Friday, October 26, 2007

Math and Knitting

I went to a graph theory conference in Detroit a few weeks ago, and while there was math happening, nothing really caught my fancy. I went to talks, I listened, I zoned, I free-associated. And this happened:

This is the Petersen graph. It's a well-known and much-used example and counterexample graph. It's complex enough to be interesting, but basic enough to be easy to play with and test things on. (Trying to find a way out of ending that sentence with a preposition. Failing.)

The graph above has had the vertices (the dots) labeled by me to go with the matrix below. An adjacency matrix is a representation of a graph in which the vertices correspond to both columns and rows, and the entries are the number of edges (lines) connecting adjacent vertices. The top matrix below is the adjacency matrix of the graph above. Below that is a chart. The entries in the matrix have been replaced with yarn overs.

The symbols along the diagonal are k3tog, the ones on the right edge are k2tog. o's are yarnovers, - is purl, and if unspecified, they're knit stitches. Since the diagonal of an adjacency matrix is a funny place, defined by whether or not you consider a vertex to be adjacent to itself (I said no in my matrix), I figured decreases would work along it. Since I didn't want to deal with triple decreases, I made them double, but since the graph is 3-regular (every vertex is adjacent to three others), I needed a third, hence the line of decreases on the right. The purl column was for definition, but I'm not sure how necessary it is.

I knit it up using the yarn and needles I had on me at the time (giant grey man socks that are still threatening me with their not-done-ness). Here's the swatch that resulted. The top is the pattern in the chart. Below is the same, except the k3tog's are instead slip 1, k2tog, psso.

It's a crappy blocking job, but I'm pretty sure a good one would still have issues with the diagonal skew - which seems like it would be predictable were I a more experienced lace knitter. In all, I'm not in love with it. I find it highly doubtful that I'll be working on Petersen socks anytime soon, but I found the idea interesting anyway. I have a hypercube matrix charted out that might have fewer issues, but then again, it might have more.


Gemra said...

Hi! I was wondering how many stitches you cast on to make your pattern work - I am writing a seminar and have produced a fair few adjacancy tables for various graphs but am having a bit of a problem turning them into work-able patterns! Any help woudl be much appreciated!

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