knitting with equations

October 15th, 2008




Warning: mathematics and knitting jargon follows.

One of my secret pleasures is knitting. Of late I mainly make beanies for my friends, but living in a hot climate means there’s not many opportunities to wear them.

I’ve been sticking to those that don’t need patterns, basic rectangles that wear as ‘ears’, pixie points with chinstraps for babies, the odd bit of Fair Isle. This week I knitted an interesting architectural beanie called ‘Folded Bloom’ for my mate Robin (obviously a fan of funny hats!) from a great book I got at the library “Folk Hats” by Vicki Square. A lot easier to knit than it looks, using only knit stitch and a little bit of counting, measuring and construction.

Possibly one of the most interesting things about knitting is the mathematical nature of the projects. Planning a new knitting project from scratch, if I haven’t made it before, I use graph paper for designing the outcome.

However, my simple projects pale in comparison to a whole world of mathematical knitters! The Mathematical Knitter Marie-Anne Belcastro designed a mathematical proof detailing why any topological surface can be knit. Her site is [here]

The Klein bottle  has many knitting applications including [beanie], [tea cosy] or object de art (links to knitting patterns). The Mobius strip is an obvious [scarf or cowl]. Boy’s surface hat, can be found [here] and a crocheted Lorenz manifold [here] complete with mathematical exposition. A hyperbolic plane can make a nice [pompom].

Fuzzy Galore has a simple explanation of how knitting can be used to solved multivariant polynomials an approach first discovered by [Ada Deitz] in the 40s. Fuzzy uses this equation to apply to a scarf:

(a + b)³ = a³ + 3a²b + 3ab² b³

(the results) come simply from doodling with the placement of the various elements. The basic idea is to define say ‘a’ as ‘knit’ and ‘b’ as ‘purl’, and a basic unit, in this case principally a=b=2 (rows). I knitted along, picking a different arrangement for the various elements each time. The result looks harmonious, while there are really no repeats of any pattern at all. …The only trick is that I varied how the same elements were arranged, and once I also changed scale, making a section where a=b=3 rows. So as not to lose track of what I was doing, I also visually separated each section with an 8-row garter stitch section, on the basis that it’s the special case a=b=1.  [pictures here]

Fuzzy also has some great patterns including [Ada Deitz patterns from polynomials], [Priscilla's Probability Pullover] and the [Fibonacci sequence] in a pair of socks. Here’s how Fuzzy applies the latter to knit a rib:

aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
aaaa bbbb aaaa bbbb
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa
bbbb aaaa bbbb aaaa

And there’s a whole biology stream of knitting I’ve yet to explore: DNA, various bodily organs. Reckon cells would make for challenging knitting. Looks like my knitting promises to get weirder and weirder…

See Mathematical Knitting for more inspirations http://www.toroidalsnark.net/mathknit.html

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