Steve Witham <sw(a)tiac.net> wrote:
> What does one call a tube looped around as if to make a torus, but
> connected clockwise-to-counterclockwise, as in this sewing pattern?
> z y>--C-->z y
> +---------+
> x|w>--A-->x|w
> v|^ v|^
> ||| |||
> B|D B|D
> ||| |||
> v|^ v|^
> y|z<--C--<y|z
> +---------+
> w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of
Fortunatus.
If you start with a square, and attach the top to the bottom without
any twists and attach the left side to the right side without any
twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to
the right with a twist, i.e. reverse the direction, you get a Klein
Bottle.
If you reverse the directions of both connections, you get a Purse
of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles
have the same sum of angles, and circles have the same ratio of
circumference to diameter, as in a flat plane). Their embeddings
in 3-space, however, have extrinsic curvature. And, except for the
torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable,
i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions,
i.e. gluing opposite sides of a cube together, flipping none, some, or
all of them. It would be interesting if one of those was the topology
of our universe. I've read that some astronomers have done searches
for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a
straight line, you'll return to Earth swapped left-to-right. That
might mean that you'd be converted to anti-matter. Or it might just
mean that pine trees would smell like lemons and you couldn't digest
the food. It would be a good defense against viruses, however.
