Is there a non-Archimedean way of measuring extrinsic curvature that allows infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)? It should be something like a combination of differential geometry and distribution theory. Jim Propp On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@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.
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