Re: [FRIAM] Fractals/Chaos/Manifolds

[Robert J. Cordingley]

In one German dictionary I found /mannigfaltigkeit/ translates to 
/variousness/ which seems pretty obtuse but indicates it may have less 
to do with the original entymology of /manifold/ 
(https://en.wiktionary.org/wiki/manifold Entymology 1). Per Dean's pdf, 
perhaps it's a made up usage inspired by Gauss/Riemann who had a concept 
about topological space but needed a word for it? That is to say 
'manifold' (in English) was a neologism in its time based on an 
appearance of the German word?


Robert C

On 3/1/17 2:21 PM, lrudo...@meganet.net wrote:

The word, as a term of Mathematical English (which is of course quite a 
distinct dialect of
English) is a calque of the Mathematical German word "Mannigfaltigkeit".  
Franklin Becher, in
the first paragraph of the lead article in the October, 1896, issue of the 
American
Mathematical Monthly, "MATHEMATICAL INFINITY AND THE DIFFERENTIAL", doesn't 
quite use the word
yet, but makes its origin clear enough.

---begin---
Mathematics, as defined by the great mathematician, Benjamin Pierce, is the 
science which
draws necessary conclusions. In its broadest sense, it deals with conceptions 
from which
necessary conclusions are drawn. A mathematical conception is any conception 
which, by means
of a finite number of specified elements, is precisely and completely defined 
and determined.
To denote the dependence of a mathematical conception on its elements, the word
"manifoldness," introduced by Riemann, has been recently adopted.
--end--

In his article on the foundations of geometry, available at
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html ,
Riemann distinguished two types of "Mannigfaltigkeit", the discrete and the 
continuous:

---begin---
cat

Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet, 
der
verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen 
Bestimmungsweisen von einer zu
einer andern ein stetiger Uebergang stattfindet oder nicht, bilden sie eine 
stetige oder
discrete Mannigfaltigkeit;

| Google Translate >

Size terms are only possible where there is a general concept, which allows 
different modes of
determination. According as, according to these modes of determination from one 
to another, a
continuous transition takes place or not, they form a continuous or discrete 
manifoldness;
---end---

In Riemann's (eventual) context, those sentences would be understood now (at 
least by
topologists of my sort, which is to say, geometric topologists, cf.
http://front.math.ucdavis.edu/math.GT) as sketching the modern concept of a 
(topological or
differentiable) manifold as a "mathematical conception" that can "precisely and 
completely
defined and determined" by a collection [called an "atlas"] of "modes of 
determination"
[called "charts"] among (some pairs of) which there are also given "continuous" 
(i.e.,
topological) or perhaps *smooth* (i.e., differentiable) coordinate changes.

I dispute, incidentally, the claim that 3-manifolds are too hard to understand; 
they're *just*
at the edge of that, but not over it (whereas 4- and higher dimensional 
manifolds are
DEFINITELY over that edge, in various well-defined mathematical ways; e.g., the 
problem of
determining whether two explicitly-given n-manifolds, n greater than 3, has 
been known for a
long time to be computationally intractable [you can embed the word problem for 
groups into
the manifold classification problem for n greater than 3], and much more 
recently has been
shown to be doable in dimension 3).

The French word for (something a little more general than a) manifold is 
"variet", by
the way; same sort of reason, I assume.




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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove




--
Cirrillian
Web Design & Development
Santa Fe, NM
http://cirrillian.com
281-989-6272 (cell)
Member Design Corps of Santa Fe


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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove

Re: [FRIAM] Fractals/Chaos/Manifolds

[Steven A Smith]

Lee -

Great bit of detective work there...


"Mannigfaltigkeit"

   manig -> many

   faltig -> wrinkle or fold

   kelt ->  having the utility of or "ness"

   many folded ness

I'd like to hear more about your own intuitive conception of 3-manifolds...

I have been a "mathematical thinker" in an intuitive sense from my 
earliest memories, so I tend to bias my expectations of other's 
intuitions with that in mind.   What 3 manifolds do you find "easy" to 
conceptualize and when does it become "hard" in your mind?  Do you find 
that non-mathematical people find 3 manifolds obvious/easy?  Do you have 
conceptions of "exotic" 3-manifolds that you can put a compelling 
description to for non-mathematical thinkers?


My earliest introduction to 3-manifolds formally came from my 
(relatively non-mathematical) father asking me to consider whether the 
universe was infinite or finite, and if finite, did it end (like a 
flat/disk-earth would) or did it "wrap back on itself" (like a 
sphere).   I don't think he offered either a sphere or a torus as an 
example, but I do think they both came to me roughly at the same time...


Reimannian 3-manifolds are within reach for me, but I don't know how to 
"give" them to non-mathematical thinkers.


