Glen - Well corrected. I think it IS important to expand the common-sense notion of "closure" here, to the more general ideal of "closure under an operation" such as "closure under taking a step". I think the basic idea of "having no holes to fall through" was (mostly) good enough for everyday thinking.
- Steve On 3/11/19 10:42 AM, uǝlƃ ☣ wrote: > I *almost* violated my standing directive to unplug on the weekends because > of all the little beeps and buzzes from my phone. It's fantastic to see so > much traffic. > > I only have one comment on "closure", as used here. I think it's a bit > misleading to talk about turning a shroud into a balloon/sphere as "closing" > it. I think the only closure needed for a manifold is closure under > particular operations (like the normal ones, +, -, *, /), where the point > being operated on and the result are both *inside* the space. So, while it's > reasonable to think of a coastline and (imagine walking along the beach), if > "step" is the operation, then you're on the coast before you step and still > on the coast after you step. While it makes intuitive sense to loop the > coast around like Steve suggests to ensure that you're always still on the > coast after taking a step, a manifold need not be a cycle in that way. You > might have, say, an infinitely long coastline and as long as you're stepping > along the coast, you're still on it, it's closed under stepping, but it's not > a cycle. > > To be clear, I don't think anyone said anything wrong. I just wanted to > distinguish cycle from closure. > > > On 3/9/19 3:17 PM, Steven A Smith wrote: >> Nick - >> >>> All I can say is, for a man in excruciating pain, you sure write good. >>> Your response was just what I needed. >>> >> Something got crossed in the e-mails. *I'*m not in excruciating pain... >> that would be (only/mainly/specifically) Frank, I think. But thanks for the >> thought! >> >> Any excruciating pain I might be in would be more like existential angst or >> something... but even that I have dulled with a Saturday afternoon Spring >> sunshine, an a cocktail of loud rock music, cynicism, anecdotal nostalgia, >> and over-intellectualism. Oh and the paint fumes (latex only) I've been >> huffing while doing some touch-up/finish work in my sunroom on a sunny day >> is also a good dulling agent. >> >>> Now, when I think of a manifold, my leetle former-english-major brain >>> thinks shroud, and the major thing about a shroud is that it /covers/ >>> something. Now I suspect that this is an example of irrelevant surplus >>> meaning to a mathematician, right? A mathematician doesn’t give a fig for >>> the corpse, only for the properties of the shroud. But is there a >>> mathematics of the relation between the shroud and the corpse? And what is >>> THAT called. >>> >> Hmm... I don't know if I can answer this fully/properly but as usual, I'll >> give it a go: >> >> I think the Baez paper Carl linked to has some help for this in that. I >> just tripped over an elaboration of a topological boundary/graph duality >> which might have been in that paper. But to be as direct as I can for >> you, I think the two properties of /shroud/ that *are* relevant is >> *continuity* with a surplus but not always irrelevant meaning of *smooth*. >> In another (sub?)thread about /Convex Hulls/, we encounter inferring a >> continuous surface *from* a finite point-set. A physical analogy for >> algorithmically building that /Convex Hull/ from a point set would be to >> create a physical model of the points and then drape or pull or shrink a >> continuous surface (shroud) over it. Manifolds needn't be smooth >> (differentiable) at every point, but the ones we usually think of generally >> are. >> >>> So, imagine the coast of Maine with all its bays, rivers and fjords. >>> Imagine now a map of infinite resolution of that coastline, etched in ink. >>> I assume that this is a manifold of sorts. >>> >> In the abstract, I think that coastline (projected onto a plane) IS a 1d >> fractal surface (line). To become a manifold, it needs to be *closed* which >> would imply continuing on around the entire mainland of the western >> hemisphere (unless we artificially use the non-ocean political boundaries of >> Maine to "close" it). >>> Now gradually back off the resolution of the map until you get the kind of >>> coastline map you would get if you stopped at the Maine Turnpike booth on >>> your way into the state and picked a tourist brochure. Now that also is a >>> manifold of sorts, right? In my example, both are representations of the >>> coastline, but I take it that in the mathematical conception the potential >>> representational function of a “manifold” is not of interest? >>> >> I think the "smoothing" caused by rendering the coastline in ink the width >> of the nib on your pen (or the 300dpi printer you are using?) yields a >> continuous (1d) surface (line) which is also smooth (differentiable at all >> points)... if you *close* it (say, take the coastline of an island or the >> entire continental western hemisphere (ignoring the penetration of the >> panama canal and excluding all of the other canals between bodies of water, >> etc. then you DO have a 1D (and smooth!) manifold. >> >> If you zoom out and take the surface of the earth (crust, bodies of liquid >> water, etc), then you have another manifold which is topologically a >> "sphere" until you include any and all natural bridges, arches, caves with >> multiple openings. If you "shrink wrap" it (cuz I know you want to) it >> becomes smooth down to the dimension of say "a neutrino". To a neutrino, >> however, the earth is just a dense "vapor" that it can pass right through >> with very little chance of intersection... though a "neutrino proof" shroud >> (made of neutrino-onium?) would not allow it I suppose. >> >> This may be one of the many places Frank (and Plato) and I (and Aristotle) >> might diverge... while I enjoy thinking about manifolds in the abstract, >> I don't think they have any "reality" beyond being a useful >> archetype/abstraction for the myriad physically instantiated objects I can >> interact with. Of course, the earth is too large for me to apprehend >> directly except maybe by standing way back and seeing how it reflects the >> sunlight or maybe dropping into such a deep and perceptive meditative state >> that I can experience directly the gravitational pull on every one of the >> molecules in my body by every molecule in the earth (though that is probably >> not only absurd, but also physically out of scale... meaning that >> body-as-collection-of-atoms might not represent my own body and that of the >> earth and I think the Schroedinger equation for the system circumscribing my >> body and the earth is a tad too complex to begin to solve any other way than >> just "exisiting" as I do at this >> location at this time on this earth.) >> >> If you haven't fallen far enough down a (fractal dimensioned?) rabbit hole >> then I offer you: >> >> >> https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on >> >> Which to my reading does not answer the question, but kicks the (imperfectly >> formed, partially corroded, etc.) can on down the (not quite perfectly >> straight/smooth) road, but DOES provide some more arcane verbage to decorate >> any attempt to explain it more deeply? >> >> - Steve >> >> PS. To Frank or anyone else here with a more acutely mathematical >> mind/practice, I may have fumbled some details here... feel free to correct >> them if it helps. > ============================================================ 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
