Since my first knitting blog in March, I’ve been occupied with a much bigger project: a “log-cabin” blanket. The log-cabin design is based on traditional American quilting, translated into knitting: each square panel in the blanket is built up from a series of “logs” around a reddish central square, supposedly symbolising the hearth, surrounded by a pattern of light and shade.

The idea is relatively straightforward in principle (as illustrated in this basic template on Ravelry.com), but I had to learn and adapt a number of techniques to achieve my intended design. **I also had to solve a problem of symmetry.**

I decided I wanted a symmetrical blanket – 24 panels consisting of 4 blocks of 6, with both horizontal and vertical symmetry. But I noticed that most of the designs posted online weren’t symmetrical …

The problem, I discovered, was that all of the instructions I could find on the net for log-cabin blankets (such as these delightfully precise videos from Texas) involved knitting the “log” strips in an **anti**-clockwise direction around the central square. That only gave me one half (or rather, two disjoint quarters) of my blanket: I could knit 6 anti-clockwise squares for the bottom right quadrant, plus a further 6 identical squares to be placed upside-down to form the top left quadrant. But for the other two quarters, I needed to find a way of producing squares with logs spiralling **clockwise** from the central square.

At this point I almost gave up with the idea of a symmetrical design. I could find no solution to my problem in the vast resources on knitting available online.

Then I unearthed a memory from a book by Martin Gardner (science writer and former Scientific American columnist). In his New Ambidextrous Universe Gardner describes how, for someone living in a two-dimensional “Flatland”, an asymmetric shape (say R) and its mirror image (say Я) could not be transposed into each other. This is exactly the problem I faced with my blanket squares: I couldn’t make an anti-clockwise log pattern into a clockwise one simply by turning it round in two dimensions.

But Gardner goes on to make the fairly obvious point that, in our three-dimensional world, we can transpose the shapes onto each other simply by flipping them over, through the third dimension. Of course, one can also use a looking-glass – though that’s not of much practical use in making a blanket.

So how could I achieve the same effect in knitting? Couldn’t I knit another 12 squares just like the others and then flip them over? That seems like an obvious answer, and if it had been so simple I probably wouldn’t have thought there was a problem in the first place. But I’d already knitted some squares and I knew that, given the technique I was using to join each log together, it wouldn’t work. The underneath of each blanket square didn’t look quite the same as the right side.

Then – from somewhere, from nowhere – the solution came. Anyone who has learnt to knit will know that stitches come in two fundamental varieties, “knit” and “purl”. But purl is effectively the reverse of knit: a purl stitch produces the same effect on the back of a piece of knitting as a knit stitch produces on the front, and vice versa. The squares I had produced so far were wholly in knit stitch. So, to produce reversed or “flipped” squares, with logs going clockwise rather than anti-clockwise from the centre square, I had to construct squares wholly from purl stitch.

And it worked: I simply had to follow every step of the procedure I’d learnt from the Texan videos, but using purl rather than knit, and reversing the orientation of the logs. It took a few trial runs to get right, but my mistakes were generally due to failing to apply the principle of reversal rigorously to its logical conclusion.

**Tension headache**

There was one problem that wasn’t solved by rigour alone: the first square I produced in purl stitch turned out about 10% bigger than the ones I’d already produced in knit stitch. This, I suspect, was due to the fact that human beings aren’t simply reversible: I’m right-handed, and the movements in purl stitch rely more heavily on the left hand, so purl turns out looser (hence bigger) than knit.

The answer was obvious, once I’d got over my frustration, stepped back through the looking-glass, and put my brain into knitting gear. I just had to use a smaller needle for the purl squares.

**The final product**

After a gestation of 5 months (or around 400 knitting-hours), the blanket has been delivered in a suitcase to my daughter’s London flat. It weighs over 4kg and has stretched to 250cm x 160cm after being left to hang on the bed.

My design template is here, with instructions, though I’d strongly recommend the satisfaction of designing your own.

**Postscript**

After my first blog on knitting and mathematics, mathematician and twitter-friend Richard Elwes challenged me to knit a Klein bottle. I haven’t (yet) risen to that challenge, but he may be amused to see that the symmetries of my blanket form a “Klein 4-group”, also named after Felix Klein.