Ariel's craft journal (dragoncrafter ) wrote in knitting ,

Klein bottles

These are my stuffed Klein bottles. They're what you get if you sew together two Möbius strips along their edges. In four dimensions, you can do this, but in three-dimensional space, you have to make the Möbius strips pass through themselves and each other.

There are two standard ways to do this.

Image hosted by Photobucket.com
Image hosted by Photobucket.com

The blue tube is the most common. I just made a tube, then pulled one end of the tube through the wall of the tube. This is the general shape seen in places like http://www.kleinbottle.com/, although they usually make the tube bulge out so you can see it better. Since I can't knit transparent fabric, I didn't bother.

I did do about six short rows at the end, to make the little "lip" near the self-intersection, and a few short rows along the way, to make it more willing to bend into a circle.

I pulled the tube through itself just by putting the working stitches on a lifeline, taking out the needles, poking a crochet hook through the fabric, grabbing a working loop, and pulling it through. This is one of the two common ways to make knitted fabric pass through itself. The other way I used for the other Klein bottle. A lot of knitted Klein bottles just have little holes to pass the tube through. I wanted to try doing knitted self-intersection.

Image hosted by Photobucket.com
Image hosted by Photobucket.comImage hosted by Photobucket.com

The pink tube is the alternative, "figure-8" embedding. It almost has to be made using either two circs, or magic loop. The yarn travels in a figure-8 pattern, so you have to have the other needle lying there between two stitches as you work. This works a lot better with a thin little cable than with a thick DPN.

I cheated. One lobe of the figure-8 was made by knitting every stitch, and one lobe was made by purling every stitch. This made the entire outside appear to be stockinette. (I think it would look better in reverse stockinette because then the join in the middle would look sharper, but ah well.) If I'd just knitted it, one lobe would be stockinette and one lobe would be reverse stockinette, and I'd have had to graft the start of the stockinette lobe to the end of the reverse stockinette lobe.

If you were willing to put up with "pinch point" singularities, then you could make a Klein bottle really easily: knit a tube, put in a lifeline, interlace the active stitches, and start knitting it again, and then graft the ends of the tube normally.

I was inspired by several other knitted klein bottles, which are linked to from this entry.
  • Post a new comment

    Error

    Comments allowed for members only

    Anonymous comments are disabled in this journal

    default userpic

    Your reply will be screened

    Your IP address will be recorded 

  • 12 comments