james@umcp-cs.UUCP (09/22/83)
Today we got started on topological projections spaces, which are indentification spaces of friendly things like spheres, intervals, and mobius strips. An identification space is made by taking a nice, friendly old thing like, for instance, an interval, and mapping it by identifying certain points of it to be the same (in the new space). The simplest example would be identifying the two endpoints of a closed interval...then you get the equivalent of a circle! That wasn't too exciting, but there are some pretty nice ones, like (and I'll just give a loose, cut/sew description): 1) Take a unit disc (pancake) and identify the boundary (perimeter), thus drawing all the edge up into a single point. 2) Take a circle, and identify any two diametrically opposite points as one. 3) Take a ball, and identify any two diametrically opposite points on the surface as one. 4) A mobius strip's edge is a closed curve. A disc's edge is a closed curve. Sew these two edges together. 5) Sew together the edges of two modius strips. Now, what do you think you get when try the above 5? (1) produces a sphere. (2), (3), and (4) produce things which aren't all drawable, definitely not buildable, not easy to visualize, and nameless (as far as I know). I think (5) gives you a Klein bottle, but I haven't proved it yet. Mathematics sure is fun! --Jim O'Toole
ljdickey@watmath.UUCP (09/25/83)
i recognized two of the things you mentioned in your list of things. Number (2) is a projective line, and (3) is a projective plane. These things have been around for a long time, long before topology came into vogue. > From james@umcp-cs.uucp > Newsgroups: net.math > Subject: Projection spaces (topology) > Organization: Univ. of Maryland, Computer Science Dept. > > Today we got started on topological projections spaces, which > are indentification spaces of friendly things like spheres, > intervals, and mobius strips. An identification space is made > by taking a nice, friendly old thing like, for instance, an > interval, and mapping it by identifying certain points of it > to be the same (in the new space). The simplest example would > be identifying the two endpoints of a closed interval...then you > get the equivalent of a circle! That wasn't too exciting, but > there are some pretty nice ones, like (and I'll just give a > loose, cut/sew description): > > 1) Take a unit disc (pancake) and identify the boundary > (perimeter), thus drawing all the edge up into a single point. > > 2) Take a circle, and identify any two diametrically opposite > points as one. > > 3) Take a ball, and identify any two diametrically opposite > points on the surface as one. > > 4) A mobius strip's edge is a closed curve. A disc's edge is > a closed curve. Sew these two edges together. > > 5) Sew together the edges of two modius strips. > > Now, what do you think you get when try the above 5? (1) produces a > sphere. (2), (3), and (4) produce things which aren't all drawable, > definitely not buildable, not easy to visualize, and nameless (as far > as I know). I think (5) gives you a Klein bottle, but I haven't proved > it yet. > > Mathematics sure is fun! > > --Jim O'Toole -- Lee Dickey, University of Waterloo. (ljdickey@watmath.UUCP) ...!allegra!watmath!ljdickey ...!ucbvax/decvax!watmath!ljdickey
levy@princeton.UUCP (09/28/83)
Identifying each two diametrically opposite points of the boundary of a closed disc gives the projective plane. This is easily seen by considering the disc to be the top half of the two-dimensional sphere; the projective plane is by definition what you get when you identify opposite points in the sphere. You also get the projective plane when you sew together a Mobius strip and a closed disc along the boundary. You do indeed get a Klein bottle when you sew together two Mobius strips (just think of a "symmetric" Klein bottle and split it up into two equal parts). As for identifying the opposite points on the surface of a closed three- dimensional ball, you get... the three-dimensional projective space! It's like the first paragraph above: the three-ball is the "top" half of the three-sphere, and identifying opposite points on the three-sphere gives you (by definition) the three-dimensional projective space. This space is encountered in practice as the space of all possible orientations you can give to a set of three coordinate axes in Euclidean 3-space; remember the Euler angles? Always glad to help, -- Silvio Levy