On Thursday, March 13, 2003, at 11:31 AM, Eugene Salamin wrote:
Perhaps the universe is connected, not as a cylinder or torus, but as a Moebius band or Klein bottle. Then parity transformations would be physically realizable, by transporting a system around the loop.
Yup. When Jeff Weeks got involved with the "circles in the sky" people, he was delighted to learn that they were already (!) planning to look for circles of temperature correlation running in both directions, seeking orientation-reversing connectivity also. After a talk on this subject at MSRI ('way back before MAP was even launched), the Berkeley grad students had a long discussion about this orientation-reversing question. Recall that the universe is *not* its own mirror image, ie that the laws of physics do give a preferred handedness to space. At the time, we talked to the physics grad students, who assured us that the relevant (non- reflection-invariant) experiment involved particles travelling only a very short distance, but on larger scales there was no known way to detect handedness. So the experiments only indicate that the universe is *locally* orientable... what a relief! --Michael Kleber kleber@brandeis.edu