[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