Having identified deficiencies in popular math writing, the obvious next step is to do better. If the people doing the writing have better examples to work from, maybe the result will be more accurate. Anyone care to tackle Dan's comment
(such as assuming that n-manifolds are subsets of n+1 space)
and come up with an accurate explanation of manifold, understandable at the level of a college graduate who majored in, say, chemistry? Rich rcs@cs.arizona.edu
The surface of a donut is a torus embedded in 3-space, but chances are no one thinks of the playing field of the videogame 'Asteroids' as being embedded in 3-space even though it behaves identically--it's just a 2D space where objects that go off one side show up on the other. Manifolds are like 'Asteroids'--they don't need embedding in anything else to have properties different from Euclidean space. Rich wrote: Anyone care to tackle Dan's comment
(such as assuming that n-manifolds are subsets of n+1 space)
and come up with an accurate explanation of manifold, understandable at the level of a college graduate who majored in, say, chemistry? -- Mike Stay staym@clear.net.nz http://www.xaim.com/staym
1. An n-dimensional manifold is a topological space whose points have neighborhoods homeomorphic to an n-dimensional ball. That's too technical, so I'll say the same thing without the jargon. 2. An n-dimensional manifold is such that every point has a region around it like an n-dimensional ball. Dropping the jargon makes it possible to equivocate about what "region around it" and "like ball" mean. Understanding the idea of a manifold requires at least a few examples, and the torus as a re-entrant rectangle is a good one. It is necessary to assert, though perhaps not to prove, that the neighborhoods of the edge points and the corners behave just like the neighborhoods in the middle of the rectangle. The Mobius strip provided another easily understood example, and identifying opposite points of the one edge to get a Klein bottle provides an example with enough difficulty in visualization to show that the idea is non-trivial.
Does anyone have some thoughts about generalizations of hex to various manifolds? On the torus, it seems to work out fairly nicely, although I haven't actually played any games with anyone. Take any tiling of the torus by hexagons (or for that matter, any tiling by polygons such that 3 polygons meet at any vertex. One reasonable method is to use a standard hex board, and "wrap it around" into a torus. Partition the possible "slopes" in first homology H_1(T^2) into two sets, for instance by using some coordinate system at an irrational slope and assigning the first and third quadrants to white, the second and fourth quadrants to black. If the torus is a wrapped hex board, a more direct rule is to say a slope is black if there is a line segment of that slope connecting the two black sides---you just need to be a little careful about the exact idetification when it's wrapped <--> the exact rule for acceptable slopes. Players alternate putting black and white stones on the hexagons, until one of them has a closed path in a homology class of their color. It's a nice little puzzle to extend the usual analysis of hex, to prove that if the board is totally filled with black and white stones, exactly one player has a curve in a homology class of the designated color. I think Dan Asimov brought up with me the question of doing it on higher-dimensional disks, long ago ... there were good definitions for even-dimensional disks, which I won't try to explain just here and now, since the boundary conditions make it a little more complicated than doing it on a closed manifold. What other manifolds have reasonable rules for hex generalizations? I think other surfaces should have good rules, but I have a hard trouble making simple definitions for these rules. For CP^2, a good rule seems to be to declare the winner to be anyone such that H^2(their colors) is non-zero. On the other hand, aI don't think there's any rule analogous to standard hex for T^3 ... anyway, I threw this out as a question in the topology class that I'm teaching, and I'd welcome any thoughts or insights. Bill Thurston
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