I have a small needlepoint project that I never finished. It’s a wreath, front and back. To finish it, I would sew it so that the stitched sides are together, leaving a small section unstitched, then turn it inside out, stuff it, and sew the final section.
But every time I think of trying to turn a torus inside out, I am stymied. If it’s possible, where do I join the pieces? Or is it impossible?!
It is possible, if the material it’s made from is flexible enough, and an inside-out torus is also a torus. But the catch is that the hole isn’t the same hole any more.
Your wreath is probably shaped more or less like a long tube, wrapped around a circle, and with the ends connected, right? Well, now picture that that long tube is like a sock, and turn it halfway inside out like you would a sock, and connect the ends together.
If that’s not clear enough, I can try to draw a picture.
Here’s an animation. Note that the orientation of the stripes is different at the end.
Those are cool! I might have to do something similar, although I won’t be able to do it with this project - nowhere near stretchy enough!
The way I’vehad this explained is that the hole that was the “doughnut hole” of the inner tube becomes the space for the air inside the inner tube, and where the air was, becomes the hole.
The problem is, you violate the laws of topology when you turn a torus
inside out. See here for details.
(starts halfway down the page.)
You also violate the laws of topology when you make the torus to begin with. We’re talking about sewing, here, not topology.