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
"varieté", by
the way; same sort of reason, I assume.
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