Re: Characteristics of multi-dimensional spaces?

On Apr 4, 12:09 pm, EskWI...@xxxxxxxxxxxxxxxxxxx wrote:
This is a vague sketch of an idea, ill-formed due to ignorance. But I'll
throw it out anyway, and I'll ask for your indulgence of a poor liberal
arts grad...

So anyway, I was thinking about a two-dimensional world, much like that
which is depicted in the book Flatland. The inhabitants of flatland exist
in a three dimensional context, but are only aware of two. They live on a
plane, extending in two of the available three dimensions. Anyone who is
familiar with the book knows what I mean.

I'm thinking that if their world is a plane, it extends to infinity in
each of the two dimensions into which it is extended. As such, it slices
the three dimensionsal world of which it is a part into two distinct
pieces. In the three dimensional context, a point cannot exist such that
it is on both sides of the plane, it must necessarily be on one side or
the other.

Ok, so this is trivial.

Actually I think you should not trivialize this detail about the
plane. Does the plane have two sides or one? If this is possible of
the plane then what about 3D space? Does it have a dual space as the
plane does? Or is it a tripled space? Especially if the general
dimensional setting is being studied then this distinction may be
important. The visualization of the 3D extension to this distinction
you raise is not easy to visualize since we need at least a 4D space
to study it in. In this vein if we consider the complex plane then it
already comes with this reflective quality built in on one side which
it mimics in reverse on its other side. But I'm pretty sure that's not
what you want to talk about.


But - if our three dimensional world is in a four (spatial) dimensional
context (please remember I said "if" :), would we similarly slice the four
dimensional context into distinct spaces? Into three spaces? Another
number?

The easy answer to this part is that you would have an infinite number
of three-dimensional slices in a continuous 4D space. If you would
prefer to work in a discrete lattice then you can picture a series of
3D sublattices.

The other way to look at this problem is in terms of projections.
Projections are more dynamic since there are lots of possible
projections. For instance back in your plane you could have projected
all of the information in your 3D space into that singular plane. Your
eyes portray a 2D projection of the real world and the occlusion that
we experience should be considered. So for instance beyond your coffee
mug is a desk surface and all of these layers of information could be
superposed. We generally only receive the shortest distance
information like a stack. Your plane that you initially built will
have very little information in it since as you pointed out most
objects are on one side of it or the other. This is a mathematical
concept to do with limits. You could give your plane projection some
thickness and get back toward a lattice description in which case you
would have frames with a good amount of information in them.

The standard matrix projection (dimensional reduction) takes a form
like

[ A, B, C ]
times
{ [ e, f ], [ g, h ], [ i, j ] }
equals
[ X, Y ]
and winds up superposing all of the objects in the ABC (3D) space into
XY (2D) without regard for depth.
These are projections of the simplest kind. The projection itself is
the e,f, ... part in the middle and each letter is literally a real
value like
+ 0.87245 so six such values are enough to do a 3D to 2D
projection.
The reverse projection is not possible because information has been
lost. All of the depth information is gone.

Sorry if I just confused you more. This is how I look at the basics of
what you are considering and there is no one right way to slice it.
The thin slice you've proposed likely contains little information and
the full projection carries too much information. High dimensional
data is very troublesome to look at via the simple projection. I
don't mean to sound discouraging; I believe a general dimension
solution will be found that yields physics. Orthogonality and its
relation to dimensionality is another important informational concept
that might fill out the picture some.

-Tim


I suppose this is more a geometry question than a physics question, but
any insights would neertheless be appreciated.

--
The whole problem with the world is that fools and fanatics are always so
certain of themselves, but wiser people so full of doubts.
-- Bertrand Russel


.

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