Friam

Re: Manifold Enthusiasts

Classic

List

Threaded

15 messages Options

Carl Tollander

Re: Manifold Enthusiasts

Nick,

This may help with manifold analogies. Or should I phrase that differently....

http://math.ucr.edu/home/baez/rosetta.pdf . See esp table 1, though most of the paper is probably more than you want.

Carl

On Sat, Mar 9, 2019 at 10:20 AM Nick Thompson <[hidden email]> wrote:

Ok, so: consider a corpse. Is the skin of a corpse a manifold? Now. Drop
a shroud over that corpse, is the shroud a manifold? Now, shrink wrap the
corpse and carefully seal the edges. Is it now a closed manifold?

No, huh? Well, ok.

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

-----Original Message-----
From: Friam [mailto:[hidden email]] On Behalf Of
[hidden email]
Sent: Saturday, March 09, 2019 5:10 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] excess meaning alert? (was, Re: are we how we behave?)

Nick et al., "surplus meaning" was the term I was misremembering.

Further replies to Nick's further questions later.

============================================================
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

============================================================
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

Steve Smith

Re: Manifold Enthusiasts

Carl -

This may be a bit more than Nick is prepared for, but it IS an interesting/useful paper and table... and perhaps somewhat relevant to the discussion around embodiment and mathematics and whether understanding through analogy/metaphor grounds out in sensorial experience or in something more platonic like Frank's "Right Triangles" and such.

Nick -

Like all good answers, mine to your shroud/manifold starts with "it depends". You are capturing *part* of the essence of a Manifold with your "shroud" and yet another with your "shrink wrap".

If the "corpse is complete with skin/tissues/etc. and we don't imagine stuffing the shroud or shrink-wrap material through the gastrointestinal track, then the shroud you drape over it provides a continuous surface, but of course it is not closed. When you come to the edge (hemmed or not) you would need to flip over and walk "the other side" or *fall off*. Your "shrink wrap" goes one further and *closes* the shroud. which then makes it a simple manifold topologically equivalent to a sphere (as the decomposing body emits gas, the shrink wrap may inflate to a roughly spherical shape).

There are a number of examples of how your shrink-wrap manifold might have a more complex topology. The aforementioned GI tract represents a hole-through which if shrink-wrapped fully/properly/vigorously (perish the image!) yields a torus (donut). IF your corpse was "shot or stabbed through with holes" (or decomposed to the point of only consisting of bones and minimal connective tissue) it becomes "yet more complex" with "yet more holes". I can't think of a physically possible way said body could become a more complex topology through in principle, one might graft arms and legs (or other appendages) to one another in such a manner as to make a trefoil or more complex knot, but that verges on "just silly". If you read Science Fiction, even someone as respectable as Kurt Vonnegut (often treated more as mainstream literature in spite of his very fanciful assumptions) then you might have encountered an alternative example of such a shrink-wrap-cum-knot that is topologically equivalent to a klein bottle (or yet more interesting/complex) but the narrative leading there would probably seem gratuitously silly.

As for manifolds as used for internal combustion engines, I won't try to reproduce my painful description/speculation about the relation between those and *mathematical manifolds*. Let it suffice to say that the purpose of an intake or exhaust manifold is to route a volume of fuel-air mixture from the carbuerator (possibly more than one in some engines) to the intake ports of each of several cylinders in a smooth and continuous fashion. These are NOT closed surfaces since they are open on the carburator end as well as each of the intake port ends, but their geometric complexity is reminiscent/suggestive of mathematical manifolds. The exhaust manifold(s) on an internal combustion engine do just the opposite, collecting hot exhaust gasses from several cylinders and combining them into a single output to run through things like catalytic converters and mufflers before releasing into the atmosphere to choke pedestrians, the city, and the globe (can you tell I've become an EV snob?).

- Ettiene SHRDLU

Nick,

This may help with manifold analogies. Or should I phrase that differently....
http://math.ucr.edu/home/baez/rosetta.pdf . See esp table 1, though most of the paper is probably more than you want.

