Holy Moly, Lee. With text books like that, I coulda beena mathematician after all!
Nick Nicholas S. Thompson Emeritus Professor of Psychology and Biology Clark University http://home.earthlink.net/~nickthompson/naturaldesigns/ -----Original Message----- From: Friam [mailto:friam-boun...@redfish.com] On Behalf Of lrudo...@meganet.net Sent: Monday, February 04, 2019 9:33 AM To: The Friday Morning Applied Complexity Coffee Group <friam@redfish.com> Subject: Re: [FRIAM] Photos of popped balloon > I think they were cylinders, not spheres, so there were two holes. > This is where we start talking about homology groups. We don't absolutely *have* to. The theories of Riemann surfaces and algebraic functions got pretty far just having the (proto-homological, but very ungroupy) notions of "simple connectivity" vs. "multiple connectivity". [For those readers, possibly consisting of Nick alone, here's what that means. Suppose you produce a thin sheet of copper by electroplating onto some or all of the surface of a solid piece of wax that you then melt away. For instance, you get a cylindrical surface if you start with a solid wax cylinder and only electroplate onto its lateral surface, leaving the round disks at its two ends unplated; and it will be possible to melt the wax away without cutting a hole in the copper. On the other hand, you get a spherical surface if you start with a solid round ball of wax and electroplate onto its entire surface (let's not worry about how you do that...); in that case, you'll have to puncture the sphere (maybe cutting out a little disk around the south pole) to let the melted wax escape. Just make one hole! (And don't worry about possible difficulties draining out all the wax, okay?) For a third example, start with a piece of wax in the shape of a donut (a so-called "solid torus" or, in a charmingly antique idiom, an "anchor ring"); the resulting copper surface is a "torus" plain and simple. Again, a single hole will suffice to drain the (idealized) wax; again, don't make any others. Now take your pair of metal shears and start cutting somewhere on an edge of the copper sheet. In the cylinder example, you have two edges, each of them a circle at one end of the cylinder. In the sphere and torus examples, you have a single (circular) edge, around the hole you drained the wax through. It is a fact (which I hope you can imagine visually with no trouble, because all this electroplating would be expensive and difficult) that no matter how you the sphere-with-one-hole with your shears, starting and ending at edge points, you will cut the copper into two pieces. It is also a fact that on both the cylinder and the once-punctured (i.e., drained) torus, there are *some* ways to cut from an edge point to another edge point that do *not* cut the copper into two pieces. (On the cylinder, you have to start somewhere on one of the two circular edges and end somewhere on the other: when you've done that you can unroll the cylinder flat onto a table. On the once-punctured torus, there are many very different ways to make such a "non-separating" cut.) Riemann and Co. described this qualitative distinction between the surface of a sphere and the (lateral) surface of a cylinder (and torus, etc., etc.) by calling a sphere "simply connected" and the others "multiply connected". "Simple" here is like 0, and "multiple" like "a strictly positive integer", which began the process of refining the qualitative distinction into a quantitative distinction. Very soon the quantitative distinction was refined much more by making the various positive integers distinct (so the "cut number" of the sphere is 0, the cut number of the cylinder is 1, and--as it turns out--the cut number of the torus is 2). Rather later, this quantitative distinction became more refined. Eventually it became *so* refined that "homology groups" appeared as the best way to describe the refinements. It is quite possible that the mathematical physicist John Baez, Joan's younger cousin, wrote all this stuff up very clearly 15 or 20 years ago. If so, it would be findable with Google.] ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives back to 2003: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives back to 2003: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
