Re: Characteristics of multi-dimensional spaces?

On Apr 5, 9:47 am, EskWI...@xxxxxxxxxxxxxxxxxxx wrote:
In sci.physics, Timothy Golden BandTechnology.com <tttppp...@xxxxxxxxx> wrote:

Actually I think you should not trivialize this detail about the
plane. Does the plane have two sides or one?

I suppose it depends on what you mean by a side. One could assign labels
to whatever space is available on one side, say X, and another label, say
-x for the other.

The plane itself, being only two dimensional, has no thickness. So any
point identified on one side of the plane itself must necessariy
correspond to a point on the other side. They are the same point, given
that plane is only one point thick (or, IOW, has no thickness).

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?

ISTM that the 3D space lacks some sort of spatial characteristic which
would be available in the 4D space. Just as the plane lacks the thickness
which would render it 3D, I suppose that a 3D solid also lacks some sort
of analogous attribute, which attribute, if possessed, would make it a 4D
solid.

Does this strike you as a reaonable conclusion?

I made a long and confusing remark below about this where we get to
the thickness of a slice.
Simple answer: yes.


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.

This is the aspect of the problem that blows my mind. I cannot conceive
of, or imagine, or visualize a 4D space. IMO, the human brain is not
wired in a manner which allows such a visualization.

BUT! Nevertheless, one can easily calculate the volume of a 4D solid,
just as one can calculate the volume of a 3D cube. Please correct me if I
am wrong, but ISTM that LxWxHx4D dimension equals such volume.

So even though we cannot understand or visualize certain simple aspects of
4D solids (such as their shape), others (like their volume) can be derived
a priori. Maybe I am simple-mided, but that is a paradox to me.

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.

Nope. I am curretly puzzing about whether a 3D structure creates distinct
spaces in a 4D world.



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.

OK. You speak of "slices". This seems analogous to the slice that a 2D
plane describes in a 3D world. Thre are an infinite number of such
slices possible, due to their being an infinite number of possible
orientatins of the plane.

Well, what you say is true but even in the same orientation there are
an infinite number of these slices. They are well ordered and
expressible in the lost dimension which lays orthogonal to the
subsurface. Calculus sits right here in this continuous sense of the
slices and their infinitude. The integral of an nD conception is an (n
+1)D conception.


Same with 3D objects in a 4D world. This may be a clue - 3D reality might
slice 4D reality into discrete parts if the 3D reality is infiite in
extent. But 3D objects with dimensional extention less than infinity will
certainly NOT create distinct spaces in the 4D world, no more than a piece
of paper (if it were truly 2D) would slice our 3d world into discrete
pieces.

I think there is support for an exception from 3D to 4D under some
topological constraints. I'll dig up a link for you if you like. The
calculus type of consideration is general and is nearly arithmetic but
objects with detail at infinity are still beyond our abilities from
what I understand. I am in agreement with your statement above but the
discussion of infinite objects is not a necessary distinction is it? I
guess this was started in the context of whether or not the slice
actually contains any data and that gets around to the thickness of
the slice and I see from this statement that you see that argument so
very good. Perhaps what we should really be discussing in this realm
is motion and whether motion should be granted in the higher dimension
and in that case then the projection of a thin slice does expose
miniscule events for a particle model, yet there is a sweep here too
so we return to the confusing text above. Chuckle-barf.

Physicists find that by all of their methods the density of existence
is very low especially as they peer inside of the atomic model.


The other way to look at this problem is in terms of projections.

Please indulge me here. ISTM that shadows are truly 2D. They have no
thickness. So when one projects a shadow from a 3D object, one produces a
2D object.

Might it be possible that our 3D world is some sort of limited-information
shadow of a 4D reality? Or are there factors which we can consider to
satisfy ourselves that this is not possible? Or even if it is possible,
can we deterime that this is not actual? Or is this beyond our ability to
determine?

I read what you wrote below, but I need to read it again to be able to
understand it well enough to respond.

Thanks for your insights so far.

I argue that we should not stop even at 4D; that the general solution
is general dimensional.
Klein got sucked into a 5D bottle; the n+1 relationship is naturally
extensible.
That's cryptic but I think you'll get it. Hinton is an author you
might appreciate. "Speculations on the Fourth Dimension" is a reprint
of numerous articles by him preceding the Einstein rush by decades.
His arguments are largely sensorial and should appeal to any who like
to escape from equations. Still his thinking is mathematical and his
persistence exposes a seriousness that is deep even while his
arguments are light.

Since you want more projection consideration why not drop down to 1D
projections? We can build a line in any number of dimensions. The
standard 2D line:
a x + b y = c
can be extended to three dimensions:
a x + b y + c z = d
Just select the constants a,b,c,d and plug in any x and y and out will
come a z.
These procedures are extensible and expose the type of freedom that a
line takes yet the line remains a line in higher dimension just as
your plane remains a plane.

When we consider 1D projections we can reduce all of the dimensions
down to one dimension. We do so simply by the same math that we had
before but it is simpler yet:
[ A, B, C, ... ]
times
[ a, b, c, ... ]
equals
[ X ]
where the multidimensional space A,B,C has become a one
dimensionial X. The a,b,c are projection components which determine
the result:
x X = a A + b B + c C ...
In some regards the higher dimensional projections are merely more
general constructions of this simple reduction. Under this math
formalism we do not get hung up on any specific dimension but imposing
topological constraints does expose interesting behaviors in stepping
through a dimensional progression.

Simply consider the n-sphere in 2D and in 1D and we can see that the
continuum concept arrives in 2D. The 1D case is just a couple of
points whereas the 2D is a circle. Even here we should admit that
dimensions are more than their integral measures portray. The step up
to 4D that you are fascinated with is broader than the step up to 3D
and while you can declare the 4D 'volume' as the product you did above
can it be visualized? We are left with informational components
w,x,y,z that seem to make the system fully specified. The
meaningfulness of this level of density is somewhat beyond us. There
are strange topological behaviors even here and a breakpoint in the
volume to surface relation exhibits a breakpoint at 7 dimensions for
the hypersphere.

We see such topological peculiarities as binary charge (the 1D sphere)
and magnetic loops (fairly close to the 2D sphere) in a 3D space. Am I
bending geometry into physics or are the two unified? The
multidimensional approach that I wish to take will yield the 3D space
we inhabit as a consequence of a general dimensional basis. As such
puzzling over the sensorial geometry will not alleviate the physics
problem. Instead we are left building a general dimensional systems
that will hopefully one day yield electrons and so forth from pure
mathematics. These systems will yield spacetime as a natural
consequence rather than presume it as traditional physics has. Under
such a model physical reality is a product of the general basis and to
attempt access to the basis is suspect. This is radically different
from the observational limitations of string theory since the string
theorists do not propose such generality and have not yet yielded
spacetime as a consequence. I do admit that the denial of access is
similar to the stringers. Dimensional progression is much more
braneish than stringish but does have loops in 2D. Sorry if I've
overloaded you but these are abstract concepts that are within your
reach. Anyhow the 1D math projection above is a simplification over
the 2D case in the previous post and the generalization of these is
every bit as simple. The 2D projection is just a pair of 1D
projections. The math of the 'times' step has been instantiated in
this post.

-Tim




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

--
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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