Posted by
Steve Smith on
URL: http://friam.383.s1.nabble.com/manifold-in-mathematics-tp3385914p3393427.html
[hidden email] wrote:
Let me add another
inquiry to this - how do we reconcile this notion of manifold with the
idea of self-similarity? If Epping Forest is a manifold, but the leaves
and twigs are not, yet the leaves and twigs have some self-similarity,
is Holt truly thinking in terms of the mathematical definition of
manifold, as Roger gave us, or is the metaphor missing something (or am
I)?
Clay, et al. -
I would submit that the term (and especially concept) of
manifold
significantly predates the mathematical use. This concept and usage
would seem to have provided the motivation for the mathematical term
itself. It also might provide a little more leeway in expanding the
application to reference or accommodate self-similarity in some way, as
you suggest.
From:
English Etymology Dictionary
manifold O.E. monigfald (Anglian), manigfeald (W.Saxon), "varied in
appearance," from manig "many" + -feald "fold." A common Gmc. compound
(cf. O.Fris. manichfald, M.Du. menichvout, Swed. m?ngfalt, Goth.
managfal?¢®s), perhaps a loan-translation of L. multiplex (see
multiply).
And from:
Webster's Revised Unabridged Dictionary (1913)
Manifold \Man"i*fold\, a. [AS. manigfeald. See Many, and Fold.] 1.
Various in kind or quality; many in number; numerous; multiplied;
complicated. O Lord, how manifold are thy works! --Ps. civ. 24. I know
your manifold transgressions. --Amos v. 12. 2. Exhibited at divers
times or in various ways; -- used to qualify nouns in the singular
number. ``The manifold wisdom of God.'' --Eph. iii. 10. ``The manifold
grace of God.'' --1 Pet. iv. 10.
My working definition (sharpened in the process of this discussion,
thank you Nick!) is to "be many and varied in nature" with some
alternate but related working definitions. "a mechanical multiplexer,
as in a
gas manifold such as used for intake fuel mixture and
exhaust gas in an internal combustion engine." This usage seems to
reference the complex topology requiring something of a patching
together of various tube-shaped surfaces". and "multi-fold, in the
sense perhaps of complex origami pieces or even the simplicity of
carbon-paper copying... or multifold writing".
As for your suggestion... I submit that for my purposes, geometric
artifacts exhibiting self-similarity (often fractals) do in fact
suggest a valuable variation on the use of "manifold". One might say
that such artifacts have sub-components which are "many and varied in
nature", though the definition of "varied" is challenged a bit by the
self-similarity. They are clearly varied (being similar, not
identical) yet, are not-so-varied (being similar, not different).
Fractals (and other self-similar geometries) embedded as surfaces in 3
dimensional spaces do strike me intuitively as being "barely" or
"almost" a manifold. They are "barely" in the sense (again) of their
subcomponents (regions) being many and vairied but "almost" in the
sense of not being as diverse as I would normally want.
A *real-world* self-similar system such as your referenced _forest of
trees of branches of twigs with leaves having at least a few levels of
self-similar structure_ would seem to be very apt and perhaps where
some of the earliest referenced usage of the *concept* ("O Lord, how
manifold are thy works!") arose.
<nostalgic anecdote>
My first encounter with the term was on the 390 CI engine of my 1964
Ford Thunderbird in High School. I confronted my learned (aka
geek-nerd) friend whose father was an engineer and who bent and welded
up his own headers for the ford 289 CI in his 1963 Ford Fairlane coupe
with: "what is the difference between a manifold and a header?". His
response was somewhat like Doug's or Owen's to many of Nick's questions
here, but I persisted until his father overheard the conversation and
weighed in himself. As an engineer he was more interested in the
practical properties rather than what I was seeking of the "essential"
properties and proceeded to explain to me all about pulse-tuning of the
exhaust system, how impedence worked in complex, dynamic systems of
compressible fluids (exhaust gas in particular) down to how the cooling
of the exhaust gasses in the exhaust system effected the
pressure/volume and how that should be accounted for in the tuning
even. Once we got past the pragmatics, I was able to engage him in my
*real* question which was why a welded-up header was not also a
manifold (even then I knew how to read a dictionary and had at least
been introduced to non-euclidean geometry and topological spaces).
What we settled on (so that my friend's mother didn't kill us all for
holding up dinner with this long-winded conversation) was quite
obvious, as the headers were an array of exhaust pipes connected for
convenience by a single flange for bolting up in place of the
manifold. We agreed that perhaps the *other end* of the header-system
where they all joined to a single exhaust pipe (well 4-1 for each of 8
cylinders and 2 exhaust pipes) was a bit of a "manifold" in the
original sense but that since the joints were roughly
non-differentiable, not really. They were a "multiplex" of 2D
surfaces joined in a complex way, but not (quite) a manifold in the
more strict sense.
</nostalgic anecdote>
I shall now return to my manifold deadlines and projects (all joined
together in a singular and continuously differentiable
high-dimensional artifact embedded in a higher dimensional space)
- Steve
PS. If you read this far Gattiker, drop me an e-mail!
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