Carl

On Sat, Mar 9, 2019 at 10:20 AM Nick Thompson <[hidden email]> wrote:

Ok, so: consider a corpse. Is the skin of a corpse a manifold? Now. Drop
a shroud over that corpse, is the shroud a manifold? Now, shrink wrap the
corpse and carefully seal the edges. Is it now a closed manifold?

No, huh? Well, ok.

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

-----Original Message-----
From: Friam [mailto:[hidden email]] On Behalf Of
[hidden email]
Sent: Saturday, March 09, 2019 5:10 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] excess meaning alert? (was, Re: are we how we behave?)

Nick et al., "surplus meaning" was the term I was misremembering.

Further replies to Nick's further questions later.

============================================================
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
============================================================
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

Nick Thompson

Re: Manifold Enthusiasts

Steve,

All I can say is, for a man in excruciating pain, you sure write good. Your response was just what I needed.

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.

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. 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?

Nick

Nicholas S. Thompson

Emeritus Professor of Psychology and Biology

Clark University

http://home.earthlink.net/~nickthompson/naturaldesigns/

From: Friam [mailto:[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 2:42 PM
To: [hidden email]
Subject: Re: [FRIAM] Manifold Enthusiasts

Carl -

Nick -

Like all good answers, mine to your shroud/manifold starts with "it depends". You are capturing *part* of the essence of a Manifold with your "shroud" and yet another with your "shrink wrap".

- Ettiene SHRDLU

Nick,

This may help with manifold analogies. Or should I phrase that differently....
http://math.ucr.edu/home/baez/rosetta.pdf . See esp table 1, though most of the paper is probably more than you want.

Carl

On Sat, Mar 9, 2019 at 10:20 AM Nick Thompson <[hidden email]> wrote:
Ok, so: consider a corpse. Is the skin of a corpse a manifold? Now. Drop
a shroud over that corpse, is the shroud a manifold? Now, shrink wrap the
corpse and carefully seal the edges. Is it now a closed manifold?

No, huh? Well, ok.

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

-----Original Message-----
From: Friam [mailto:[hidden email]] On Behalf Of
[hidden email]
Sent: Saturday, March 09, 2019 5:10 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] excess meaning alert? (was, Re: are we how we behave?)

Nick et al., "surplus meaning" was the term I was misremembering.

Further replies to Nick's further questions later.

============================================================
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
============================================================
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

Frank Wimberly-2

Re: Manifold Enthusiasts

Steve is in excruciating pain too?

-----------------------------------
Frank Wimberly

My memoir:
https://www.amazon.com/author/frankwimberly

My scientific publications:
https://www.researchgate.net/profile/Frank_Wimberly2

Phone (505) 670-9918

On Sat, Mar 9, 2019, 3:07 PM Nick Thompson <[hidden email]> wrote:

Steve,

All I can say is, for a man in excruciating pain, you sure write good. Your response was just what I needed.

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.

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. 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?

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

From: Friam [mailto:[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 2:42 PM
To: [hidden email]
Subject: Re: [FRIAM] Manifold Enthusiasts

