Nick - I do know that reading my missives *can be* excruciatingly painful but I do trust those without such masochistic tendencies to use their <delete> or <next> buttons freely.
Frank - Sorry I can't commiserate better with your physical pain... but in an ironic reversal of roles, my pain is entirely abstract (existential angst) while yours sounds to be entirely embodied! - Steve On 3/9/19 4:23 PM, Nick Thompson wrote: > > Sorry, everybody, > > > > I am experiencing phantom pain in Steve’s body. > > > > Gotta read these threads more carefully. > > > > Nick > > > > Nicholas S. Thompson > > Emeritus Professor of Psychology and Biology > > Clark University > > http://home.earthlink.net/~nickthompson/naturaldesigns/ > > > > *From:*Friam [mailto:[email protected]] *On Behalf Of *Steven > A Smith > *Sent:* Saturday, March 09, 2019 4:17 PM > *To:* The Friday Morning Applied Complexity Coffee Group > <[email protected]> > *Subject:* Re: [FRIAM] Manifold Enthusiasts > > > > 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
============================================================ 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
