Posted by
Steve Smith on
URL: http://friam.383.s1.nabble.com/manifold-in-mathematics-tp3385914p3393764.html
Owen brings up an interesting (and important in it's relevance?) question...
>
>
> Where is your interest? History/Philosophy of Mathematics?
> Fascination with words and language? As a component of CAS? As a
> core mathematical theme to be mastered, somewhat like the
> epsilon/delta concept of the Limit?
What might the relevance of manifolds (in the strict mathematical sense)
be to CAS?
ABMs on arbitrary manifolds anyone?
In 1983, I had the good fortune of hanging out at the Cellular Automata
conference with the likes of Farmer, Packard, Crutchfield, Wolfram,
Langton, Fredkin, Toffoli, Gosper and even John H. Conway (and his wife
and two young children who also attended).
The agenda and topics covered were manifold (I couldn't resist). At
the time, CAs were pretty much the only/best example of what we call
ABMs today. Despite the varied and diverse approaches to CA, the one
area that was not covered (but was very interesting to me) was how CA's
behaviour might be qualitatively different if embedded in higher
dimensions, in non-euclidean spaces and more interestingly in more
complex topologies. Planes, Tori, and Spheres were the only topologies
of interest it seemed and all were considered to be locally euclidean.
I was struggling with formally specifying (so that I might implement)
something I called a "Network Automata" which today would roughly be a
formalization of ABM's in general, and was trying to understand the
relationship between the spatial embedding and the topological embedding
of these automata. Wolfram was struggling to classify all 1D, 2 state,
CA into 4 classes while Farmer (I think) was trying to demonstrate that
higher D, numerous State CA could be emulated in 1D, 2 State CA (making
Wolfram's work applicable across arbitrary spatial dimension and
(discrete) state space).
I fortuitously met an anesthesiologist who had postulated that some part
of consciousness arose out of the electro-mechanical properties of
microtubules in the neurons of the brain (and rest of the nervous
system). I was (apparently) the only one at the conference who thought
that the embedding space of a CA was interesting (despite some pretty
wide interests at that time in that domain). To that end, I struck up
a collaboration and proceeded to build a CA embedded in a 13 unit, 6-off
helical space (the way "normal" microtubules are arranged) for him.
The results were not that interesting in many respects, but we *did*
manage to demonstrate that information processing could occur in that
"small" of a space. We did not demonstrate anything more interesting
than information (soliton or glider-like) propogation along the main
axis (and soliton-anti-soliton anhillation). The constriction of one
axis (13 units) seemed to prevent any interesting lateral propogation
(the head of one propogating structure would encounter it's own tail, or
more aptly, the hip encountering the opposite shoulder, do to the helix).
Sadly this work is over the aft (redshifted into near invisibility)
information horizon and exists only on paper (archaic concept?)... I
have a scanned PDF of it somewhere if anyone might be entertained by
trying to read fuzzed-out grey words originally mastered on a typewriter
(yup, those were the days!) with many hand-drawn figures.
The recent ABM art-piece that Ilan and Ben (and Stephen) built for a
show at the Complex began to approach the same question, though
technically, their ABMs were confined to a complex 2D geometric region
in euclidean space. I believe that their original intent was to model
the complex surface (topologically a 3 hole torus?) with it's geometric
(polygonal cross-sections) properties to constrain and inform the ABMs
directly. Surely there has been some work in the area of ABMs and
their ilk embedded in more complex manifolds than the euclidean plane?
Any references, insights, ideas?
- Steve
>
> -- Owen
>
>
> On Aug 4, 2009, at 11:12 AM, Nicholas Thompson wrote:
>
>> I wonder if anybody has any comment to make on the following passage
>> from EB holt? (Remember, I am the guy who tends to ask questions of
>> PEOPLE when he should look them up, so feel free to ignore me here.)
>>
>> Holt (1914) writes: "If one is walking in the woods, and remarks
>> that "All this is Epping Forest," one may mean that this entire
>> manifold of some square miles is the forest; or else, that every twig
>> and leaf which one sees, in short, every least fragment of the whole
>> is Epping Forest. The former meaning is the true one; the latter
>> meaning is absolutely false. Everyone admits that while a circle is a
>> manifold of points, a single point is not a circle; while a house is
>> a manifold of bricks, boards and nails and any single brick is not a
>> house. "
>>
>> I am interested in this concept of "manifold" . Can anybody make the
>> metaphor come alive for me? Is it like a shroud?
>>
>> Nick
>
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