Conference hydra::dejavu

About a year ago, I read the book  _The Planiverse_  by A.K.Dewdney  which is a
whimsical, yet mathematically sound (seemingly sound, anyway) look at life in
two dimensions, seen both through the eyes of a two dimensional resident and a
three-dimensional watcher.  I became so engrossed in the book and its setting at
one point, that when my husband said he was leaving for hockey, my first
reaction was "no.  Hockey's not possible.  You'd have to crawl over each other."
[not that that isn't done in hockey... :-)] 

For a short period therafter, I found four dimensions far easier to imagine.
Part of the illusion created by the reading, I suppose.  I'm planning on reading
the book again in another month or two, and am interested in experiencing that
effect again. 

I have just obtained a copy of Edwin A. Abbott's  _Flatland_  which is an
earlier look at a two-dimensional existence which inspired _Planiverse_.
                                                                        
 
                                              Kathy  (the other Reinig)

Lots of thoughts:

1) Any measurement can be a "dimension."  That includes temperature, capa-
citance, color, brightness, etc.  "Time" is but one "fourth" dimension.
(there are more than five "senses" too.)

2) Any model that can help you perceive something is useful.  But confusing
the model with reality can be dangerous.  A shadow can be perceived as
two-dimensional, or, on a foggy night, three-dimensional.  But just because
a shadow "is" two dimensional, extrapolating the analogy ("what casts the
three-dimensional hypershadow?") leads one down garden paths.

N-dimensional geometry is treated thoroughly in some mathematics books, if
you are interested.

Steve Kallis, Jr.

I am interested in N-dimensional geometry.  Could you suggest an particular
books?

		Dave

re .3:

I'll try to look up some titles.  Dover Press has several in their "Mathemat-
ics" section.  

Steve Kallis, Jr.

Many of Martin Gardner's collections have articles and stories on 
"extra-dimensional" universes.  This has long been a popular (and 
entertaining) topic in science fiction and fantasy.  Two of the better 
stories based on "extra dimensions" are "And He Built A Crooked House" 
and "A Subway Named M�ebius".  (I don't recall the author of either, 
although Bob Heinlein's name comes to mind in relation to the first.)

Tom

Heinlein wrote "crooked house"; an architect created the house in the
shape of an unfolded tessaract.  An earthquake "folded" it.  Cute story.
"Subway named Moebius" appeared in _Astounding Science Fiction_ and
I recall it was later anthologized in Fadiman's _Fantasia Mathematica_.
Author's name escapes me, too.

Steve Kallis, Jr.

	    I find it more useful to think of time as the *first* dimen-
	sion because, to my human mind at least, for anything to exist
	(as my mind knows existence) it has to occupy an expanse of time
	as well as space.  I can't experience a line or plane or cube if
	it's not there long enough for me to get a look at it!
			<_Jym_>

I recently read an article in Sky and Telescope (Nov 1985) called
"New Physics and the New Big Bang".  In a nutshell it states that
someone has found a way to make Einstein's Unified Theory work....
but it's Mathematics involves 11 dimensions.

It's one of the better "Gee Whiz" articles I've read.  No formulae but
there's enough stuff there to make my head really spin.

If there's interest, I could reread it and give a more complete synopsis.

Happy New Year!
Tom

    re .0
    
    You should be aware that a configuration in an N dimensional space
    may come from many different N+1 dimensional configurations, by
    projection.
    
    There are some theories out now, "super string theories" that postulate
    physical space is really 11 dimensional but most of it is mashed
    down to the four extensive dimensions that we all know and love.
    

    re. Time as the 4th dimension.
    
    Have a look at the explaination of time in the first chapter(I think.)
    of H.G Wells "The Time Machine".-Although the book is fictional
    the explaination of time as THE fourth dimension seems reasonable.
    I found it interesting ,at any rate.
    
                            The Mad Eavesdropper.

    re .10:
    
    >...the explaination of time as THE fourth dimension seems reasonable.
    
    I don't believe anybody questions treating time as _a_ dimension,
    it's the "THE" that many disagree with.
    
    Steve Kallis, Jr.
    
