Big news! The exciting new book Making Mathematics with Needlework (in which, incidentally, I have an article) is listed for preorder on Amazon and Barnes and Noble among other places. For details of the many projects and mathematical curiosities shared by mathematical fiber artisans, take a look at sarah-marie's page for the book.
The Sierpinski triangle, or Sierpinski gasket, shown above, is a fairly well-known fractal. It can also be described in terms of a cellular automaton, or even as the mod 2 representation of Pascal's triangle. Also, since one can start at the top, and stop at any row that's a power of 2, it's an easy thing to construct and stop whenever you feel like you're done. For all these reasons, it's one of a small number of fractals which is amenable to crochet.
My work in this area isn't entirely original: the variation which got me started (given as "basic filet variation" below), originally comes from Mary Pat Campbell's online journal. The other variations are, as far as I know, original.
To jump straight to examples of this work, check out the Sierpinski gallery!
I call this the "filet" variation since filet crochet is the particular branch which deals with creating shapes via open blocks and closed blocks of double-crochet. While this is not strictly a filet work, it bears certain similarities to filet.
Row 1: ch 4 (counts as 1 dc), 2 dc in 4th ch from hook, turn.
Row 2: ch 3, dc in first dc, ch 1, skip next dc, 2 dc in last dc.
Row 3: ch 3, dc in first dc, ch 1, skip next dc, dc in ch 1-sp, ch 1, skip next dc, 2 dc in last dc.
Row 4 and on: start with a ch 3, dc in first dc, and in each stitch except the last, follow the following method: look to the left and right of the next stitch or space. Are they both the same thing (i.e. both dc or both ch-1), or are they different? If they're the same, put a dc in the next stitch or space. If they're different, make a ch-1 and skip the stitch or space. Finally, in the last dc of the row, work 2 dc.
Here are a few hints to let you know if you're doing it right:
The angle at the head of the work should be between 45 and 60 degrees, depending how tightly you crochet.
If you use different size stitches (for instance sc or hdc for a shorter triangle, trc for a longer one), the triangle will have a different shape and a different head angle; if you're going to do a project involving stitching them together, this sort of control over the shape may be useful.
To accentuate the visual aspect of the self-similarlty, one might change colors after every (2n+2)th row.
The 2dc at the end of the triangle form a minimal edging. For fancy work, one might opt to replace these with something a bit more ornate, or, at the end of the work, to work an edging around the entire triangle.
The filet method can also be worked in the round: this has the advantage of edgelessness (which may be neater), but the disadvantage of being "biased": since each stitch is roughly L-shaped, the head of the stitch is off to one side of the stitch's post (which side depends on the handedness of the artisan). As a result, work in the round tends to "swirl" a bit; this may, however, be a desirable effect. Below is an example of how the round-work might be done (assuming a head-angle of 60 degrees, so that 6 triangles together lie flat):
Chain 3, sl st in first st to make a loop.
Round 1: 6sc in loop.
Round 2: (2 hdc in next sc) 6 times.
Round 3: (ch 3, skip next hdc, dc in next hdc) 6 times.
Round 3: ch 3, dc in first ch-3 space, [dc in next dc, (dc, ch 3, dc) in next ch-3 space ] 5 times, dc in next dc and in next ch-3 space.
Round 4: (ch 3, dc in same ch-3 space as last dc, ch 1, skip next dc, dc in next dc, ch 1, skip next dc, dc in next ch-3 space) 6 times.
Round 5: (ch 3, dc in same ch-3 space as last dc, dc in next 5 stitches, dc in next ch-3 space) 6 times.
Round 6: (ch 3, dc in same ch-3 space as last dc, ch 1, skip next dc, dc in next 5 stitches, ch 1, skip next dc, dc in next ch-3 space) 6 times.
This pattern continues in the same manner as the basic filet version, except instead of increasing by 2dc at the edges, one increases with a dc-ch 3-dc in the corner ch-3 spaces.
The procedure given here, in which the status of a stitch on a row depends on the stitches in the row directly above it, mirrors the evolutionary aspects of an elementary cellular automaton. Stephen Wolfram has studied and classified these automata, and has determined 256 different rules (each cell is influenced by 3 predecessors, so a "rule" is a function mapping 8 possible inputs to 2 possible outputs). The stitch-design given above is in fact Wolfram's Rule 90, with white squares considered as stitches and black squares as chain spaces; other rules interpreted as crochet may also produce interesting results, such as Rule 30 or Rule 150. A full list of the rules is available at the Wolfram Atlas of Simple Programs; the only caveats to consider are that odd-numbered rules do not generally yield triangles, and that excessively long black sections would correspond to long chains, which may be structurally unsound or simply unattractively non-rectangular.
The relief variations were initially conceived as a way to avoid the bias effect produced when working in the round, but they produce a strikingly different sort of effect from a Sierpinski-triangle design.Unfortunately, their applicability to general automata is limited.
Row 1: ch 3 (counts as 1 hdc), 2 hdc in 3rd ch from hook, turn.
Row 2: ch 2 (counts as 1 hdc), skip first hdc, 2 FPdc around next hdc, hdc in head of ch-2, turn.
Row 3: ch 2 (counts as 1 hdc), skip first hdc, BPdc around next dc, hdc between the dc's, BPdc around next dc, hdc in head of ch-2, turn.
Row 4: ch 2 (counts as 1 hdc), skip first hdc, 2 FPdc around next dc, skip next hdc, 2FPdc around next dc, hdc in head of ch-2, turn.
Row 5: ch 2 (counts as 1 hdc), skip first hdc, BPdc around next dc, hdc in the next 3 spaces between the hdc's, BPdc around next dc, hdc in head of ch-2, turn.
Row 6: ch 2 (counts as 1 hdc), skip first hdc, 2 FPdc around next dc, hdc in the next 2 spaces between the hdc's, 2FPdc around next dc, hdc in head of ch-2, turn.
The general pattern to be followed is as such: each row begins with an hdc and ends with an hdc. In between we use relief stitched (either FPdc or a BPdc, depending on which side the relief design is on) and nonrelief stitches (hdc); however, we consider these stitches as corresponding not to the stitches below them but to the spaces between the stitches below them. If the space below a stitch is flanked by two identical stitches (2 relief stitches or 2 nonrelief stitches), we work a nonrelief stitch. If the space is flanked by one relief and one nonrelief stitch, we work a relief stitch around the post of the nearby relief stitch. So, if this pattern is followed, row 7 should be
Row 7: ch 2 (counts as 1 hdc), skip first hdc, BPdc around next dc, hdc between next two hdc's, BPdc around next dc, hdc between next two hdc's, BPdc around next dc, hdc between next two hdc's, BPdc around next dc, hdc in head of ch-2, turn.