Saturday, April 15, 2017

Permutation cardi

There's some maths behind this one!  The idea here is that there are exactly 6 ways to permute 3 elements, and each of the cable represents one of these two permutations.

We can describe a permutation by writing down what it does to the elements 1, 2 and 3 - so the two cables on the front represent (1, 3, 2) and (2, 1, 3).

The ones on the back are slightly more complex - these ones represent (2, 3, 1) and (3, 1, 2), but in these cases you've got one element moving two places, which distorts the knitting a little more, so I've broken these into two steps each containing a single crossing.

The arms are the bit I'm most excited about - of the 6 possible permutations, there are two pairs which are mirror images (the ones we've seen), but the remaining two are their own mirror images, so they don't have a relationship - (3, 2, 1) and the trivial permutation (1, 2, 3), which leaves everything unchanged.  These give this nice asymmetry between the two arms.  (3, 2, 1) again has elements moving two places, but it's hard to break this up into two steps, so here I've added an extra chain stitch to extend the cables.

Also exciting, the length of the repeats varies between permutations -  this gives their order in the symmetry group: (1, 3, 2), (2, 1, 3) and (3, 2, 1) all have order 2, (2, 3, 1) and (3, 1, 2) are order 3, and (1, 2, 3) is order 1, so the lengths of their repeats is different.  It's cool too that these all divide 6, so if you could fit in exactly 6 repeats, all the cables would end up in the right place after the same number of rows.

You could go much further with this too!  These permutations are at the heart of bell-ringing, although taken much further - for example here.  I love the intricate patterns this gives, but I would be wary of doing this with intarsia cables - as much as I love the colours, this ends up with the yarn getting crossed and needing to be constantly untangled.  Still, imagine that beautiful scarf with rainbows weaving together!