    P.S.:  I enjoyed the story thoroughly.

    re: .0
    
    Try this one on for size:
    
    given a point:
    	add time and you get a line (first dimension)
    	add time and you get a plane (second dimension)
    	add time again and you get the familiar third dimension
    
    now-- if you continue to add time do you respectively get the fourth
    and consecutive dimensions?
    
    anything to add?
    
    GRoss
    

    Re .12 n(GRoss):
    
    >given a point:
    >	add time and you get a line (first dimension)
    >	add time and you get a plane (second dimension)
    >	add time again and you get the familiar third dimension
     
    Whoops!  How are you defining "add time"?  If you treat time as
    a (linear spatial) dimension, the second "time" you "add time,"
    that second step must be done at right angles to the first; thus,
    "time" in the second case would haver a different meaning than "time"
    in the first.  The same would hold true with the third step.
    
    What is "time at right angles to time"?  Some have suggested it's
    the probability dimension.  Then what is "time at right angles to
    time and time_probability"?
    
    Also, if you have a time/time/time component, you might have a
    time-solid with no spatial component.
    
    Steve Kallis, Jr.

    RE: .12
    
    Where the heck does time enter into Euclidian geometry?
    
    
    John M.

    Re: .14
    
    Time or space (in the normal sense of the word) does not "enter
    into" any geometries.  Geometries, like most (all?) mathematical
    systems are independent of interpretations (models) placed on them.
    In some sense this is one means by which mathematics seems to have
    such an extraordinary versatility.  With respect to this there is an
    interesting connection between mathematical systems (especially
    algebras and logics) and the story form known as parable.
    
    /Jon

    Re .16:
    
    Well, the fourth right-angles-spatial-motion gives you a hypercube
    (tesseract).  The Heinlein house was an unfolded tesseract that
    an earthquake somehow caused to fold up completely.  Cute story
    though.
    
    >                 -< viz. "The Crucifixion" by Dali >-
          
    Dali's name for the work was _Corpus Hypercubus_; I saw the original
    when living in New York.  Very effective if you see it in full
    splendor.
    Steve Kallis, Jr.

RE: .16,.17
    
    Actually, you *don't* get a hypercube -- you get what is called
    a 4-dimensional Euclidean manifold, symbolized as E4 (the 4 is
    superscripted and the E is bolded).
    
    If you start with a 0-manifold (which when imbedded in a higher
    dimensional manifold is also a point) and extend it indefinitely
    in any direction (both ways) you will get a 1-manifold (which when
    imbedded in a higher dimensional manifold is also a line).  If you
    extend the 1-manifold indefinitely in any direction perpendicular
    to itself (both ways) you will get a 2-manifold (which when imbedded
    in a higher dimensional manifold is also a plane).  If you extend
    the 2-manifold indefinitely in any direction perpendicular to itself
    (both ways) you will get a 3-manifold (which has no simple geometric
    term since traditionally higher than 3 dimensional manifolds in
    which to imbed it were not considered.).
    
    If you start with a point and extend it a finite distance in any
    direction (one way), you will get a line segement.  If you extend
    the line segement the same distance in any direction perpedicular
    to the direction of the line segment, you will get a square planar
    region.  If you start with a square planar region and extend it
    the same distance in any direction perpendicular to its generating
    directions, you will get a cubical region.  If you extend *that*
    the same distance in any direction perpendicular to its generating
    directions, you will get a *hypercubical region*.
    
    If leave out the interiors in the above you will get a point, two
    points (sometimes called, somewhat confusingly, a segment), a square,
    a cube, and a hypercube, respectively.
    
    One more, just for good measure -- Start with a point: call it
    simplex-0.  Add another, different point and all points between:
    that's simplex-1, also called a line segment.  Add another point
    which is not in the line (1-manifold) defined by the first two
    points, connect the points: call it simplex-2 or a triangle.  Now
    add another point which is not in the same plane (2-manifold) as
    simplex-2: call it simplex-3 or a tetrahedron.  Finally, add another
    point which is not in the same 3-manifold as the simplex-3: call
    it simplex-4, a hyper-tetrahedron, or a pentalope.
    