Carl -
This may be a bit more than Nick is prepared for, but it IS an interesting/useful paper and table... and perhaps somewhat relevant to the discussion around embodiment and mathematics and whether understanding through analogy/metaphor grounds out in sensorial experience or in something more platonic like Frank's "Right Triangles" and such.
Nick -
Like all good answers, mine to your shroud/manifold starts with "it depends".   You are capturing *part* of the essence of a Manifold with your "shroud" and yet another with your "shrink wrap".
If the "corpse is complete with skin/tissues/etc. and we don't imagine stuffing the shroud or shrink-wrap material through the gastrointestinal track, then the shroud you drape over it provides a continuous surface, but of course it is not closed.   When you come to the edge (hemmed or not) you would need to flip over and walk "the other side" or *fall off*.   Your "shrink wrap" goes one further and *closes* the shroud. which then makes it a simple manifold topologically equivalent to a sphere (as the decomposing body emits gas, the shrink wrap may inflate to a roughly spherical shape).
There are a number of examples of how your shrink-wrap manifold might have a more complex topology. The aforementioned GI tract represents a hole-through which if shrink-wrapped fully/properly/vigorously (perish the image!) yields a torus (donut). IF your corpse was "shot or stabbed through with holes" (or decomposed to the point of only consisting of bones and minimal connective tissue) it becomes "yet more complex" with "yet more holes". I can't think of a physically possible way said body could become a more complex topology through in principle, one might graft arms and legs (or other appendages) to one another in such a manner as to make a trefoil or more complex knot, but that verges on "just silly". If you read Science Fiction, even someone as respectable as Kurt Vonnegut (often treated more as mainstream literature in spite of his very fanciful assumptions) then you might have encountered an alternative example of such a shrink-wrap-cum-knot that is topologically equivalent to a klein bottle (or yet more interesting/complex) but the narrative leading there would probably seem gratuitously silly.
As for manifolds as used for internal combustion engines, I won't try to reproduce my painful description/speculation about the relation between those and *mathematical manifolds*. Let it suffice to say that the purpose of an intake or exhaust manifold is to route a volume of fuel-air mixture from the carbuerator (possibly more than one in some engines) to the intake ports of each of several cylinders in a smooth and continuous fashion.   These are NOT closed surfaces since they are open on the carburator end as well as each of the intake port ends, but their geometric complexity is reminiscent/suggestive of mathematical manifolds.   The exhaust manifold(s) on an internal combustion engine do just the opposite, collecting hot exhaust gasses from several cylinders and combining them into a single output to run through things like catalytic converters and mufflers before releasing into the atmosphere to choke pedestrians, the city, and the globe (can you tell I've become an EV snob?).
- Ettiene SHRDLU
Nick,

This may help with manifold analogies. Or should I phrase that differently....
http://math.ucr.edu/home/baez/rosetta.pdf . See esp table 1, though most of the paper is probably more than you want.

Carl

On Sat, Mar 9, 2019 at 10:20 AM Nick Thompson <[hidden email]> wrote:
Ok, so: consider a corpse. Is the skin of a corpse a manifold? Now. Drop
a shroud over that corpse, is the shroud a manifold? Now, shrink wrap the
corpse and carefully seal the edges. Is it now a closed manifold?

No, huh? Well, ok.

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

-----Original Message-----
From: Friam [mailto:[hidden email]] On Behalf Of
[hidden email]
Sent: Saturday, March 09, 2019 5:10 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] excess meaning alert? (was, Re: are we how we behave?)

Nick et al., "surplus meaning" was the term I was misremembering.

Further replies to Nick's further questions later.

============================================================
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
============================================================
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

Steve Smith

Re: Manifold Enthusiasts

In reply to this post by Nick Thompson

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.

Nick Thompson

Re: Manifold Enthusiasts

In reply to this post by Frank Wimberly-2

Yes. I don’t know if it’s nerve pain, or not. But come to think of it, isn’t all pain nerfve pain? What does that distinction MEAN?

Nick

Nicholas S. Thompson

Emeritus Professor of Psychology and Biology

Clark University

http://home.earthlink.net/~nickthompson/naturaldesigns/

From: Friam [mailto:[hidden email]] On Behalf Of Frank Wimberly
Sent: Saturday, March 09, 2019 4:01 PM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] Manifold Enthusiasts

Steve is in excruciating pain too?

On Sat, Mar 9, 2019, 3:07 PM Nick Thompson <[hidden email]> wrote:

Steve,

All I can say is, for a man in excruciating pain, you sure write good. Your response was just what I needed.

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.

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. 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?

