double-knit moebius

It all started at the Island Retreat a couple of weeks ago. Some well-meaning knitter - I forget the exact context now - said something about double-knitting a moebius. I immediately wanted to do just that. Because I never had. And it sounded really cool. And you could have a moebius that had no purl side, but only a smooth knit side. And different colors. And what would those colors do when swirled around a non-orientable surface that has only one side and one edge?

I immediately headed to Island Wools for yarn. Cat, who is wiser than I, perhaps saw the gleam of scientific inquiry in my eyes because she smiled and nodded and said nothing but sounds fun.

Yeah.

Thus began a quest for how to cast the darn thing on. It kept twisting around itself in strange and other-dimensionally ways. After much frustration and frogging bad words were said in my out-loud voice several false starts, I finally hit on a method that seemed to sort of work: I cast on two individual moebii, knit 1 ring on the white one and purled 1 ring on the green one. Remember it takes two rings to make a round, so I had worked 1/2 round on each. Then I threaded both on to a single long cable needle by alternating stitches.

Gentle reader, I have learned more about topography in the last week than I ever expected to in my wildest dreams.

I turns out that when you take two moebii and attach them along the edge, essentially what I did when I threaded the two on to one needle, the result is a torus (that’s a fancy word for a doughnut) with a moebius twist in it. And that’s one representation of a non-orientable surface called a Klein Bottle. And such beasties do not live in 3 dimensional space without some contortion. And I haven’t yet figured out the knack of knitting in 4-space, where they live happily but I do not.

So what I knit in my little experiment is a Klein Bottle represented locally in my particular patch of 3-space by letting what I thought would be a moebius intersect with itself - which it wouldn’t do in 4-space. You can see the insertion point - it’s the point at which the green suddenly changes to white.The green has actually crossed through the white to reappear on the other side (if it could be said to have sides, although it really only has one), and the white has crossed through the green. But if you trace either color by itself, ignoring the other color, you will find that each is, really, a moebius. And together they appear to be a moebius. Sort of. If it weren’t for that pesky insertion point.

I don’t think there’s any way to get around the color change, short of knitting the whole thing in one color. But then what would be the point of double-knitting? I’ve decided instead to make the intersection a design feature by adding in random ones scattered about various places. We’ll see how well that plan works out. I may end up with a mess. And I have no idea how I’m going to bind this thing off, but I’m open to suggestions.

I hope I don’t sound like I understand what I’m talking about, here, because really my math is not good enough to understand it in more than lay-person’s terms. All of this stuff makes my brain hurt. But, nonetheless, this is the sort-of-moebius that’s on my needles and that I will be knitting today with healing intentions.

I do think it’s cool that, even though there are two colors, there are really no sides. And although the green and white started out as completely separate entities, they have now merged in new and unexpected ways.

Knit on!