[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, lrudolph at 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 "varieté", by
> the way; same sort of reason, I assume.
>
>
>
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