Pattern for fractal shawl
This pattern is adaptable to any yarn, any size hook. Once you get started, you just need to remember a rule. This pattern is based on the concept of =cellular automata=, in which one looks back at the previous row of interest to figure out what you're doing on the current row. With the particular rules I used, the pattern produced will be the fractal called the Sierpinski triangle (if you don't make a mistake... but mistakes can produce neat patterns of their own!) and I got the idea from an old issue of Knitter's magazine, and I adapted it for crochet.
So here it is:
(and I'm using american definitions of double crochet.)
Row 1: Ch 4. 2 dc in 1st ch - 1st row, counts as 3 dc. (the turning chain is 3 ch, and counts as one dc). Ch 3, turn.
Row 2: 1 dc in 1st dc, Ch 1 & skip middle dc, 2 dc in last dc. (so looks like - 2 dc, ch-sp, 2 dc) Ch 3, turn.
Row 3: 1 dc in 1st dc, Ch1 & skip next dc, dc in ch-sp, ch 1 & skip next dc, 2 dc in last dc. (looks like: 2 dc, ch sp, dc, ch sp, 2dc) Ch 3, turn
Ok, the first couple rows go by this rule, but here it is:
You're going to add a dc at each end for each row (that's the 1 dc in the 1st, + the turning chain at the beginning, and the 2 dc in the last dc); this gives the standard triangular shawl shape. One starts at the tip and works until the triangle is as large as you wish.
Now, you're going to be looking at the previous row while you build up your current row. The rule is to look at the two "areas" neighboring your current area in the previous row. If there's two spaces, or no spaces, you simply do dc's straight. If there's exactly one space , make a ch1-sp, then dc.
So let me show how this works on the 4th Row:
So you've done the turning chain and 1 dc in the 1st dc. (You've just added a new "area")
Looking at the previous row, you see a ch-1 sp immediately to the left and two dcs to the right. So looking at my rule - exactly one space, make a space - I ch 1, skip a dc, and dc in the ch-sp in the row below. Second area done.
Now, I look below, and I see a ch-sp to the left, and a ch-sp in the right. My rule: two spaces beget no spaces, so I do 1 dc each in the next two dcs. Third area done.
Finally, I see the ch-sp to the right, and 2dcs to the left, so exactly 1 sp. So I ch-1, skip the dc and dc in the next (and last dc). Fourth area done.
Finally, one more dc in the final dc, to add an area on the end.
So the pattern of the 4th row looks like:
2 dc, ch-sp, 3dc, ch-sp, 2dc.
Just to let you know, the 5th row looks like:
2 dc, ch-sp, 1dc, ch-sp, 1dc, ch-sp, 1 dc, ch-sp, 2 dc.
2dc, ch-sp, 7dc, ch-sp, 2dc
Those are just to let you check to see if you understand the rules.
If you do this correctly, on every row there should be 1 ch-sp at the very beginning and 1 ch-sp at the very end. Some of the rows will have nothing but dcs and no spaces. Some rows will be completely full of ch-1, 1dc blocks. There's actually a mathematical pattern as to which rows this occurs upon, and if you =really= care, you can figure it out. (Hint: It's related to powers of 2)
So you end up with a complicated-looking pattern:a small version
I wish I had a digital camera to show you the results. But make your own, and you'll see.