My flatmate Julia has been busy these last couple of months, knitting and crocheting mathematical toys for me to play with. Her inspiration came from meeting Pat Ashforth at last year’s MathsJam. Pat and her husband Steve are the authors of the wonderful website Woolly Thoughts, which contains patterns for all sorts of knitted mathematical wonders. Blankets, cushions, hats, scarves, puzzles,… All of which are guaranteed to bring smiles to the friends, family or colleagues that you show your creations to!

The first thing that Julia decided to make was a flexagon cushion. A flexagon is traditionally made by folding a piece of paper into triangles (or squares) which then folds into a hexagon (or a bigger square) and can be ‘flexed’ to reveal hidden sides to the shape. It’s difficult to describe in words! I suggest you download a flexagon template and get folding – you will soon be hooked on the idea. The advantage of having a crocheted hexaflexagon is that it’s very robust and can’t be torn by playing with it too much. It’s also easier to unwind it a bit and see the structure of how it fits together. It turns out that a hexaflexagon is just a 3-twisted Möbius strip!

Here’s a short video of Julia playing with the hexaflexacushion:

Can you track all the different colours?

Of course, no education on hexaflexagons would be complete without watching the wonderful videos by Vi Hart, including a Hexaflexagon Safety Guide. See the first of them here: http://www.youtube.com/watch?v=VIVIegSt81k.

The second toy that Julia made is called an Octopush. This can be confusing if you google for it, because it’s also the name of an underwater sport. The toy is made of 8 cubes sewn together into a 2x2x2 mega-cube, and the colours are such that it is possible to flex the cube into lots of other positions. As with the flexagon, this is much easier to describe by showing you the video:

I’m not particularly impressed with Julia’s first attempt at knitting this, as the cubes aren’t perfectly cubical and it doesn’t fit together very neatly. But I guess we can’t expect humans to get it right every time. Hopefully she’ll make a better one someday. Can you figure out how it all fits together?

So, what should I get Julia to make next? Suggestions welcome!