Fractal Crochet

25 Mar 2002


I thought I'd put something a little more concrete in here.  I do tend
to spout off, philosophize, etc.  To paraphrase, opinions are like
parents -- everybody has them (and nobody particularly wants to meet
them, which extends the original simile (if you don't know the original
simile, I'm not going to tell it to you -- just think of things
everybody has)).

(Speaking of opinions, and random spouting, let me impart my latest
thought - on the phrase "blind as a bat" (also a simile, lookee that).
Now many bats are blind, but some have some limited sight, and to a
large part the bats don't need the sight to survive, thank you very
much.  Such a phrase indicates our own biases towards sight in human
civilization, and even though we may concede that blind bats seem to
manage to eat and mate without visual cues, we may say "ah, but those
bats are missing the beauty of a moonlit night."  Hey, those bats might
think us lacking the experience of the 3-d world, and world of
trajectories and surfaces... just as dogs might feel sorry for their
smelling-impaired masters. Anyway, I just think "blind as a bat" is a
weak simile; why not "blind as a busy man"?  It's even sadder when one
has a sense but doesn't ever experience the beauty of the world through
that sense.)

In any case, for some time I've been quite the crafter -- I knit,
crochet, cross-stitch.  For the most part, I use patterns made by other
people, though I invariably change them, due to my own mistakes, the
materials I'm using (or lack thereof), or personal purposes.  For
example, I just took a simple teddy bear pattern and did the bear in
green.  Then I added spikes to the back and a long lizard tail --
Bearzilla!  I like making personalized presents (Bearzilla is for my
ma).

However, a couple times I've designed my own patterns.  Once I came up
with an angel pattern, I've made a braided cross pattern, and I did a
personalized wedding gift for a high school friend, by making a pattern
where his and his wife's names were done as mirror images of each other.
I usually just sit down with some graph paper and draw out the patterns.

But sometimes I use math.  Once I did a cross-stitch pattern for a
friend, where I came up with a cellular automaton pattern based on a
nickname and using his initials from other names as seeds.  A cellular
automaton is a mathematical construct where there are cells in a regular
pattern - usually a square grid - and a discrete set of properties for
each cell, such as color, or whether a cell is "on" or "off".  The most
famous example is John Conway's "Game of Life", which is a 2-dimensional
cellular automaton that "evolves" over time, and the cells are either
"occupied" or not, which is usually represented as colored in or not
(sometimes the colors are based on how long the cell has been "alive").

Well, a pattern that changes over time isn't exactly the easiest to
replicate on graph paper for a cross-stitch pattern, so instead of a
fluctuating set of rules, I made an automaton that only grew  -- cells
never died.  I called it the "kudzu automaton" in honor of my friend's
nick "kudzu".

Years passed, and then I came across an article in Knitter's Magazine.
It mentioned using cellular automata for knitting lace patterns, cabled
patterns, fair-isle patterns, etc. It used a cellular automaton that is
1-dimensional (after all, knitting and crochet build up row by row, not
a 2-d field like cross-stitch) that I recognized to be the generator for
the Sierpinski gasket.  So I tried this with knitting, but knitting is
too easy to screw up and too hard to fixed once one has screwed up (and
besides, I'm not good at lace knitting.)  So I adapted the technique for
crochet to make a large fractal shawl.  In crochet, one has only one
active stitch at a time - you can't drop the entire row - and it's easy
to see what the neighboring cells are in the previous row in crochet.

I am willing to teach anybody to crochet, but it's much easier to do it
in person. For those interested in mathematical crochet, I can teach
people to crochet the hyperbolic plane, which is even easier to do, as
one needs only to know how to chain and do the single crochet stitch.  I
found the pattern online, and if you wish to learn to crochet, I
recommend going to the web site of the Craft Yarn Council of America
(headquartered in Gastonia, NC, I believe).

So here's the pattern: ch = chain ch-sp = chain space dc = double
crochet (the american definition)

This pattern is adaptable to any yarn, any size hook. Once you get
started, you just need to remember the cellular automaton rule.

So here it is:

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.

6th row: 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)