I've been knitting various möbiuses for some time now, and noticing a lot of curious phenomena. You use one long cable doubled up, cast on (using a method that produces what feels like weaker stitches, weaker links) a number of stitches, say 60, and end up with twice that number, 120. You have 60 stitches above and 60 below, and it looks rather like the stitches that you are most immediately working with (those on what appears to be the top) are the positive stitches, and the ones below are inverted (negative). The half twist of the möbius is almost never where the needles meet as it can get in the way as you knit, which makes it even more difficult to wrap your mind around it at first---you'll probably end up knitting 60 (the circumference of the loop, if it wasn't a strange one) and realize that the marker for the beginning of the row is on the cable below and that suddenly one row is all 120 stitches. And as you work with the first row, every other stitch is facing the opposite direction. It gets even better: the stitches grow between the needles/cable; you started knitting in the middle instead of at one end and it's growing out! (But actually between.) Moreover, by knitting every stitch, which naturally appears as a purl stitch on the back (think positive and negative again), your resulting möbius is a strip that's half knit and half purl on one "side" (and inverted on the other "side"), despite your using only the one type of stitch, and despite there being only one side. It would be the same creature if you purled the whole thing; without even turning it inside out. The only thing that would make it different would be that it was constructed with a different orientation.
You'll certainly have gone loopy by the beginning of this, and you won't know it if you think you're at the beginning. The stitches you knit are the edge of the möbius. They're the first stitches you knit, the last stitches you'll knit, and every stitch you knit between-in. You start in the middle and grow from there, though it's growing between and not out. The beginning and the end are one and the same, despite that you can make it bigger and bigger, adding more stitches "between" the two.
The topological definition of a möbius is a surface on/in which there is no point at which to orient oneself. Objectively, ultimately. Every point is relative. And yet as you knit the möbius, each stitch you make is unique and individual, separate and identifiable, made of the same never the same material, flow of material, varying in dimensions, mass, consistency, time of creation, etc. Similar to beings in our world, and to particles in general, right? So we can chose a point to orient from, and we do as we beings live life. Everything becomes relative; every measure, every separation, every individualization of anything is relative and yet is what it is, completely and as if relativity never factored in.
I like the möbius because for me it symbolizes existence in a way, making one coin out of all 'many' things, even half twists and not. It transforms an either/or, a polarization that can be a deception (in the way that dichotomies can be) and yet still be totally necessary to life, into a both/and. It even makes both/and and either/or one thing . I like the möbius because it reminds me of the middleway, of all ways and no ways, and even if we can't totally pick it apart or wrap ourselves around the paradox, we can still make one to hold and touch and live. And we can gradually become familiar with its nature.
I'm currently underway with a new knitting near-fiasco to see what comes of a hyperbolic möbius. My dad shared with me a devastatingly awesome project, movement, and eventually foundation started by these twin sisters from Australia, who, combined, have a background in mathematics, art, creative and science writing, women's studies and more, and (to boot!) like to crochet while they watch Battlestar Galactica and Xena: Warrior Princess. They defied Euclidian geometry and lots of stuffy old mathematicians by crocheting hyperbolic surfaces (previously thought to be unproducible) in the image of sea cucumbers and corals and such to draw attention to the world's suffering coral reefs, while at the same time empowering women through a handicraft that's traditionally been 'theirs' and empowering the people by putting this previously inaccessible theory of math, geometry, and physics into their hands, with a crochet needle. They are sock-knocker-offers who continue to inspire the communities of the world on the topics of environmentalism, math and geometry, physics and the sciences, feminism, education, evolution, fiber arts and more as their project continues. Look them up at:
http://www.theiff.org/
You'll certainly have gone loopy by the beginning of this, and you won't know it if you think you're at the beginning. The stitches you knit are the edge of the möbius. They're the first stitches you knit, the last stitches you'll knit, and every stitch you knit between-in. You start in the middle and grow from there, though it's growing between and not out. The beginning and the end are one and the same, despite that you can make it bigger and bigger, adding more stitches "between" the two.
The topological definition of a möbius is a surface on/in which there is no point at which to orient oneself. Objectively, ultimately. Every point is relative. And yet as you knit the möbius, each stitch you make is unique and individual, separate and identifiable, made of the same never the same material, flow of material, varying in dimensions, mass, consistency, time of creation, etc. Similar to beings in our world, and to particles in general, right? So we can chose a point to orient from, and we do as we beings live life. Everything becomes relative; every measure, every separation, every individualization of anything is relative and yet is what it is, completely and as if relativity never factored in.
I like the möbius because for me it symbolizes existence in a way, making one coin out of all 'many' things, even half twists and not. It transforms an either/or, a polarization that can be a deception (in the way that dichotomies can be) and yet still be totally necessary to life, into a both/and. It even makes both/and and either/or one thing . I like the möbius because it reminds me of the middleway, of all ways and no ways, and even if we can't totally pick it apart or wrap ourselves around the paradox, we can still make one to hold and touch and live. And we can gradually become familiar with its nature.
I'm currently underway with a new knitting near-fiasco to see what comes of a hyperbolic möbius. My dad shared with me a devastatingly awesome project, movement, and eventually foundation started by these twin sisters from Australia, who, combined, have a background in mathematics, art, creative and science writing, women's studies and more, and (to boot!) like to crochet while they watch Battlestar Galactica and Xena: Warrior Princess. They defied Euclidian geometry and lots of stuffy old mathematicians by crocheting hyperbolic surfaces (previously thought to be unproducible) in the image of sea cucumbers and corals and such to draw attention to the world's suffering coral reefs, while at the same time empowering women through a handicraft that's traditionally been 'theirs' and empowering the people by putting this previously inaccessible theory of math, geometry, and physics into their hands, with a crochet needle. They are sock-knocker-offers who continue to inspire the communities of the world on the topics of environmentalism, math and geometry, physics and the sciences, feminism, education, evolution, fiber arts and more as their project continues. Look them up at:
http://www.theiff.org/
And also watch this terrific 15 minute lecture:
I'm in thought right now about the meaning of this growing hyperbolic möbius. I haven't found that anyone else out there has tried this, but that might be the fault of internet search engines. It's working out, though the increase is slow. I'm particularly interested in the kind of... Uh, reframable amplification that's going on. Lots of seemings. The exponential growth looks strange in a möbius. It makes me think about lasers and mirrors and the conversion of matter into energy and the relationship between matter and energy, matter and anti-matter, and also egos and selfs and consciousness. And space. And the geometry of the cosmos. And also, the dimensions. (My theory is that there's a half-twist uniting the dimensions, as in a twist that folds, for example, the 3rd through the 4th dimension, and that life emerges at the intersection, interjunction of the two, if not from all of them... Still workin' on that!) More to come later. I'll be entering into a retreat in engaged Buddhism, and I'm sure that more will emerge.
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