I've been thinking about tessellations in knitting recently, since Lucy Neatby is going to be teaching at Borealis Yarns this November. As a warning, knitted tessellations require either extensive intarsia, or extensive seaming. Two-stranded knitting in a few cases might work.
First off, a how-to guide and gallery (including an extensive collection of Escher tessellations) from Tessellations.org.
Once I started looking for knitted tessellations, it was like learning a new word - you see it everywhere. My Little Fishy is a child's raglan with three bands of two stranded knitted fish. Designed by Marie Grace Smith at The Garter Belt. (This might be stretching the definition a bit, since it has more distinct bands than typically seen in tessellations - but it does satisfy the requirement of interlocking tiles that can be tiled on a plane to infinity.)
I would also like to introduce the podcast Math4Knitters. This is mostly focussed on utilizing math in knitting, in a practical sense, but the underlying appreciation of math also comes through. Highlights include an interview with Meg Swanson (Episode 8), and a guest spot on the podcast Cast-On on fibonacci sequences (Episode 23).
Understandably enough, most of the entries in the geeky knitting section come from people in STEM careers (Government-speak for science, technology, engineering, mathematics). I can't even begin to express how exciting it is to find such creativity coming from the extended geek pool.
Here we present the beginning of the Redhead Genome Project scarf from She Dances in Dragon. She researched the mutations in the MRC1 gene thought to cause red hair, and charted the alleles as shown in the top part of the photo. G is black, C is blue, T red and A green, as per standard coloring for DNA bases. The DNA is charted 30 bases across a row, so it is even possible that the codons are together.
Designed by Rachel Bishop at Math Scarves. The pattern is based on a "perfect shuffle", a shuffle that perfectly interleaves the cards. Do it enough times, and the cards will return to the original configuration - the "enough times" is represented by the number of caston stitches in the scarf. Certain cards will group together, and are represented by the same color - modular arithmetic provides the logic behind the groupings. Rachel provides the full explanation under What is a perfect shuffle?.
I think this is very exciting, since I love the underlying pattern behind the Fibonacci sequence, but this is really only a means to grade two colors. I do not find the practice of rotating different colors within the Fibonacci sequence to use more colors to be emotionally satisfying (or a true Fibonacci sequence - and it is not based on modular arithmetic, either). I have a lot of green yarn in worsted and DK weight with different shades and textures, that I think would work well in a scarf like this. Instructions to knit your own perfect scarf. Alas, *.exe files will not run on the mac, so I will be using old fashioned pencil and paper.
It is also exciting since Rachel's was the first request to be added to the geeky knitting section. So she gets the first geeky knitting update - several more in (hopefully) the near future. The good news is that geeky knitting is burgeoning on the web; the bad is that it might not be possible to archive it all. When I first started a year ago, I did extensive searches and felt pretty confident that I got nearly all the available material. Now I feel that I am barely scratching the surface. But, overall that is a good thing - better too much geek knitting than not enough.
Yes, I've heard all of the criticisms: the twist is backwards, not the right offset between the two coils, not enough base pairs per twist... But really this trumps all objections - you recognize instantly what it is.
Why yes, I have seen the knitted digestive system that has been taking the 'net by storm. Probably, so has everyone else. But I'm updating the geeky knitting section, so this is going in too. I'm also starting to put in thumbnails - I don't want to step on any copyrighted toes, but small photos fall under fair use, and the illustrations really add to the link collection.
I've already blogged Heidi Antila's design of a knitted illusion scarf featuring Linux and Tux, originally published in the Finnish online knitting magazine Ulla. In true open source fashion, the designer is generously allowing translations (which are released under a creative commons license): currently English and French, with Spanish coming soon.
I have had a fractal pattern that has been stewing for awhile (so old it is not in the notebook that my SP gifted me, but rather on random scraps of paper). I have had a real block with translating it into knitting, since the principle of fractals means increasing amounts of negative space; so fair isle would work on the small scale, but on the larger scale it would mean intarsia. Or else making smaller blocks and joining together. Ugh to both, since it seems very inelegant. I just realized, while answering a question on what to do with two colors other than intarsia or fair isle, that the solution is double-knitting! *slaps forehead* Though for larger projects, joining blocks might be the way to go.
The other issue is distortion; I was thinking of a felting project, since the typical felting shrinkage results in stitch height becoming about equal to width, instead of stretched 2:3 for stockinette. Perhaps done in garter? That would be awfully stretchy for a large project. Must think more on this issue. The distortion does not matter to the fractal, it would still be self-similar, but it might not be pleasing to the eye.
The Fractal patterns I was thinking of knitting were based on the Cantor set, where in each iteration, the middle third of the line is removed and becomes negative space. This could make a really striking cuff or neckband. Or a sideways knit scarf. I was thinking of using the Sierpinski triangle on the side of a felted bag (I'm now working on a cellular automata pattern, though). The Sierpinski carpet would make a stunning afghan. Or, the triangle could be tiled into a hexagon, which would also be gorgeous.