First up: if you don’t know what fractals are (or if you do, for that matter) don’t be scared! It’s a pretentious name for a pretty easy concept.

Difficulty: I will assume comfort with basic tatting, including rings, chains, picots, joins, etc. You will also need pretty good spatial reasoning, depending how big an object you want.

Notation: I use a pretty aggressive shorthand when writing up tatting patterns for myself. A very simple edging pattern might be: “R6j6p6p6, C6b6, repeat”; R and C of course stand for Ring and Chain; the numbers indicate numbers of double stitches (ds); p means picot, j means join, and b means a picot with a bead on it. Joins before any picots join to the previous iteration of the pattern, so the first time you do the repeat unit, do picots where it says j. For more complicated patterns, I will give the p’s and j’s subscripts, so j_{A} joins to picot p_{A} and so on. I like this shorthand because it fits on a card small enough to fit in my tatting carry-case, but will give long form as well in the blog.

Today’s pattern is actually a set of very simple procedural rules. With them, you can generate flat 45-45-90 triangles or squares of arbitrarily large size, suitable for doilies or shawls or purses or whatever. It is made of three elements: the basic motif, the turning ring, and the turning long chain. Chains and rings in this pattern always connect in the usual way, *i.e.* you never need to swap threads/change shuttles.

If you want beads, thread a bunch onto your ball thread before you start. I make beaded picots by simply sliding them up the ball thread onto the picot as I make it. There are other methods involving joined picots, but that is not how I am beading today. If you don’t want beads, simply omit them, either replacing “bead” in the pattern with a decorative picot (never join to them) or omitting entirely.

Basic motif (M):

- Short pattern: R6p6p6, C4p6b6p4, R6j6p6.
- Long pattern:
- Ring: 6ds, picot or join to another motif, 6ds, picot A, 6ds.
- Chain: 4ds, picot or join, 6ds, bead, 6ds, picot or join, 4ds.
- Ring: 6ds, join to picot A, 6ds, picot or join, 6ds.

Whether you picot or join, where unspecified, depends on where in the overall pattern you are, and will be noted later.

Turning stargate [ring with legs] (S):

- Short pattern: C4, R6p6p6, C4.
- Long pattern: Chain: 4ds. Ring: 6ds, picot or join, 6ds, picot or join, 6ds. Chain: 4ds.

Turning long chain (L):

- Short pattern: C4p6b6p4.
- Long pattern: Chain: 4ds, picot or join, 6ds, bead, 6ds, picot or join, 4ds.

I am using size 10 crochet thread, a size 5 needle, and size 6/0 glass beads. If you use a different combination of materials, you may need to alter the chain lengths to get it to lie flat–particularly the middle segments of all the long chains, which is currently 6b6.

Now we have three basic elements, we need to connect them into a larger whole. Using the abbreviations above, if you tat M-S-M (all connected; don’t cut your thread!), you will get two motifs sitting chain-side to chain-side, connected by this stargate piece. If you join the first picot of the chain in the second motif to the last picot of the chain in the first motif, it should sit flat. If on the other hand you were to tat M-L-M, the two motifs sit ring-side to ring-side, and joining the first and last ring picots in the second motif to the last and first picots in the first motif should be natural. You can imagine building up larger pieces in this way: M-L-M-L makes a small cloverleaf with chains on the outside, M-S-M-S-M-S-M-S makes a larger square with rings sticking out. (MSML)x4 makes a bigger box, and so on.

One of these arrangements is special, and now comes the time for some mathy talk. It’s special because it’s a fractal–that is, it’s self-similar at different length scales. Technically this kind of fractal should be infinite in extent, but since I don’t actually have infinite thread, time or patience, I’m going to ignore that requirement. Self-similarity is interesting, though, and can give beautiful results. It is also what’s known as a space-filling pattern; fractals like the Mandelbrot set that you may be familiar with have these gaping holes in the middle and a background where nothing is going on, but space-filling fractals (and other space-filling curves) fill up all available space.

The special arrangement is this: tat MSMLMSM. Name this pattern N, N=MSMLMSM. Now you could continue tatting: N(the piece you’ve made)SNLNSN. Notice that’s the same sequence of motifs and turning pieces, just the motif is bigger. If you want, continue this pattern: name O=NSNLNSN, and tat OSOLOSO. Note you are quadrupling the size of the piece with every step, so doing this many layers is a pretty serious investment of time. But you can see what’s going on by doing a little tatting or a little drawing.

Self-similarity just means that different parts of the object are the same as (or at least similar to) each other. All of the M units are the same, because you do the same steps to make each one. Self-similarity at different length scales is a bit more special–this means that different-sized parts of the object are the same. Motifs M and N and O are similar, despite being 4x as big or small as each other. That is, they are all 45-45-90 triangles, they all have chains along the hypoteneuse (long side) and rings along the legs of the triangle, and they all connect to their neighbors in the same way. You may protest that M is more of a circle than a triangle, which is true, but if you include half of the turning pieces on each side, it looks more triangular, and it takes up a triangular space in the larger pattern.

As the piece gets larger–MSM or MSMLMSM will be plenty–you start to see the 45-45-90 triangle pattern emerge. Another way to think about this special arrangement is this: you make a 45-45-90 triangle. If there are chains on the hypoteneuse of the triangle, do a long chain and repeat what you have done so far. If there are rings on the hypoteneuse, do a stargate and repeat what you have done so far. Your thread will always be coming out one of the sharp corners of the triangle, and you will always do what’s necessary to reflect around the closest short side and repeat what you’ve done. No matter how big your triangle gets, this is always always what you do to make it one step larger.

Bonus: you can make this arbitrarily more complex; (M=R6p6p3p3,R3j3p6p3p3,R3j3p6p6; L=C12p9,R3j3p3,C9p12; S=C12,M,C12) joined in MLMSMLM fashion for instance has clover-leaf motifs joined into dense clusters, in the same fractal framework. For the love of whatever, though, don’t try this until you’re comfortable with the way the normal one joins.

Disclaimer: this pattern owes a lot to finding this but because I am not a member of intatters I can’t linkfollow and give proper attribution. It’s not quite the same, as I’ve changed numbers around to fit my own aesthetic purposes and run away with the fractal concept.

needlesnshuttlesReblogged this on needlesnshuttles.

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seesawyerPost authorPaul: that is beautiful, and I think I can even tell what’s going on! I take it the quilt is the first- through fourth-order Hilbert curves overlaid each other in different colors. Have you actually made a quilt like this, and if so, are there pictures?

Other readers: the Hilbert curve is another space-filling fractal that’s pretty closely related to the space-filling fractal my lace is based on.

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Paul ZagYes, this quilt has been made, but it’s at the quilter’s right now and I don’t think we took any pictures. Sorry! I’ll make sure to send some when we get it back. Not sure when that will be, though. The quilting is the 5th and 6th order curves overlaid in the same way, so it’s… kind of complicated.

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Paul ZagWe finally got the new one back from the quilters! It looks AMAZING. Dad took pictures to send to a contest, so here you go. http://zag.net/quilts/Kathy/

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seesawyerPost authorThat is lovely. Thanks for the pictures!

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Paul ZagHere’s a fractal quilt I designed based on Hilbert’s Curve. If you can figure out the patterns. Have fun!

https://drive.google.com/file/d/0B-fMiUIOrWSXVTNSWkhXbXpfRjQ/edit?usp=sharing

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