Sunday, October 4, 2009

Is there a topologist in the house?

Seriously. Because I would really like to know what is the proper name for this form:
It's a double-knit moebius. But what's the mathematical name of this thing? It's got a half-twist, so it must be a moebius. But it has an inside and an outside, which means it has volume, so it can't be a moebius. So what is it? (My best guess: some kind of twisted toroidal form). If there are any topologists out there reading this, I hope you know the answer, and that you will let me know.

Here is my latest video, which explores these topics.
video
(View in YouTube).

Here's where this saga began...

Once upon a time I learned from Cat Bordhi's book A Treasury of Magical Knitting how to knit a moebius. Pretty cool. Then I heard through the grapevine that Judy Becker had made a doubleknit moebius, which naturally made me want to try to make my own. Only I wanted mine to be continuous, with no insertion point. The image above is what I came up with. The red crochet chain shows where the stitches came off the needles.

While I was knitting it, I was vastly curious what I would end up with if I bound off the edges separately and unfurled it, instead of grafting the edges together. So, here's what you get if you pull off the crochet chain...












Pull the sides apart...












Untwist, and voilá!
To my complete surprise, I found it to be the *exact same* form that you get if you cut an ordinary moebius down its center axis: a tube with 720 degrees (2 full turns) of twist.

Ergo, you can take this form and re-shape it into an ordinary moebius (note how the knit and the purl sides meet in the middle here):
















I don't know about you but I thought, oh man, this is way cool!!

If you stuff the closed form to give it some volume, you get this:


Note how the half-twist goes away as it gains volume, and instead you get a spiral running through the middle. Awesome!




I've continued to explore this topic and found that there are yet more ways to create a doubleknit moebius, some of which may surprise you. Stay tuned...

3 comments:

  1. It reminds me of the klein bottle hat that was popular a few years back...

    http://www.woolworks.org/patterns/klein.txt

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  2. Sorry to disappoint you, but I think that's, topologically speaking, just a standard torus.

    What makes the mobius special is, indeed, its lack of volume (and hence the fact that it has edges). Now, with a metric that preserves the twist, it might be some sort of special topology, but...

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  3. This is a comment from Sarah-Marie Belcastro, relayed to you through me because of a snafu with my comment settings (now corrected).

    "Hi Jeny,

    The twisted torus you made is geometrically different from the standard torus but topologically the same under a homeomorphism. You might be interested in Amy Szczepanski's chapter in _Making Mathematics with Needlework_ as it contains several Mobius band explorations along the lines of those you've been doing.

    --sarah-marie."

    I might add, anyone reading this MUST check out Sarah-Marie's blog, http://toroidalsnark(dot)net.

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