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

From: Friam [mailto:[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 2:42 PM
To: [hidden email]
Subject: Re: [FRIAM] Manifold Enthusiasts

Carl -
This may be a bit more than Nick is prepared for, but it IS an interesting/useful paper and table... and perhaps somewhat relevant to the discussion around embodiment and mathematics and whether understanding through analogy/metaphor grounds out in sensorial experience or in something more platonic like Frank's "Right Triangles" and such.
Nick -
Like all good answers, mine to your shroud/manifold starts with "it depends".   You are capturing *part* of the essence of a Manifold with your "shroud" and yet another with your "shrink wrap".
If the "corpse is complete with skin/tissues/etc. and we don't imagine stuffing the shroud or shrink-wrap material through the gastrointestinal track, then the shroud you drape over it provides a continuous surface, but of course it is not closed.   When you come to the edge (hemmed or not) you would need to flip over and walk "the other side" or *fall off*.   Your "shrink wrap" goes one further and *closes* the shroud. which then makes it a simple manifold topologically equivalent to a sphere (as the decomposing body emits gas, the shrink wrap may inflate to a roughly spherical shape).
There are a number of examples of how your shrink-wrap manifold might have a more complex topology. The aforementioned GI tract represents a hole-through which if shrink-wrapped fully/properly/vigorously (perish the image!) yields a torus (donut). IF your corpse was "shot or stabbed through with holes" (or decomposed to the point of only consisting of bones and minimal connective tissue) it becomes "yet more complex" with "yet more holes". I can't think of a physically possible way said body could become a more complex topology through in principle, one might graft arms and legs (or other appendages) to one another in such a manner as to make a trefoil or more complex knot, but that verges on "just silly". If you read Science Fiction, even someone as respectable as Kurt Vonnegut (often treated more as mainstream literature in spite of his very fanciful assumptions) then you might have encountered an alternative example of such a shrink-wrap-cum-knot that is topologically equivalent to a klein bottle (or yet more interesting/complex) but the narrative leading there would probably seem gratuitously silly.
As for manifolds as used for internal combustion engines, I won't try to reproduce my painful description/speculation about the relation between those and *mathematical manifolds*. Let it suffice to say that the purpose of an intake or exhaust manifold is to route a volume of fuel-air mixture from the carbuerator (possibly more than one in some engines) to the intake ports of each of several cylinders in a smooth and continuous fashion.   These are NOT closed surfaces since they are open on the carburator end as well as each of the intake port ends, but their geometric complexity is reminiscent/suggestive of mathematical manifolds.   The exhaust manifold(s) on an internal combustion engine do just the opposite, collecting hot exhaust gasses from several cylinders and combining them into a single output to run through things like catalytic converters and mufflers before releasing into the atmosphere to choke pedestrians, the city, and the globe (can you tell I've become an EV snob?).
- Ettiene SHRDLU
Nick,

This may help with manifold analogies. Or should I phrase that differently....
http://math.ucr.edu/home/baez/rosetta.pdf . See esp table 1, though most of the paper is probably more than you want.

Carl

On Sat, Mar 9, 2019 at 10:20 AM Nick Thompson <[hidden email]> wrote:
Ok, so: consider a corpse. Is the skin of a corpse a manifold? Now. Drop
a shroud over that corpse, is the shroud a manifold? Now, shrink wrap the
corpse and carefully seal the edges. Is it now a closed manifold?

No, huh? Well, ok.

Nick

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/

-----Original Message-----
From: Friam [mailto:[hidden email]] On Behalf Of
[hidden email]
Sent: Saturday, March 09, 2019 5:10 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] excess meaning alert? (was, Re: are we how we behave?)

Nick et al., "surplus meaning" was the term I was misremembering.

Further replies to Nick's further questions later.

============================================================
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
============================================================
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

Nick Thompson

Re: Manifold Enthusiasts

In reply to this post by Steve Smith

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:[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 4:17 PM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
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!

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:

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.

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?

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

- 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.

Steve Smith

Re: Manifold Enthusiasts

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 [[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 4:17 PM
To: The Friday Morning Applied Complexity Coffee Group [hidden email]
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

Frank Wimberly-2

Re: Manifold Enthusiasts

My pain was unbearable until I saw my neurologist who prescribed Gabapentin and then my primary care physician added Cymbalta. Both relieve nerve pain. My left arm is partially paralyzed. I can't raise it above my chest. All of this is because of an impingement of a nerve on C6, left side. It was amazing how the neurologist diagnosed that. It involved tiny needles and mild shocks.