With our current administration being a "ship of fools" in many ways, I 
expect Trump to whip out the old idea of "legislating Pi to be rounded 
off to (redefined as?) 3"  which we all love to find ridiculous... but 
we could instead imagine that he is imagining that such legislation 
could curve space appropriately to make it literally true?


- Steve

On 3/1/17 2:21 PM, lrudo...@meganet.net wrote:

The word, as a term of Mathematical English (which is of course quite a 
distinct dialect of
English) is a calque of the Mathematical German word "Mannigfaltigkeit".  
Franklin Becher, in
the first paragraph of the lead article in the October, 1896, issue of the 
American
Mathematical Monthly, "MATHEMATICAL INFINITY AND THE DIFFERENTIAL", doesn't 
quite use the word
yet, but makes its origin clear enough.

---begin---
Mathematics, as defined by the great mathematician, Benjamin Pierce, is the 
science which
draws necessary conclusions. In its broadest sense, it deals with conceptions 
from which
necessary conclusions are drawn. A mathematical conception is any conception 
which, by means
of a finite number of specified elements, is precisely and completely defined 
and determined.
To denote the dependence of a mathematical conception on its elements, the word
"manifoldness," introduced by Riemann, has been recently adopted.
--end--

In his article on the foundations of geometry, available at
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html ,
Riemann distinguished two types of "Mannigfaltigkeit", the discrete and the 
continuous:

---begin---
cat

Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet, 
der
verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen 
Bestimmungsweisen von einer zu
einer andern ein stetiger Uebergang stattfindet oder nicht, bilden sie eine 
stetige oder
discrete Mannigfaltigkeit;

| Google Translate >

Size terms are only possible where there is a general concept, which allows 
different modes of
determination. According as, according to these modes of determination from one 
to another, a
continuous transition takes place or not, they form a continuous or discrete 
manifoldness;
---end---

In Riemann's (eventual) context, those sentences would be understood now (at 
least by
topologists of my sort, which is to say, geometric topologists, cf.
http://front.math.ucdavis.edu/math.GT) as sketching the modern concept of a 
(topological or
differentiable) manifold as a "mathematical conception" that can "precisely and 
completely
defined and determined" by a collection [called an "atlas"] of "modes of 
determination"
[called "charts"] among (some pairs of) which there are also given "continuous" 
(i.e.,
topological) or perhaps *smooth* (i.e., differentiable) coordinate changes.

I dispute, incidentally, the claim that 3-manifolds are too hard to understand; 
they're *just*
at the edge of that, but not over it (whereas 4- and higher dimensional 
manifolds are
DEFINITELY over that edge, in various well-defined mathematical ways; e.g., the 
problem of
determining whether two explicitly-given n-manifolds, n greater than 3, has 
been known for a
long time to be computationally intractable [you can embed the word problem for 
groups into
the manifold classification problem for n greater than 3], and much more 
recently has been
shown to be doable in dimension 3).

The French word for (something a little more general than a) manifold is 
"variet", by
the way; same sort of reason, I assume.




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

Re: [FRIAM] Fractals/Chaos/Manifolds

[lrudolph]

The word, as a term of Mathematical English (which is of course quite a 
distinct dialect of 
English) is a calque of the Mathematical German word "Mannigfaltigkeit".  
Franklin Becher, in
the first paragraph of the lead article in the October, 1896, issue of the 
American 
Mathematical Monthly, "MATHEMATICAL INFINITY AND THE DIFFERENTIAL", doesn't 
quite use the word 
yet, but makes its origin clear enough. 

---begin---
Mathematics, as defined by the great mathematician, Benjamin Pierce, is the 
science which 
draws necessary conclusions. In its broadest sense, it deals with conceptions 
from which 
necessary conclusions are drawn. A mathematical conception is any conception 
which, by means 
of a finite number of specified elements, is precisely and completely defined 
and determined. 
To denote the dependence of a mathematical conception on its elements, the word 
"manifoldness," introduced by Riemann, has been recently adopted.
--end-- 

In his article on the foundations of geometry, available at  
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html , 
Riemann distinguished two types of "Mannigfaltigkeit", the discrete and the 
continuous:

---begin---
cat

Grössenbegriffe sind nur da möglich, wo sich ein allgemeiner Begriff vorfindet, 
der 
verschiedene Bestimmungsweisen zulässt. Je nachdem unter diesen 
Bestimmungsweisen von einer zu 
einer andern ein stetiger Uebergang stattfindet oder nicht, bilden sie eine 
stetige oder 
discrete Mannigfaltigkeit;

| Google Translate >

Size terms are only possible where there is a general concept, which allows 
different modes of 
determination. According as, according to these modes of determination from one 
to another, a 
continuous transition takes place or not, they form a continuous or discrete 
manifoldness;
---end---

In Riemann's (eventual) context, those sentences would be understood now (at 
least by 
topologists of my sort, which is to say, geometric topologists, cf. 
http://front.math.ucdavis.edu/math.GT) as sketching the modern concept of a 
(topological or 
differentiable) manifold as a "mathematical conception" that can "precisely and 
completely 
defined and determined" by a collection [called an "atlas"] of "modes of 
determination" 
[called "charts"] among (some pairs of) which there are also given "continuous" 
(i.e., 
topological) or perhaps *smooth* (i.e., differentiable) coordinate changes.

