Polynomial riff scarf

We've already inflicted upon you ways to use solutions to multivariant polynomials to design patterns, an approach first discovered by Ada Deitz in the 40s. We thought it'd be fun to give you a simple example on how to do this, in knitting instead of the weaving used by Ada.

So to make it fun let's consider something a bit complex but not difficult:

	  
	  (a + b)³ =  a³ + 3a²b + 3ab²  b³

Like the results? They 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.
This would take 3 balls of Merino Frappe from Crystal Palace, in 016 fern color. Knitted on US size 11 bamboo needles, straight or circular.

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. You can sort of see all that in this bad picture.

Let's review these variations in turn:

a³ + 3ab + 3a²b² + b³
start off with a mistake! should have been:
a³ + 3a + 3b + 3a² + 3b² + b³

3a²b + a³ + b³ + 3ab²

3a²b + a³ + 3ab² + b³

3a²b + b³ + 3ab² + a³
again, but with a=b=3
(doesn't all show)

a³ + 3ab² + 3a²b + b³

Is that all starting to make sense? Of course you could choose whichever variation suits you best, and do only that one from one end of the scarf/sweater to the other. And I'm sure there are other variations I didn't happen to think of before I had a long enough scarf. You could mix and match different size a and b in the same section for instance. So go ahead :-)...