textfiles/magazines/HOE/hoe-0255.txt

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>> A Crackpot Mathematician Analyzes The Fifth Dimension IN Layman's Terms <<
                              by -> Swiss Pope

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     Fellow colleagues, I have startled upon an important discovery.
In order to elaborate, one must have an elementary understanding the
principles involved.

 First, let me take you on a quick tour of the dimensions known to man.

        DIMENSION ZERO.  It's simply a point.  This point represents an
empty set {} because it is the smallest possible unit we can imagine,
and you can't very well put anything in the smallest set imaginable
because everything else would be much bigger.

 [This is why we can't divide by zero.  You can't split something
  up into nothing.]

                  Illustration:    *


        DIMENSION ONE.  This can be contained in a segment which connects
two points called vertices.  The vertices are denoted by a star (*).

                  Illustration: *-----*

        We may represent this dimension as a linear array.  It can be as
big as you want or as small as you want.  For the purposes of
explaining, I will assume that this set has a finite number of elements.
Each finite element is a point, which we stated above.  Suppose we twist
around our illustration to look like this: *******, since a line is
merely a collection of points.  Therefore, we may define our dimension
as a set of, in this case, 7 elements.  Our set looks like this:

                          { 1, 2, 3, 4, 5, 6, 7 }

In mathematical concept, a line is an infinite collection of points and
assuming that each point (as a unit) exists, we shall say this first
dimension is a set of all of the integers:

                 { -Infinity, ..., 0, ..., +Infinity }

 [Note: We can save talking about irrational numbers for another day,
        because you can't count an infinite number of irrational numbers,
        whereas you can count an infinite number of integers.  You
        can count an infinite number of rational numbers, too, but
        I don't even need to bother talking about rational numbers
        because if each element is the smallest point you can imagine,
        to take fractions would be ludicrous.]


        DIMENSION TWO.  This is a plane.  How do we draw this?  We can't
very well draw a true plane, which in theory would have Infinity^2
points, but we can draw a portion of it.  How many vertices will it have?
Well, the line had 2, so this one will have 2^2 = 4.

                 Illustration:  *-----*
                                |     |
                                |     |
                                |     |
                                |     |
                                |     |
                                *-----*

        So, we can bound this particular portion of a plane into a collection of points by saying that
by saying that our 2-dimensional plane begins at the upper left hand
corner (1,1) and ends at the lower right hand corner (7,7).  Notice that
this portion takes _two_ elements from the first dimension to define
each element in the two dimensional set.  These elements are called ordered
pairs.

                  { (1,1), (1,2), ..., (7,6), (7,7) }

                  This set might also be described as two one
                  dimensional sets if you pretend that

                        { 1, 2, 3, 4, 5, 6, 7 }

        Here, each number 1, ..., 7 really corresponds to a line instead
of a point, and that line (1) would consist of points 1, ..., 7.

        Indeed, no further explanation is necessary.

        DIMENSION THREE.  I need not draw a diagram.  You need only
imagine a cube.  But as you can reason, a three dimensional set of
points would consist of three numbers in the ordered pair, like so.
Sticking with the idea that we are bounding a portion of three
dimensional space that starts with 1 and ends with 7, we show a three
dimensional set like so:

              { (1,1,1), (1,1,2), ..., (7,7,6), (7,7,7) }

        Again, each ordered pair represents a singular point.

        DIMENSION FOUR.  Einstein suggested that the fourth dimension is
time, but really a dimension can be anything you want it to be.  Our
assignment of the dimensions is arbitrary to logicians, yet of utmost
importance to physicists who attempt to describe our natural world.
Logicians describe a fourth dimensional object as a hypercube.  If a
line segment has 2 vertices, a square possesses 4, and a cube possesses
8, then a hybercube possesses 16.  Do not attempt to visualize such as
an object, as your weak minds will be damaged by the strain.  Only I,
through deep zazen meditiation, have been able to picture this new
wonder of mathematics.

        I will now attempt to apply these fascinating concepts to
describing our natural world.

        If you could freeze time, you could cut just about anything up
into three dimensional space and describe the contains therein by saying
that that content is a function of the three variables in the ordered
pair that describe the three dimensional set.

        For instance, cartographers map the world by dividing it up into
sections made up of degrees, minutes, seconds.  From anywhere on the
earth's surface, you can describe your location by using this system.

        In actuality the coordinates that are used in maps have
direction, making them vectors, and it would be quite complicated
to describe our theoretical section of space using this system.  So,
we're going to play the role of the mapmaker and develop our own bounded
coordinate system for a portion of three dimensional space.

        If you have been paying close attention, you might say that if a
three dimensional set describes an array of two dimensional sets, and a
two dimensional set describes an array of one dimensional sets, and a
one dimensional set contains of points, which according to the previous
definition, are really empty sets which contain nothing, then a three
dimensional set must just be an overcomplicated way of describing a
whole lot of nothing.  You couldn't be any more wrong, as I have
outsmarted you.  The idea that I now unravel is that dimension zero may
not only consist of _nothing_ but it may consist of _anything_.

        Let's assume that my office is the center of our coordinate
system, and each unit of space consists of a cubic foot.  So, the center
of my office will be at (0,0,0), and a cubic foot of space atop my desk
will exist at (3,-6,-2).  Anyone who has constructed a two dimensional
plot using the Cartesian coordinate system knows that there are two
axes: x and y.  y is a function of x, i.e., y=f(x).  Suppose set S
consists of whatever is in this cubic foot of space above my desk.
S, at a singular instant in time, will be a function of x, y, z.  In
this case, S = f(3,-6,-2).  S may contain anything I desire.  S is a
subset of _everything_ in the universe.  Here, S = { pencil, paper, cup }.
But, that space might not _always_ contain those items (elements).
Thus, we introduce the fourth dimension. Suppose that time began 10
minutes ago, and that a singular instant in time is an earth minute (the
smallest unit of time in my universe). Therefore, S = f(3,-6,-2,10).  If
this seems imprecise, we may approximate units of space and time to
whatever value we please.  As those units of space and time get smaller,
so does the set of everything in the universe.  We might suggest that x,
y, and z are all lengths of an atom, narrowing the set of everything in
the universe to the number of elements on a periodic table.  You could
narrow even further when considering subatomic particles.

        So what, then, do you ask, if the fifth dimension?  How does one
cube time?  This is a philosophical argument in itself and assumes that
the fourth dimension as time.

        If the fourth dimension is the linear progression of time, i.e.,
what happens, then the fifth dimension must be what _might_ happen.
When you consider that three dimensional space is a countably infinite
way of describing points in all known directions, and four dimensional
hyperspace is where those points exist in the countablely infinite time
continuum, then fifth dimensional space must be the time continuum in
an infinite number of directions-- probability.

        Probability is entirely dependent on the number of
possibilities.  We determine the number of possibilities based on what
_exists_, and if we have narrowed dimension zero down to containing the
set of everything in the universe (say, a finite number of elements),
then the fifth dimension will be entirely dependent upon what exists in
dimension zero.  In other words, if you say that the set that exists in
space (1,5,3) can be Hydrogen through Uranium, then a fifth dimensional
variable to describe what _may_ exist in space and time would be
finitely bounded by the set of all natural elements: its cardinality
would be that of the set of natural elements.

        In conclusion, I shall note that I have not gotten laid in a
very, very long time.

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   * (c) HoE publications.  HoE #255 -- written by Swiss Pope -- 7/15/98 *