Still working out the details, but seriously, rows of stitches are like sets, and knitting patterns are like groups or functions that act on the sets! It's just like transformation theory in music!
Say you have a tone row, where the notes C through B are represented as pitch classes 0 to 11.
Random tone row: 3 7 6 2 1 9 5 8 11 10 4 0
Function: T3 (transpose up 3 chromatic steps).
Elements in the domain (original tone row) map to elements in the range (resulting tone row) according to the function x + 3 (add 3).
Resulting tone row: 6 10 9 5 4 0 8 11 2 1 7 3
In other words, 3 maps to 6, 7 maps to 10, etc.
You could write this as three cycles: (0, 3, 6, 9), (1,4, 7, 10), (2, 5, 8, 11)
These are called orbits: an element can't map to anything outside its orbit, no matter how many times you apply the function.
Anyhoo, the function turns one tone row into another that is a rearrangement of the same elements. Similarly, in knitting, any function (pattern) will take a row of knits and purls and produce another row of knits and purls in some other order. And that's how you knit cool textiles. Just two different stitches, but mix them up and the possibilities are endless!
Off to geek out. I mean, sleep.
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