[FRIAM] Manifold Enthusiasts

[Steven A Smith]

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.


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