[FRIAM] Fractals/Chaos/Manifolds

[Steven A Smith]

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
>

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