen. Then have a play to come up with a few layouts that you find pleasing. I came up with two simple designs: a large central 'snowflake' and a set of three decreasing in size. However, a card filled with snowflakes of different sizes would be stunning if you have the time and patience for it. Then just go for it! And be prepared for a few wobbly lines here and there - they will just add to the handmade charm of the cards :)

❄️❄️❄️❄️❄️❄️❄️❄️❄️❄️ Happy Christmaths! ❄️❄️❄️❄️❄️❄️❄️❄️❄️❄️

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! ❄️❄️❄️