On Sat, Mar 9, 2019, 4:28 PM Steven A Smith <[hidden email]> wrote:

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 [[hidden email]] On Behalf Of Steven A Smith
Sent: Saturday, March 09, 2019 4:17 PM
To: The Friday Morning Applied Complexity Coffee Group [hidden email]
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

gepr

Re: Manifold Enthusiasts

In reply to this post by Steve Smith

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.

--
☣ uǝlƃ
============================================================
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

uǝʃƃ ⊥ glen

Steve Smith

Re: Manifold Enthusiasts

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.
>

Frank Wimberly-2

Re: Manifold Enthusiasts

Perhaps annoying clarification. The word closure has two important meanings.

In a topological space a set is closed if it is the complement of an open set.

In the more concrete case of Rn, which we're talking about, a set is closed if it contains it's limit points (boundary).

The other use of closure is the one Glen(?) mentioned. The sum of two elements in the set is in the set and the same for other operations.

Barry, help!

Frank

Steve Smith

Re: Manifold Enthusiasts

Frank -

> Perhaps annoying clarification. The word closure has two important
> meanings.
>
> In a topological space a set is closed if it is the complement of an
> open set.
>
> In the more concrete case of Rn, which we're talking about, a set is
> closed if it contains it's limit points (boundary).
>
> The other use of closure is the one Glen(?) mentioned. The sum of two
> elements in the set is in the set and the same for other operations.
>
> Barry, help!

Yet more good elaboration/correction. I think for Nick's benefit, the
second one is *most* relevant, comes closest to your own original
common-sense idea of "walking around on a surface without falling off"?
Cannot 2 be derived from 3 with the right assumptions?

Also, I hope your nerve pain is subsiding... I guess this bodes poorly
for your tennis game? At least overhead slams and solid serves, unless
it is your off-hand that is afflicted?

I'm unfamiliar with the pain meds you mentioned (really all outside of
the basic salisylic acid, acetomenophen, and ibuprofen.   I was hoping
your report of the two med's efficacy would lead to some elaboration on
the question of change of behaviour without change of brain state?
Where do these meds act? In the brain tissue itself? On the pinched
nerve? Somewhere in between?   And does that just beg the question over
into whether the spinal chord and the entire nervous system are
nominally "part of the brain"?

- Steve

============================================================
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

Frank Wimberly-2

Re: Manifold Enthusiasts

Pregabalin reduces neuronal calcium currents by binding to the α2δ subunit of calcium channels, and this particular mechanism may be responsible for effects in neuropathic pain, anxiety, and other painsyndromes. The exact mechanism of analgesic effect for gabapentin has not been defined.

Duloxetine inhibits the reuptake of serotonin and norepinephrine (NE) in the central nervous system. Duloxetine increases dopamine (DA) specifically in the prefrontal cortex, where there are few DA reuptake pumps, via the inhibition of NE reuptake pumps (NET), which is believed to mediate reuptake of DA and NE.^[53] Duloxetine has no significant affinity for dopaminergic, cholinergic, histaminergic, opioid, glutamate, and GABA reuptake transporters, however, and can therefore be considered to be a selective reuptake inhibitor at the 5-HT and NE transporters. Duloxetine undergoes extensive metabolism, but the major circulating metabolites do not contribute significantly to the pharmacologic activity.^[54]^[55

On Mon, Mar 11, 2019, 11:57 AM Steven A Smith <[hidden email]> wrote:

Frank -
> Perhaps annoying clarification. The word closure has two important
> meanings.
>
> In a topological space a set is closed if it is the complement of an
> open set.
>
> In the more concrete case of Rn, which we're talking about, a set is
> closed if it contains it's limit points (boundary).
>
> The other use of closure is the one Glen(?) mentioned. The sum of two
> elements in the set is in the set and the same for other operations.
>
> Barry, help!

