From: "W. Dale Hall" Newsgroups: sci.math Subject: Re: Klein bottle question (a much belated correction) Date: 21 Oct 1998 02:45:15 PDT Dr. Michael Albert wrote: > Here is a simple non-orientable (compact) three space. Take a solid ball > of radius R. As a point exists through the surface at radius R, > let it re-enter at the antipodal point, ie., the point at distance R > in the oposite direction (ie., identify antipodal points on the surface-- > to get a feel for how this game is played, think about "PacMan"--you > play on a square but if you leave the top you enter the bottom, and > if you leave the right you enter on the left--you essentially play > on the surface of a doughnut). This space in non-orientable. If I have > a pair of gloves and I hold onto one and my brother takes the other > on a trip out "through" the surface and back in the other way, when > he returns I might have two right-hand gloves. He will also have > his heart on the opposite "side" of his body, and be reversed. > At the risk of closing the barn door after the horses have escaped to the same side,I must point out that the space described here (most commonly referred to as RP^3, "real projective 3-space") is most assuredly orientable. Other readers of this group may already be acquainted with RP^3 from its alternate description as SO(3), the group of (ordinary) rotations of R^3. Once you know that identification (which is fairly simple, it's just too late here now for me to spend the time to show how it's done), orientability is an easy consequence (in fact, one shows that translations produce a trivialization of its tangent bundle, from which orientability follows). OK, you guys can continue. > Best wishes, > Mike And to you as well. Dale.