hi there.  came across another quote in a different book that mentions
a mathematical structure called a Riemannian manifold.  Here is the
quote:  " Viewing our consciousness in terms of its dimensions, we see
that it is not a three - or four-dimensional field, but a
multidimensional manifold, a manifold in the sense that it is
characterized by a non-linear -- Riemannian  -- geometry, in that all
the dimensions open up to all the others in nonlinear ways."
how are at explaining math to the mathematically challenged?  are
there any websites that you know of that give a visual representation
of what this author is talking about?
curious,
ts

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Clarification of Question by timespacette-ga on 27 Apr 2005 14:36 PDT

sorry, that sentence should read "how are YOU at explaining math to
the mathematically challenged?"

Judging from gorib-ga's comment, this looks like a crash course that's
worth considerably more than three bucks.

see below . . .

Hi Timespacetter,

If you are going to be trying to wrap your mind around Riemannian
manifolds you need to first understand the concept of a metric tensor.

http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey=Metric+tensor&gwp=8&curtab=2222_1

In order to be a complete Riemannian manifold, the metric d(x,y) must
be defined as the length of the shortest geodesic curve between x and
y.

Does that clear things up?

Check it out:
http://mathworld.wolfram.com/RiemannianManifold.html
http://www.answers.com/main/ntquery?method=4&dsid=2222&dekey=Riemannian+geometry&gwp=8&curtab=2222_1
http://www.answers.com/Riemannian%20manifold

thanks for trying gorib,

as I've said before, I am a bear of little brain when it comes to math

I was hoping for some kind of visual representation, as in a moving
graphic of some kind, just so I could get a general understanding

thanks

ts

Hi, timespacette-ga:

The approach I intended to take for this Question is ironic, given
that you are looking for "some kind of visual representation".

A manifold is the mathematician's version of this famous Buddhist parable:

[The Blind Men and the Elephant]
http://www.cs.princeton.edu/~rywang/berkeley/258/parable.html

wherein the blind men are unable to find agreement as to what the elephant is like.

We define a manifold as a topological space which is "covered" by an
"atlas" of "charts" that each map out part of that space.  Each chart
corresponds to viewing its own "local" portion of the space as if it
were in ordinary n-dimensional space, where n is the same dimension
for all charts, and whenever two charts overlap, they "agree" (which
means that on the overlap between two charts, conversion from the
coordinates of one chart to the other is "continuous" and also
consistent, whichever direction the conversion is done).

For a more humorous take on the parable, check out the follow on link
in this version:

[The Blind Men and the Elephant]
(see link at bottom of page)
http://www.buddhistinformation.com/blind_men_and_the_elephant.htm

wherein the blind elephants are able to find agreement about what a human is like.

*  *  *  *  *  *  *  *  *  *  *  *  *  *  *

A Riemann manifold adds more "structure" to the concept of a manifold.
 As gorib-ga has said, what this extra information amounts to a
"geometry" in the local charts, so that lengths and angles are defined
there (and are consistent across overlaps between charts).

Regular n-dimensional space is "flat" (like the human in the follow on
parable), and we call this Euclidean space after the basic geometry
learnt in high school, where every line must have its "parallel" lines
that never meet it.

Curved spaces provide a more interesting setting.  I will try to step
carefully among the Web pages and locate some helpful graphics!

regards, mathtalk-ga

I am an expert on neither topology nor cognitive science, but I will
venture a guess that they are about as closely related as cosmology
and cosmetology, and that the quote above is wilfully and
unnecessarily abstruse.

Maybe you are just curious about the math, but if your ultimate goal
is to try to understand human consciousness, you might want to
consider the possiblity that there's about one person on the planet
who will tell you with a straight face that it's a Riemannian
manifold.

I think the best approach to understand what a reimannian manifold is
without much math is thousands of examples:
Stupid example:
the plane (a flat manifold) everything works as with euclidean geometry

Interesting examples
the surface of earth you can choose as coordinate system longitude and
latitude. this manifold is not euclidean, in fact triangles' angles do
not sum up to 180° (take the triangle with a right angle on the north
pole and basis on the equator, it has 3 right angles)

the surface of a saddle where there are no parrallels

you can devise  many more  in higher dimensional spaces but you cannot
visualize them. The point of the author of the quote is that the
geometry of the our counsciousness is not bounded by the axioms of
euclidean geometry, but can have many more sutle and remarkable
properties.
And IMHO most of the time this kind of authors just add references to
maths and physics just to give the text a more scientific suond but
most of the time and I think this is the case they don't have the
slightest idea of what they are talking about

hi racecar and k4r10,
thanks for your collective input on this problem.  It remains a
problem, I think for the reason that k4r10 mentioned: "you can devise 
many more  in higher dimensional spaces but you cannot
visualize them."  Reminds me of the book Flatland.  Anyway, I'd like
to make two points ... racecar, it's called a metaphor. No one ever
said consciousness IS a Riemannian manifold, but the implication is
that certain experiential views of consciousness resemble this
structure, and it's simply an attempt to describe something that is
damn near indescribable. I would venture to guess that the number of
people who have attempted to describe it are many, far more than your
estimated one, as in the ancient Tibetan Buddhist and Hindu
descriptions of certain states of consciousness; it's just a matter of
finding these references.  Actually, their descriptions point to
something more like a hologram (see 'net of Indra' in Hindu mythology)
 Also, I would tend to disagree with k4r10 about what appears to be a
gimmick on the part of the author, as he holds a PhD in math, and from
what I can discern he's not your run-of-the-mill woo-woo lecture
circuit guru, not a guru at all even.   If you want to look it up the
book is called The Inner Journey Home by A.H. Almaas.

cheers!

ts