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<p>Nick -</p>
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<p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">All
I can say is, for a man in excruciating pain, you sure write
good. Your response was just what I needed. <o:p></o:p></span></p>
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<p>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!<br>
</p>
<p>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.<br>
</p>
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<p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">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 <i>covers</i> 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. </span></p>
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<p>Hmm... I don't know if I can answer this fully/properly but as
usual, I'll give it a go:</p>
<p>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 <i>shroud</i>
that *are* relevant is *continuity* with a surplus but not always
irrelevant meaning of *smooth*. In another (sub?)thread about <i>Convex
Hulls</i>, we encounter inferring a continuous surface *from* a
finite point-set. A physical analogy for algorithmically
building that <i>Convex Hull</i> 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. <br>
</p>
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<p class="MsoNormal"><span
style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p>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. <br>
</span></p>
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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).<br>
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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? </span></p>
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<p>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.</p>
<p>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.</p>
<p>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.) <br>
</p>
<p>If you haven't fallen far enough down a (fractal dimensioned?)
rabbit hole then I offer you:</p>
<blockquote>
<p><a class="moz-txt-link-freetext" href="https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on">https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on</a></p>
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<p>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?</p>
<p>- Steve</p>
<p>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.<br>
</p>
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