Yet more good elaboration/correction. I think for Nick's benefit, the
second one is *most* relevant, comes closest to your own original
common-sense idea of "walking around on a surface without falling off"?
Cannot 2 be derived from 3 with the right assumptions?

Also, I hope your nerve pain is subsiding... I guess this bodes poorly
for your tennis game? At least overhead slams and solid serves, unless
it is your off-hand that is afflicted?

I'm unfamiliar with the pain meds you mentioned (really all outside of
the basic salisylic acid, acetomenophen, and ibuprofen.   I was hoping
your report of the two med's efficacy would lead to some elaboration on
the question of change of behaviour without change of brain state?
Where do these meds act? In the brain tissue itself? On the pinched
nerve? Somewhere in between?   And does that just beg the question over
into whether the spinal chord and the entire nervous system are
nominally "part of the brain"?

- Steve

============================================================
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

Frank Wimberly-2

Re: Manifold Enthusiasts

Some people have suggested I meditate with the goal of perceiving the pain as being separate from me. Or something like that.

The Duloxetine (Cymbalta) has certainly changed my behavior. I am much more patient, considerate, loving, empathic, etc. Some might think I am more like the ideals of Abrahamic faith traditions which raises the question of whether you can be a better person by using meds.

Frank

On Mon, Mar 11, 2019, 12:08 PM Frank Wimberly <[hidden email]> wrote:

Pregabalin reduces neuronal calcium currents by binding to the α2δ subunit of calcium channels, and this particular mechanism may be responsible for effects in neuropathic pain, anxiety, and other painsyndromes. The exact mechanism of analgesic effect for gabapentin has not been defined.

Duloxetine inhibits the reuptake of serotonin and norepinephrine (NE) in the central nervous system. Duloxetine increases dopamine (DA) specifically in the prefrontal cortex, where there are few DA reuptake pumps, via the inhibition of NE reuptake pumps (NET), which is believed to mediate reuptake of DA and NE.^[53] Duloxetine has no significant affinity for dopaminergic, cholinergic, histaminergic, opioid, glutamate, and GABA reuptake transporters, however, and can therefore be considered to be a selective reuptake inhibitor at the 5-HT and NE transporters. Duloxetine undergoes extensive metabolism, but the major circulating metabolites do not contribute significantly to the pharmacologic activity.^[54]^[55

-----------------------------------
Frank Wimberly

My memoir:
https://www.amazon.com/author/frankwimberly

My scientific publications:
https://www.researchgate.net/profile/Frank_Wimberly2

Phone (505) 670-9918

On Mon, Mar 11, 2019, 11:57 AM Steven A Smith <[hidden email]> wrote:
Frank -
> Perhaps annoying clarification. The word closure has two important
> meanings.
>
> In a topological space a set is closed if it is the complement of an
> open set.
>
> In the more concrete case of Rn, which we're talking about, a set is
> closed if it contains it's limit points (boundary).
>
> The other use of closure is the one Glen(?) mentioned. The sum of two
> elements in the set is in the set and the same for other operations.
>
> Barry, help!

Yet more good elaboration/correction. I think for Nick's benefit, the
second one is *most* relevant, comes closest to your own original
common-sense idea of "walking around on a surface without falling off"?
Cannot 2 be derived from 3 with the right assumptions?

Also, I hope your nerve pain is subsiding... I guess this bodes poorly
for your tennis game? At least overhead slams and solid serves, unless
it is your off-hand that is afflicted?

I'm unfamiliar with the pain meds you mentioned (really all outside of
the basic salisylic acid, acetomenophen, and ibuprofen.   I was hoping
your report of the two med's efficacy would lead to some elaboration on
the question of change of behaviour without change of brain state?
Where do these meds act? In the brain tissue itself? On the pinched
nerve? Somewhere in between?   And does that just beg the question over
into whether the spinal chord and the entire nervous system are
nominally "part of the brain"?

- Steve

============================================================
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