    How about: in 2 dimensions there are an infinite number of regular
    figures: the equilateral triangle, the square, the regular pentagon,
    etc. up to the circle.  In 3 dimensions there are only 6: the
    five platonic solids: the regular tetrahedron, hexahedron (cube),
    octahedron, dodecahedron, and the icosahedron, plus the sphere.
    In 4 and higher dimensions there are only 3: the simplex-n (pentalope
    when n=4), the n-cube (hypercube when n=4), and the n-sphere
    (hypersphere when n=4).
    
    					Topher

RE .12, .13

    >given a point:
    >	add time and you get a line (first dimension)
    >	add time and you get a plane (second dimension)
    >	add time again and you get the familiar third dimension
     
    > "time" in the second case would haver a different meaning than "time"
    > in the first.  The same would hold true with the third step.
    
This is not necessarily true either!  The "new time" you make mention of
is really the product of two units of the "time" in the first step.
In other words "time squared"  Pretty tricky eh?  The same is true in
the 3d dimension also, "time cubed".

    > What is "time at right angles to time"?

Now you're trying to use a concept (right angles) in relation to a quantity
which cannot be described in such terms (to the best of my knowledge).  It
just happens to be be a convenient (but limited) way of expressing (or
visualizing) something we lack complete ability to define in other terms.

---
I am by no means an "expert".  I just had an idea that I thought had merit,
and wanted to hear reactions.  I've heard the reactions, but I am not
completely satisified with anything discussed up to this note.  Some of
the comments got quite "heavy" with abstract mathematical terminology
which was completely meaningless to me.  Obviously there are no *easy*
answers or we wouldn't be having this discussion, but it is entirely 
possible we are overlooking a *simple* answer in favor of the overly
complex which we find so necessary today.

Gordon Ross, Jr. (GRoss)

    Re .19 (Gordon):
    
    >This is not necessarily true either!  The "new time" you make mention of
    >is really the product of two units of the "time" in the first step.
    >In other words "time squared"  Pretty tricky eh?  ...
    
    Lest we get lost in definitional difficulties, let's start over.
     
    
    When you're referring to a "dimension," you're speaking of a unit
    of measure.  In _simple_ terms, any measurable quantity (spatial
    displacement, brightness, temperature, capactiance, time, mass,
    color, flux, charge, spin ....) _can_ be called a "dimension." 
    In the popular press, time has been tagged as "the" fourth dimesion;
    it isn't.  It is "a" fourth dimension.
    
    On the "right angles" business: that was a poor analogy, but it
    was better than defining the second unit of time as a vector cross
    product or some such thing.  No, :"right angles" is a hard analog
    to use, but it's a workable one.  I appreciate about "time squared,"
    which could be thought of as the inverse of acceleration in some
    models, but in spatial math, a square unit is the product of two
    orthogonal measures; and I thought your original question had to
    do with the "familiar third dimension," which includes this definition
    for volume within it.
    
>I am by no means an "expert".  I just had an idea that I thought had merit,
>and wanted to hear reactions.  I've heard the reactions, but I am not
>completely satisified with anything discussed up to this note.  Some of
>the comments got quite "heavy" with abstract mathematical terminology
>which was completely meaningless to me.
    
    Two points:  First, I'm sorry you've found little or no satisfaction in the
    answers; however, you did ask a question that required delving into
    some fairly abstruse areas.  Second, your question, whatever else,
    had a mathematical (by which I don't mean arithmetic) base, and
    there's no way _not_ to get into abstract mathematical terminology
    when trying to respond to it.
    
    Steve Kallis, Jr. 

    Try this on for size.  If you read Flatland (which I have not),
    you may be able to draw analogies from 2-dimensions and apply them
    to 3-dimensions.  For example, imagine Flatland as a 2D expanse
    like a piece of paper.  Now imagine taking a pencil and passing
    it through Flatland (the paper) at right angles.  What would a
    Flatlander see?  As the point of the pencil contacted the paper
    he would see a point.  As the pencil passed through the paper, he
    would see a growing circle until it reached the diameter of the
    pencil.  At this point it would be a just a circle in his world.
    When it came to the end of the eraser of the pencil, poof, the 
    circle would suddenly disappear.  Similar analogies apply to a 
    hypervolume passing through our 3D world.  We would see a volume.
    It would appear to us to appear and disappear "magically".  Have
    any of you had such an experience?  Not I.  Would such a hypervolume
    passing through our 3D world be a conduit to other 3D worlds? Just
    as a bullet passing through a "phonebook" of Flatlands would be?
    Put me on the next hypercube to...???
    
    What about black holes?  Here we have a mathematical singularity
    in our nice neat 3D universe.  In "reality" there is a lot of matter
    at the core of the black hole and all matter with in "vacuuming"
    range gets sucked in and adds to the matter there.  Eventually all
    matter will be sucked into a black hole, because as matter is
    added the "vacuuming" distance gets larger and more is sucked in.
    Presumeably, then there is a critical volume and mass to a black
    hole, it explodes and recreates a new universe (a la the "Big Bang").  
    In that sense our matter has been through millions of universes and 
    will go through millions more, assuming time is infinite (what happens
    when time stops?  A little boggler there, eh?).  Also if a black 
    hole has a critical volume and mass then it is finite.  Therefore, 
    there are multiple universes being created and destroyed throughout 
    time and space.  Maybe a universe can be defined as a galaxy with a 
    black hole at the core that periodically sucks in and ejects all the 
    matter in the galaxy?  This is probably not exactly macrocosmic theory
    but is my interpretation.  I thought some in this notes file might
    be amused by it.
    
    What about gravity?  In a black hole, gravity is so strong even
    light is sucked in.  What if gravity is a way to be sucked from one 
    universe to another via a black hole (the doorway?).  Suppose there
    are two (or more) universes connected by black holes.  We could
    have a constancy of universes (matter is neither created or destroyed)
    (matter and anti-matter where one is the other passed through a
    black hole and acted on by the tremendous gravitational forces?)
    where when some matter is sucked into one universe other matter
    is ejected (sucked) into another.  This seems inherently unstable 
    and a universe which had the wrong size doorways (more matter going 
    out than in) would be sucked entirely into another universe and in 
    the end there would only be one universe.  Unless the process itself 
    were unstable and doorways (eddies) could occur spontaneouly and a
    new universes could be suck off the current one until it became
    unstable and was sucked back or another was sucked off it.
    
    Just something to think about on the drive home tonight.
    
    An engineer,
    Stan

RE: .21
    
    > Presumeably, then there is a critical volume and mass to a black
    > hole, it explodes and recreates a new universe (a la the "Big
    > Bang").
    
    Could be, but I wouldn't presume it.  According to current theories
    the bigger a black hole gets the less likely it is to explode.
    
    Black holes radiate energy according to Hawking's well accepted
    arguments about the effects of Quantum Mechanics on black holes.
    Small ones radiate quickly, large ones radiate much more slowly
    and gently.
    
    Matter which falls towards a black hole will tend *not* to be swallowed
    by it.  Rather it will tend to be whipped around it and out again
    in a parabolic or hyperbolic orbit.  As matter slowly collects in
    the black hole, the probablility of meeting matter decreases.
    
    The effect of this on things was revealed by some fascinating
    calculations done a few years ago on "The Ultimate Fate of the
    Universe".  Assuming that the density of the universe is not more
    than 10 times higher than we observe, then after a suffient number
    of terrayears (periods of a million million years each) things will
    settle down to a steady state of many, many medium size black holes
    coliding and evaporating at roughly the same rate, with a rich
    mix of elementary particles filling space between them (or that's
    how I remember it).
    
    					Topher

    Topher,
    
    I admit my theory of black holes is severly limited.  I got most
    of it from my fertile imagination.  I tossed it out to see what
    kind of responses I would get.  I am glad I got yours (reasonable).
    
    There is one thing of which you never will convince me.  That is
    the "ultimate fate of the universe" you mentioned.  Some mechanism
    must exist to "recycle" the universe.  I offer the following simple
    "proof".  The universe (as we know it) exists.  This implies a
    probability for such a universe to exist.  If there is a probability
    and one exists (ours) then there is a mechanism.  If there is a 
    mechanism and a probability then it will exist again given a long 
    enough time.  This is not rigorous, but it is good enough for me.
    
    Stan

RE: .23 (Stan)
    
    Interesting argument there -- it may even be valid.  Two open issues
    in it occur to me:
    
    It assumes that, in technical terms, that the probability of two
    "universe creation events" are "independent".  In less technical
    terms, it assumes that the creation of one universe doesn't prevent
    any future universes from being created.  One could say using the
    same logic "This light-bulb burned out.  It must therefore be possible
    for a light-bulb to burn out and so if we only wait long enough
    *this same light-bulb* will burn out again."
    
    The other issue is a rather subtle one in probability theory and
    I for one do not have the rigorous grounding in the mathematics
    of probability theory needed to solve it.  It revolves around the
    odd paradox of probability theory that events of probability
    (technically, "probability measure") zero are not impossible, i.e.,
    can occur (I'll give an example in a moment).  In that case, and
    if the creation of the Universe is such a zero probability event
    then what is the probability that the Universe will be created *again*
    even given "infinite" time?
    
    The example: if you are allowed to pick any integer between 1 and
    10, then the probability of picking any particular number is 1/10.
    If you are allowed to pick any integer between 1 and 100 then
    the probability of picking any particular number is 1/100.  If
    you are allowed to pick any positive integer at all then the
    probability of picking any particular number is 1/infinity or
    zero (this is not a rigorous mathematical statement -- it can
    be said rigorously and the result is the same).  Nevertheless,
    *some* integer, which must have had zero probability of being
    picked, *is* picked and so an event with zero probability has
    occured.
    
    Actually, in some versions of the Grand Inflationary Universe theory
    (the current "hot" theory for the creation of the universe, it
    can either be considered as the successor to the Big Bang theory
    or a more specific version of it) Universes such as ours are
    created spontaneously by quantum fluctuation within a containing
    (I almost said larger, but it is not clear that that concept is
    meaningful here) super-universe.  The analogy given is like bubbles
    appearing in a glass of soda-water or champagne.  Although these
    theories imply fairly strongly that each such universe will tend
    to be permanant and non-recycling, the whole super-universe is
    steady-state with new, isolated bubble-universes appearing out
    of nothing all the time (although it is not clear exactly what
    that last word means here, either).
    
    					Topher

    Re: .24

>    the probability of picking any particular number is 1/100.  If
>    you are allowed to pick any positive integer at all then the
>    probability of picking any particular number is 1/infinity or
>    zero (this is not a rigorous mathematical statement -- it can
    
    I think there is a subtle difference between this example and the
    previous two (though they may suffer from a different problem...).
    In order for the probability to be "1/infinity" the choice must
    be *random*.  Since no one "knows all of" the natural numbers it
    doesn't seem that the choice *could* be random (not even counting
    psychological factors influencing the choice - in fact to eliminate
    this problem let's suppose we have some strictly mechanical means
    for choosing).  Similarly for any mechanical picking, it is not
    physically possible to pick any such number for essentially similar
    reasons (impossible to represent all such numbers).  Now, I agree
    that we can show the *existence* of some function which will return
    an arbitrary random number but we can't *construct* it (this is
    an example of logicism vs. constructionism).  So the issue is if
    we can't construct such a thing can we still claim that such *events*
    *can* occur (much less *do* occur)?  In other words, 0 probability
    events are *logically* possible, but not say "physically" (actually?)
    possible.
    
>   Acutally, in some versions ...
    
    "You too, can create a universe in your own basement..." :-)
    
    /Jon

RE: .25

    Finally found some time to reply to this.

    Your comments are quite apropos: physical processes which can be
    described by infinite discrete uniform distributions are not
    plausible.

    I did not, however, intend the example as a description of such
    a plausible physical process -- I was only trying to illustrate why
    in probability theory the probability of an event can be zero and
    yet still occur.

    Here is a more appropriate example of a zero probability event which
    may occur:

    A continuous distribution is a description of a random process which
    may take on any real value within a given (possibly infinite) range
    of values.  One example of this is the "continuous uniform
    distribution", e.g., any real value between zero and one is equally
    likely and no value less than zero or more than one is possible.
    Another is the familiar "bell shaped curve" of the "normal"
    distribution.

    The probability that any *particular* value will be produced from
    a random process described by a continuous distribution is zero.
    Obviously, of course, the probability that *some* value will be
    produced is greater than zero -- specifically it is one -- some
    value *will* be produced.

    To see why this is we can look at a particular example: we can
    ask what the probability of getting the value .5 from the continuous
    uniform distribution between 0 and one (i.e., any real value between
    0 and one is equally likely and no other value will ever be produced).

    First lets ask how likely it is that we will get some value between
    0 and 1.  The answer is obviously "1", since no other values are
    possible and we always get a value we are certain to get a value
    between 0 and 1.

    Now lets ask how likely it is that we will get a value between one
    quarter and three-quarters.  To answer this we can divide up the
    region from 0 to 1 into four pieces: 0 to 1/4, 1/4 to 1/2, 1/2 to 3/4
    and 3/4 to 1.  Each of those four pieces are equally likely -- there
    is no reason to expect one to be chosen more often than the others --
    so the probability that the value will be in any particular one of
    them is 1/4.  We want to know what the probability is that the value
    will either be in the second or the third.  The probability for each
    of them is 1/4 and they are distinct (the value cannot be in both
    of them at the same time) so we can simply add the two together to
    get a probability of 1/2.

    We could do the same thing to find out the probability that value was
    between 3/8 and 5/8 and would find that the probability is 1/4, and
    so on.

    What we would find is that for this distribution is that for any
    region starting at zero or more and ending at or before one, the
    probability that the value is in that region is simply the width
    of that region (which can be found by subtracting the upper-bound
    from the lower-bound).

    We can mark off narrower and narrower regions around the value we
    are interested in (1/2) and as we do so the probability that the
    value will be in that region becomes less and less.  Finally, we
    will get to a region which contains *only* 1/2, but its width (and
    therefore the probability of it being selected) will be zero.

    So the probability of getting *exactly* 1/2 (or exactly any other
    value) is zero.  The same thing happens for other continuous
    distributions except not quite so neatly.  The probability of the
    value being in a particular region is proportional to the width
    of the region (along with being proportional to some more complex
    value) and to specify a *single* value the width is zero, and so
    the probability is zero.

    Now, that established, back to the original situation --

    The following statements are all at least plausible about at least
    some physical processes:

    1) Classical physics says that many underlying physical quantities
    are real-valued.  That is they can take on any arbitrary real value
    within some range.  Examples are time, position, direction, force,
    electrical field string at a position, gravitational force at a
    position, energy of a photon, etc.  Contrary to some descriptions,
    quantum mechanics does not contradict this -- it only says that
    for some of those quantities only certain discrete values will occur
    under specific conditions.  Note that these underlying physical
    quantities are not the same as our laboratory measurements of them.
    The latter, due to finite precision, can only take on values from
    within a finite set of rational values.

    2) Quantum mechanics says that a system can "jump" from one value
    for a physical quantity to another without ever having any of the
    intermediate values.

    3) Chaos theory says that the *exact* value of a quantity can be
    significant for a physical process.

    The following scenario is thus plausible (not likely, perhaps,
    but plausible, which is all that is needed to demonstrate an
    incompleteness in the previous proof.  To "plug the hole" it
    is necessary to show that this scenario is not plausible or not
    possible):

    The "archuniverse" within which ours was born can be described, at any
    given "archetime", by some number of quantities (its state), some of
    which are real-valued.  The archuniverse stays in a particular state
    for a random amount of archetime and then one or more of its
    state-variables jump to a new value, without going through any
    intermediate values.  A "universe" is created only when some set of
    these state-variables take on a specific value.

    The probability, then, of a universe being created is zero, even though
    it is not "impossible" for a universe to be created.

    As I said, the subtleties of statements about how likely it is that
    the universe will be *re*created given "infinite" amounts of archetime
    are beyond me.  My intuition is that the universe will be recreated
    over and over again (assuming that it being created once doesn't
    preclude its being created again) but intuition is a poor guide to
    infinity (gee, I should get a button made of that last statement,
    or perhaps have it put on my tombstone :-).

				Topher

    
    If memory serves me aright, the author of "A Subway Named Mobius"
    was Asimov.
    
    Also, there was an article in Scientic American within the past
    couple years on the Klatu-Klein* theory that "space" actually contains
    11 dimensions. The unseen dimensions fold up so tightly that they
    form "tubes" with subatomic radii and are undetectable.
    
    *not the "Gort! Baravada Nictu!" Klatu...I may not have the first
    of the two names right.

    Re .27:
    
    >If memory serves me aright, the author of "A Subway Named Mobius"
    >was Asimov.
     
    The name of the author eludes me at the moment, but it wasn't Isaac.
    The story is in Clifton Fadiman's anthology, _Mathematical Magpie_.
    
    >*not the "Gort! Baravada Nictu!" Klatu...I may not have the first
    >of the two names right.
     
    Actually, the quote's "Gort!  Klaatu verada nicto."  
    
    Now that all that literary stuff's out of the way ...
    
    Steve Kallis, Jr.

    Gee, I thought it was "Gort, Klaatu barada nikto."
    
    						Ann B.

    
    No, it's Gort from SKIING.
    
    Hi there!  How are you doing?  *<(8*)|
    
    Cindy

    I believe the 11-dimensional unification theory mentioned some time
    back is called "Kaluza-Klein," for the two people principally
    responsible for it.  The general idea is that space-time has eleven
    dimensions; four of these (normal space and time) are infinite or at
    least very large in extent; the other seven are curled up on themselves
    microscopically small.  The net effect is to associate a
    seven-dimensional ball, smaller than a proton, with every point in
    space and time.  This ball is the seven-dimensional cross-section of
    11-space at that point.  The exact shape of the ball determines the
    forces in operation at that point.  If more forces are discovered,
    we'll need more dimensions.  This theory and its variations is often
    combined with "string theory," according to which elementary particles
    are little loops of string, vibrating in different ways.  Adding seven
    additional dimensions gives them more ways to vibrate.
    
    Earl Wajenberg

    re.31 You rang? Hi Cindy.
    
    -j
    

                
    Re.-1 - Hi Jerry!
    
    Trip report:
    
    Just back from a week's vacation on the island of St. Lucia in the 
    Caribbean (another dimension where the sun actually shines 7
    consecutive days in a row....a rare occurrence in the northeastern
    US), and am just now catching up on the happenings in DEJAVUland.
                      
    Learned how to snorkel (what a WONDERFUL experience that was!) and 
    spent most of my days with my mask in the water exploring the reefs 
    nearby.  Utterly breathtaking.  Wish my camera were waterproof -
    the best pictures were to be had underneath the surface....however
    the rest of the island was beautiful as well and managed to shoot
    a couple of rolls of slide film anyway.
    
    Good to be back though......  Be seeing you!
    
    Cindy
                            

I haven't read through all the replies in this note, so excuse me if
I repeat something already said.  I read a very good book by a Russian
philosopher named P. D. Oudspensky called "The Fourth World, a New 
Model of the Universe".  In it he attempts to visualize the 4th
dimension.  He believes that it is not possible for us to do so
entirely.  He imagines that if there were beings living in a two
dimensional world they would not be able to percieve us in the three
dimensional world.  In his example, he assumes a line in a 2D plane
having eyes along it's width.  It encounters a circle, but cannot
percieve it as such.  It can move around it and get a feeling of a
circle but can't actually experience a circle.  He goes on to suggest
that this circle could actualy be a sphere that intersects this beings
2D plane creating the image of a circle that it can only slightly intuit.

Could it be we are forms of a higher 4th dimensional plane and all we can 
really know about who/what we are is that part that intersects 3D reality?

Any thoughts?

Terry

    
    How about this?  suppose we are in the third dimention, but this
    dimention is actually a dream from the fourth dimention.  And when we
    go to sleep at night, and dream, we move down one more level into the
    second dimention.  Then when we awaken... back to 3D.
    
    We would return to the fourth dimention only when we wake up from this
    crazy "dream" we live in now -- in other words, die.
    
    Then we exist in that plane for a time, call it heaven if you will,
    eventually get a new life and body to live in, and its a long fourth
    dimentional sleep for us.  A new lifetime here on Earth (or some planet
    like ours where physical laws must be obeyed.)
    
    After so many of these lives, (dreams from the 4th dim.) we die (or
    wake up depending on how you look at it) and return to the 5th
    dimention.  Kind of like nested subroutines.
    
    Maybe this could continue on for infinity, or maybe we'd reach an upper
    dimentional limit from which this all started from, and realize that we
    are God!
    
    Oh well, Ive wasted enough peoples time with this rubbish... Just a
    trippy thought!
    
    Ken
    
    in, and

Note 13.36                      
KRAPPA::ALI                                          

    >Kind of like nested subroutines.

Good thinking Ken.  
Have you ever seen 'fractals'?  Fractal images are a result of 
repetitive mathematical computation and are used to model phenomena in fields
ranging from astronomy to zoology.  What if the fractals represent places or
dimensions which one could freely enter or leave at will.
    
Mary

    From what I understand, there are 4 demenations.  1st. physical length
    2nd physical width, 3rd physical depth. Everything consist of these
    first three.  The 4th is time, inwhich everything exist.

    Re .38 (Swenson)
    
    >From what I understand, there are 4 demenations.  1st. physical length
    >2nd physical width, 3rd physical depth. Everything consist of these
    >first three.  The 4th is time, in which everything exist.
    
    In .2 of this string, I pointed out:
        
>1) Any measurement can be a "dimension."  That includes temperature, capa-
>citance, color, brightness, etc.  "Time" is but one "fourth" dimension.
>(there are more than five "senses" too.)
>
>2) Any model that can help you perceive something is useful.  But confusing
>the model with reality can be dangerous. ....
                                       
    That was/is a little simplistic, but it gets the basic idea across.
    
    Steve Kallis, Jr.

    dimensional theorizing can be obfuscating. If a 3 dimensional being
    enters a 2 dimensional universe only one plane at a time will appear.
    If the being has a complex shape it will change form as it passes
    through 2 dimensional universe. It can also disappear completely
    from the stand point of the 2 dimensional observers. If we extend
    the analogy to 3 and 4 dimensions what direction does one look that
    is perpendicular to the 3 familiar dimensions? Throughout my life
    my culture has presented to me hundreds of stories of beings termed
    angels. In most cases they appeared and behaved in a fashion that
    provided little evidence that they were not human. In many cases
    the main evidence people had that they were nonhuman was the manner
    of their disappearance. People would glance away and the visitor
    was gone. In one case a invalid was brought wood during the alaskan
    winter. When she looked out the window the snow on her wood pile  was
    untouched and the tracks ended a few feet from her door. It would
    be outside the scope of this discussion to deal with the validity
    of these accounts. I thought to bring them up in the context of
    dimensions. In almost every case the beings did not allow themselves
    to be obseved while vanishing. Perhaps they merely moved slightly
    in the 4th dimension passing beyond what we can see. If humans have
    the capacity to see in the direction of the 4th dimension, disappearing
    while being whatched might train a human in how to see all the time
    in this way. In the bible Elisha was promised a blessing if he was
    present when Elijah vanished. Apparently ever after he could see.
    At one point when the Syrian armies surrounded Samaria, he prayed
    that his servants(student) eyes would be opened so he could see
    the host of protectors in some spiritual plane.
    
    	Also Ezekial's semi incomprehensible description of the Merkava
    with its 4 aspect creatures might be an attempt to render in 3
    dimensions something existing in 4.
    
    	Just as 3 D's infinitely multiply the possibilities of 2 D's,
    a 4th would do the same to 3. If our intellect, morals and training
    are inadequate for 3 D's probably insanity would be the result of
    making the jump prematurely to 4
    
    					john turner

    
    Perhaps the astral plane is the 4th dimension?  
    
    One book I was reading the other night stated that in Atlantean times,
    the astral plane was revealed but not understood.
    
    Cindy

    
    
    RE 13.40
    
                    In his book _A_New_Model_of_the_Universe_ (ISBN394-
    71524-1, Vintage Books) P.D. Ouspensky spends some time (39 pages) 
    discussing the Fourth Dimension. Fascinating reading. The entire book
    is a collection of his essays translated from Russian and deals with
    many of the topics found in this file.

    I'd be interested in reading that book.  I'll pick it up. 
    
    Thanks,
    
    Mary