The Fractal Thing. Caution: math content.

Last Tuesday (so I'm slow... I am responding!), Laurie left a comment about Seed Magazine's inspirational potential, and asked whether or not I had ever seen fractals, and what did I think? (To paraphrase. This is not a direct quote.)

In a word, I think fractals are cool. But I can't figure out how to knit one (at least, how to knit one without going insane). In a nutshell, a fractal is a graphic representation of a math equation, just like the line-on-a-grid stuff you did in algebra class, but more complicated and prettier. (I am dumbing this down, it's a knitting blog, not a calculus class. And I hate higher math. But that's the bottom line.) Depending on the equation, you can get a repeating pattern in smaller and smaller detail (it's the same concept as when you do, say, a division problem and get an answer with a repeat, like 1.66666 to infinity. All a fractal is, is a picture of that answer, repeating itself smaller and smaller, to infinity). So, anyway, you get stuff that looks like this:

Fractal from here; they have many others that are equally interesting.

As for knitting something like it... Well. In some ways it's impossible, and in other ways, we knit a fractal every time we pick up our needles. You could certainly take those colors (pretty, aren't they?) and knit something Fair-Isleish and call it a success. But to TRULY do a fractal, you'd need to start off on broomstick-size needles and get smaller and smaller and shift from superbulky all the way down through to sewing thread, using some kind of increasing-but-getting-smaller series of stitch patterns. (I could do this, stuff it, and call it a pillow, but the very idea smacks of math and makes my head hurt.)

However. Every time we do one of those "decrease every second row three times" deals, all we're doing is knitting a complex (sort of complex) equation, and creating, you guessed it, a graphic representation of it. So by that view, everything we knit is a fractal, or a graph, or... something like that. Don't believe me? Here's a fractal.

I SWEAR that's a fractal, and not a doily. Food for thought, no? (Image from somewhere on the Yale website, the page loaded weird and I lost the link.)

AND THEN, we have Debbie New of "Unexpected Knitting", who has already done her own version of fractal knitting, called "Cellular Automaton Knitting" (chapter seven in her book). You make up a simple rule based on your knitting, and then follow through with it on a scarf, a hat, a sweater, whatever. (I don't wanna get into too much detail because it's her book, and her idea.) What you get looks and behaves mathematically just like this:

Graphic and details on how this works (in math, not knitting), from here.

My conclusion? I don't think I'll ever knit a fractal. And I do it all the time. Music is math, nature is math, and even though we don't want to admit it, knitting is, alas, math.

All this pure theory has made my head hurt. I'm going to go lay down.


FOOTNOTEISH:
My ability to converse about math at all is due to a series of excellent math professors (including one who was absolutely mad about graphs, and this kind of stuff) at Leeward Community College. I would take in 'knitting math questions' and they would discuss and ponder and puzzle and not think I was in the least wierd. Since then, my mother-in-law (a retired math teacher) has bravely learned to knit, and we can discuss knitting math together.

If you found anything I wrote here even remotely interesting, get a copy of "Unexpected Knitting" by Debbie New. I am a dabbler in this stuff. She is the master. The only thing I have ever found wrong with the book is that it's too big for me to take into the bathtub to read there.

Downloads are available here, to create your own fractals and all sorts of other math craziness. It includes some downloads about halfway down the page entitled "Create Patterns with Math" under the heading "Images".

Oh, and Trish? You can't make me learn to crochet.