I dispute, incidentally, the claim that 3-manifolds are too hard to understand; 
they're *just* 
at the edge of that, but not over it (whereas 4- and higher dimensional 
manifolds are 
DEFINITELY over that edge, in various well-defined mathematical ways; e.g., the 
problem of 
determining whether two explicitly-given n-manifolds, n greater than 3, has 
been known for a 
long time to be computationally intractable [you can embed the word problem for 
groups into 
the manifold classification problem for n greater than 3], and much more 
recently has been 
shown to be doable in dimension 3).

The French word for (something a little more general than a) manifold is 
"variet", by 
the way; same sort of reason, I assume.




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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove

Re: [FRIAM] Fractals/Chaos/Manifolds

[glen ☣]

FWIW, Penrose describes it: "a space that can be thought of as 'curved' in 
various ways, but where /locally/ (i.e. in a small enough neighbourhood of any 
of its points), it looks like a piece of ordinary Euclidean space." -- The Road 
to Reality


On 03/01/2017 12:26 PM, Steven A Smith wrote:
> Robert C -
> 
> I did a tiny bit of research, as I have also been curious, but found no 
> specific 
> etymology beyond the "obvious" many-foldedness origins from early anglo-saxon.
> 
> 1 dimensional manifolds are nearly trivial and 3+ dimensional manifolds are 
> nearly incomprehensible intuitively, leaving only the 2 dimensional manifold 
> as 
> an interesting, intuitive example.   In practice, the "hydrological manifold" 
> which is roughly used to channel one to many (or less common, many to one) 
> fluid 
> flows, has from it's form/function. These would seem to be the first 
> *examples* 
> of geometric spaces with locally euclidean properties but significant 
> global/topological complexity.  2-dimensional surfaces with continuous 
> deformations away from euclidean.  From a form-function duality, the need for 
> "smooth flow" of fluid through volumes bounded by continuous (and smooth) 
> surfaces, convolved with an obvious method of fabrication (distorting and 
> folding ductile surfaces such as metal or clay until the surfaces 
> self-intersect) seems to reference "many folds" or "manifold".
> 
> This is merely speculation that has developed over decades with very little 
> input.
> 
> The range of more "interesting" 2D manifolds is obscure to me... donuts and 
> "knots" (like gerbil habitrails or loop-de-loop roller coaster envelopes?) 
> are 
> the only obvious ones for me, with a Klein bottle being the lowest order 
> "exotic" example?  In my research I tripped over a recursive "Matrushka-Klein 
> example":
> 
> 
> which I haven't taken the time to properly sort thorugh in my head to know if 
> it 
> is topologically (as well as geometrically) different than a regular Klein?  
> And 
> are there even-odd species?   I don't think they have Chirality?  Puzzling!
> 
>> OK, why are mathematical manifolds called that?  It seems such a weird and 
>> out 
>> of place term.  I've tried to find out without success.
>>
>> Robert C

-- 
☣ glen


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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove

Re: [FRIAM] Fractals/Chaos/Manifolds

[Dean Gerber]

With respect to the word "manifold" somewhere around here:
http://www.quantum-gravitation.de/media/3a2a81c0493b7f728061fff0.pdf

--Dean Gerber 

On Wednesday, March 1, 2017 12:35 PM, Robert J. Cordingley 
 wrote:
 

 OK, why are mathematical manifolds called that?  It seems such a weird 
and out of place term.  I've tried to find out without success.

Robert C

-- 
Cirrillian
Web Design & Development
Santa Fe, NM
http://cirrillian.com
281-989-6272 (cell)
Member Design Corps of Santa Fe



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
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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove

[FRIAM] Fractals/Chaos/Manifolds

[Robert J. Cordingley]

OK, why are mathematical manifolds called that?  It seems such a weird 
and out of place term.  I've tried to find out without success.


Robert C

--
Cirrillian
Web Design & Development
Santa Fe, NM
http://cirrillian.com
281-989-6272 (cell)
Member Design Corps of Santa Fe



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
FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove