# some thoughts I considered for book 3 of possibility thinking explorations in logic and thought

These are some thoughts that I considered for book 3
of possibility thinking explorations in logic and thought and
many of them are probably flawed so the burden of understanding
lies entirely on the reader and gossip is not allowed.
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This is an unfinished writing and I disclaim all liability.
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Book III:
Math Ideas:
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Infinity
Something that is infinite in one context may be finite in another
context. For the re-definition of "infinite" is something that goes on
forever along the dimensional framework of a given context. But once new
axioms are applied to the context where that something went on forever,
the context is changed, and thus so the definition of many if not all
things that existed in the former context, and in many cases infinite
objects may become quitet definable (finite).
---------------------------------- 3/2/2005 update by Justin Coslor
Also, it's important to not that perspectives changes (such as
recontexualizations), may come with different axiom sets than the
original context. 10/19/2004 Justin Coslor Public Domain, free for well
intended use only. The upper limits of NP-Completeness Polynomial time
computations' upper limit can be described by saying "infinity^x", and
that has finitely many dimensions of context, but infinite scope along
those dimensions. Non-polynomial time computations can be described by
"x^infinity", and that has finite scope, but infinitely many dimensions
of context. As you can see, cannot exactly equal np, however, it can
approximate an incomplete abstract summary of some parts of np, using
part of p's scope. This is because exponents stand for the number of
perpendicular or symmetrical dimensions that the variable exists in. So
saying that p=np is like trying to say that infinity^x=x^infinity, which
it clearly is not; but p can be composed of a selection of np's
dimensions, as long as they have a common base for forming selective
perspective. Copyright 10/17/2004 Justin Coslor Qualifying & Quantifying
Dimensionality In equations such as AnX^n + A{n-1}X^n-1 + . . . + A1X +
A0 = 0, the coefficients (An to A1) can be considered to be quantifiers,
and X^n to X or Y's etc, can be considered to be qualitative variables.
When the variables X, Y, etc have exponents or are multiplied
together, each combination of exponents and variables defines the
dimensionality of the planes that the equation is holding in relation to
one another, and the coefficients define the size or length or quantity
or magnitude of each dimensional/qualitative structure in the equation
that is held in relation to each other dimensional/qualitative structure
in the equation. Now some dimensional structures are best described by
equations that have more than two sides to the equal sign, such as those
that exist on higher prime and prime composite levels of balance than
most of current mathematics is based on. So we can only approximate
descriptions of those structures in a duality format if at all.
I guess a computer array or database or arrays of arrays can be used
to depict higher dimensionality, but past a certain number of dimensions
it surpasses the human brain's neuro- hardware's ability to visualize
the relations and dimensional complexity. Arrays can be used to list out
infinitely many dimensions categorically and quantitatively. However,
nobody as of yet has discovered a way to think of a way to bound the
classification of objects or situations using more than two extremes,
using dualities such as maximum and minimum to balance an equation.
Triality, or quintality, etc, along the prime numbers may indeed be
possible, though our brains don't seem to interpret the universe along
those dimensions as of yet. Perhaps eventually we will learn to adapt
higher logical foundations. Copyright 10/4/2004 Justin Coslor Spirals
(See the photos of the pictures depicted by this text on this date.) A
number that has exponents contains one perpendicular or symmetrical
dimension per exponent , so f^5 in this equation might look something
like the multidimensional picture of a spiral within a spiral within a
spiral within a spiral within a spiral. This is how my math invention
dimensionality (see journal entry dated 7/10/2004 Justin Coslor). The
line going through the center of the spiral might actually be a spiral,
a circle, an elliptical loop, a curve, or some combination of those.
This kind of visual notation ("Exponential Sine Spiral Graphing" I call
it) can be used in conjunction with conical orbit graphing I call it)
can be used in conjuntion with conical orbit graphing to simplify the
interaction visualizations of multiple spinning and/or orbiting bodies
that have at least one plane of rotation in common.
------------------------------ Update: Copyright 2/10/2005 Justin Coslor
The optimal structure of nanotechnology parallel-processing
supercomputer memory structure could be something like this f^5
composite exponential spiral, except with ribbons of memory units and
have vertical pipelines interconnecting each exponential layer of the
composite spiral, and have a brick made out of short columns of these
f^4 or f^5 or f^n spirals that are laterally connected on the ends of
each column and stack multiple columns on top of each other in sheets of
intensely interconnected spirals, like slices of a tree trunk. 7/10/2004
by Justin Coslor Light Spirals
For several weeks now I have believed that light (and other
emissions of convecting energy) particles/packets/quanta travel not in
waves, but in spirals and flocks of spirals. I came to this conclusion
after figuring out how to visualize Balmer's frequency equation (the one
with the Rydberg constant and electron shell radiuses: f=R(1/Nf^2 -
1/Ni^2) where Nf is the outermost shell and Ni is the initial shell) in
terms of sine-spiral graphing (Sine-spiral graphing is something of my
own invention, and is a 3D resentation of circular motion, where the
sine-waves or cosine waves represented for all points in time as a
spiral (cosine of a point is X, sine at that point is Y, and time at
that point is Z in the 3D coordinate system....remember the unit
circle?) through time (or through a 3rd dimension if time is irrelevant
or instantaneous or if motion is uniform)). See pg 67 of the comic book
"Introducing Quantum Theory" by J.P. McEvoy and Oscar Zarate - Copyright
1996 (2003 reprint) ISBN: 1840460571 for Balmer's frequency equation.
*Note: Waveforms only look like that from a perpendicular side-view, and
I think this because, interestingly enough, 3D spirals look exactly like
that when they are looked at from a perpendicular side view, which
essentially is a 2D perspective. That is part of the basis of my
sine-spiral graphing methods (I came up with the math for it when I got
way behind in 10th grade Math-Analysis class). 7/11/2004 update by
Justin Coslor Light Spirals Continued
To visualize it I juggled the equation around a little, and figured
out the intent that went into creating the algorithm. In Nf^2 and Ni^2,
f^2 and i^2 just means that the variable f exists in a two- dimensional
plane where one f axis is perpendicular or symmetrical to every other
variable in the composite of the multiplicative parts; and when numbers
or values get plugged into those variables, the visualization depicts a
specific graph within the context of that combined dimensionality. That
is why multiplication is used in algorithms to combine variables that
are proportional to each other. *Multiplication shows that they have a
proportional relationship. **Multiplication can also show that
variables' dimensionality can share the same space, by perceiving of
them in the broader context of their dimensions' combined contexts
(whether it be symbolic, semantic, algebraic, or geometrical). ***One
variable=1 dimensional representation. Two variables=2 dimensional
representation. Three variables=3D . . . There is a limit to our neuro
hardware's dimensional ability. ****If a variable is squared it exists
within a two-dimensional context, if it is cubed, it exists within a 3D
context, etc. Copyright 5/6/1997 Justin Coslor Sine Spiral Graphing
A new method of graphing motion called "Sine Spiral Graphing" was
developed by me when I was 16. It allows for simultaneously graphing the
sine and cosine curves of an object in motion, three-dimensionally. Sine
and cosine, when graphed simultaneously in two dimensions, look like two
staggered intersecting waves traveling in the same general direction.
(Fig. 1) There has been a need for developing better methods of graphing
an object's two-dimensional (flat) motion through space over a period of
time that more clearly shows the progression of travel. At present,
mapping three-dimensional motion using different variables is more
complicated, but could be a further application of the principles
presented in the "Sine Spiral Graphing" method. The "Sine Spiral" is
based on the spiral shape of two-dimensional circular motion graphed in
three dimensions using this new graphing technique. The name is derived
from the general name of the sine wave combined with what the actual 3D
graph looks like: a spiral. This technique could be helpful for
scientists and students alike in many applications. Some possible
application for the Sine Spiral could be: - Plotting the motion of a
bead in a hula hoop as it spins around one's waist. - Calculating the
position of various atomic/subatomic particles moving in relation to
each other over time. - Plotting the velocity and position of a point on
an automobile wheel as sit spins down a runway or curvy hilly road. -
Plotting the motion of a baseball spinning through the air as it travels
forward to the catcher over a period of time. - Calculating the motion
of a point on a bowling ball as it rolls down the lane over time. -
Calculating the speeds and positions of a set of points, on various
gears at work, in a clock in relation to each other over time. -
Calculating the motion of a point on a rocket ship, or of a point on a
space satellite as it orbits a planet. - Plotting the movement of a
All of these examples listed present graphing difficulties when
depicted on a normal graph. The motions in these examples could be
calculated on a computer and represented in a simulated fashion to show
the actual movement in space for one point in time at a time. Concurrent
Sine Spiral graphs can also be drawn for comparison of points on
multiple moving objects. However, it would be difficult to graphically
represent these motions for all points in time all at once. A simulation
could be like a video, where one can only view one place on the video at
a time. Viewing forward and reverse at the same time is not logistically
possible on a video. However, when motion is three-dimensionally graphed
on a computer using a Sine Spiral, it is possible to view these motions
for all points in time all at once. A very effective way to manipulate
and browse three-dimensional graphs (such as a Sine Spiral) on a
computer is with Virtual Reality equipment. With Virtual Reality
equipment, the perspective of the viewer can freely move around in space
(on the graph) and see the 3D objects in one's graph from any
perspective. In a Virtual Reality graph, the user can have total control
over what is viewed and how it is viewed.
Understanding the trigonometric functions of sine, cosine, tangent,
and their inverse counterparts is a necessity for understanding Sine
Spiral Graphing. Trigonometric functions of real numbers, called
"Circular Functions" (or Wrapping Functions), can be defined in terms of
the coordinates of points on the unit circle with the equation x^2+y^2=1
having its center at the origin and a radius of 1. (Fig. 2)
There are three elements in a two-dimensional trigonometric
function: the angle of rotation (sigma), the radius of the rotation r,
and the (x,y) position of the point at that angle and radius. As can be
seen in Figure 3, the x and y portions of the graph are always
perpendicular to each other. Thus a right triangle is formed between the
x, y, and radius sides. Right triangle rules can therefore be applied to
this point in space (Brown/Robbins 190).
Such trigonometric functions as sine and cosine can be applied to
the triangle formed by rotation. These functions, sin and cos, are of
fundamental importance in all branches of mathematics. One can use
points other than those on the unit circle to find values of the sine
and cosine functions. (Fig. 4) If a point Q has coordinates (x,y), and
it is at angle sigma in reference to the origin, (cos sigma) = x/r and
(sin sigma) = y/r. To obtain a rough sketch of a sine wave, plot the
points (t, sin t) (Fig. 5), then draw a smooth curve through them, and
extend the configuration to the right and left in periodic fashion. This
gives the portion of the graph shown in Figure 5 (Swolowski 78).
A cosine can be graphed in the same fashion by simply shifting the
graph 90 degrees to the right. (Fig. 6) An object's circular motion can
be described by either a sine wave or a cosine graphed in the same
fashion. Such a wave is composed of the object's radius of rotation and
the vperiod (number of degrees in on cycle) per unit of time that it
rotates. Seeing an object's sine and cosine graph simultaneously greatly
helps in visualizing the object's motion analytically compared how it
found in real life. Watching an animation of an object spinning is the
same as seeing the x and y coordinates (cosine and sine) of the object
for each frame of the animation, one frame at a time. This is because
one could see a scale view of its whole two-dimensional motion over a
period of time. Visualizing an object's true motion in nature from
merely looking at a graph of its sine or cosine can be difficult to
conceptualize. For this reason, the Sine Spiral may be an improvement in
current co-linear graphing (Fig. 7).
Velocity over the period of one rotation on a sine curve can be
measured by dividing the distance traveled in one rotation by the amount
of time it takes to complete that one rotation. Velocity = change in
distance/change in time + direction.
Any change in velocity (a change in time) will change the distance
between peaks of the spiral. The whole Z-axis around which the spiral
revolves represents time passed. When the velocity is constant, the
distance from peak to peak in the spiral is constant or each distance
from one peak to another peak is the same. (Fig. 6) Therefore, if the
distance from one peak to another changes somewhere in the spiral, this
indicates that the velocity has changed at that point in time.
Within the Sine Spiral, some of the variables that can change in the
object's motion are velocity, radius of rotation, position of axis of
revolution, and the scale upon which measurements are based. The shape
of this spiral is an indication of any and all of these variables. The
change in the shape of the spiral correlates to the change in one or
more of these variables. (Fig. 7)
Webster's Third New International Dictionary defines a spiral as "A
three-dimensional curve (as a helix) with one or more turns around and
axis." In current circular motion, the sine of the angle of rotation
provides a Y value (Sine=Y/Radius of Rotation), while the cosine of that
same angle provides and X value (Cosine=X/Radius of Rotation). These X
and Y values are all that is needed to draw the two-dimensional models
of rotation known as the sine curve and the two- dimensional models of
rotation known as the sine curve and cosine curve (or sine wave). To my
knowledge it has not been thought possible to graph this same motion in
three dimensions though, because one needs an X, Y, and Z coordinate in
order to graph in 3D. There can be an X and Y coordinate by finding the
sine and cosine of a unit circle. All that is needed is a Z coordinate
to make the circular motion graphable in three dimensions.
That Z coordinate could be representable by time, or speed of
rotation, or even the period of degrees it takes for one complete
rotation. In a sine wave, the period is 360 degrees. Using the period of
degrees in one rotation, one can find a constantly increasing Z
coordinate by dividing the current number of degrees traveled by the
period of degrees it takes to complete one rotation. In short,
degrees/period. The period can be depicted by a set amount of time.
Finding a ratio between something that can be used as a reference point
(one second vs the number of degrees in one rotation) to one's current
progress in that measurement scale (number of seconds that have passed
vs number of degrees that have been traveled) determine where one is on
the Z- axis.
By dividing one's progress by a predetermined scale of reference, a
new dimension can be generated in which to plot on a graph in order to
illustrate this in three-dimensional fashion. This new dimension can be
called the "Z-axis". Now that there is an X, Y, and Z dimension
available, a three-dimensional model of an object's progress through its
path of circular motion is possible.
For 3D motion, one can draw three spirals over the same T axis and
where two of the spirals intersect, plot a point. Connecting the dots
between the points gives one a tri-spiral (a spiral or shape that
represents 3D motion over time). One can continue plotting the points
with several objects and where the tri-spirals intersect, the objects
intersect. One can break down the tri-spiral to find out where the X, Y,
and Z coordinates are in space and the time coordinates of the
intersection.
To use the Sine Spiral to map the 3D motion throughout time, one
could mark the spiral with tags (or color code it) that tell one when
and how far down the Z-axis it travels. Then to graph several objects to
compare their motions and positions to each other, one can have a
computer draw lines of the same color of the Z-tag, linking all of the
objects that intersect on the two planes like the ZX plane, or the ZY
plane. That way, one could identify when objects like planets line up on
a plane or intersect.
There is much to benefit from in being able to graph an object's
progress at the same time as its position in space. One can see time
from an outside perspective and also see how an object's motion,
position, and speed relate to any point in time. In many circumstances,
it may be very useful to finally be able to get to see the general shape
of an object's travel through all points in time all at once. This new
method of graphing circular motion in three dimensions is the "Sine
Spiral".
The graph forms a regularly spaced spiral whose axis is a straight
line equidistant from the perimeter of the spiral. Changing the radius
of rotation around a center axis changes the radius of the spiral around
the Z axis. Changing the center of rotation in two-dimensional space (X,
Y coordinates) makes the Z axis of the sign spiral curve up, down, or to
the sides when graphed (instead of the normal straight line Z-axis).
For instance, an air hockey puck pinning in place would have a
regular sign spiral that represents a point on the puck's perimeter that
is traveling in a circle. Now if the spinning puck were to be slid
across an air hockey table, that same point (on the perimeter of the
puck) would have an irregular sine spiral whose radius would be
constant, but the Z-axis around which the graph spirals would
instantaneously bend at a ninety degree angle.
A computer can easily generate this three-dimensional picture of an
object "N" at point "T" in time if the speed of travel is irregular (or
at the ratio of degrees traveled to the period of one complete rotation
if the speed is constant). (Fig. 8)
Graphing any two-dimensional motion (motion that moves in any
direction on a flat plane), or rotation in three-dimensions using time
or progress as the third dimension allows one to look at time from an
outside perspective. The Sine Spiral can be used to graph any such
two-dimensional motion, or any number of combinations of such motion. It
can be used to graph several objects moving around in 2D (flat) space on
the same plane. The Sine Spiral can be used to graph an object which has
a rotation within a rotation, and so on (Fig. 9). In this case, each
next level of rotation is on an incrementally larger scale. To view some
of the higher levels of rotation, one must graph the object's motion
over a longer period of time. This concept can relate to complex motions
of a longer period of time. This concept can relate to complex motions
of a large scale found in, for example, the universe. Sine Spiral
graphing can literally be used to graph the motion of every particle in
perceivable universe for all points in observable time, simultaneously
(by bending the Z-axis appropriately to accommodate changes is axis
orientation). Using the Sine Spiral, graphing motion in the Z-axis, or
time, requires one to employ a means to mark or reference the Sine
Spiral in order to distinguish how deep down the Z-axis the motion has
traveled.
Without a Sine Spiral, one can only pick three-dimensions to see on
a graph for all points in those dimensions. One could have X, Y, and Z
coordinates on a 3D graph all at once, but only for one point in time
per graph. Or one could illustrate motion in any two dimensions for all
points in time using the Sine Spiral. Here are some of the dimensions
from which one can choose: X, Y, Z, and Time. One can have four or more
dimensions on a graph by selecting 3 variables form out of the X, Y, Z,
and Time, as well as any number of descriptive, qualitative,
categorical, computational, or other quantitative dimensions. These
kinds of dimensions may appeal/apply to one's senses and could be
described in "real" dimensions such as the Z-axis and others.
With 3D applications using this concept (once improved methods of
graphing 3D motion with the sine spirals are better developed), other
more complex spirals can be mapped. Such 3D applications could include
the universe in their motion through space throughout all time to see
where certain ones meet or line up), and graphing the motion of
particles of a sun during a supernova (the spiral would look similar to
a tangent spiral as described below). The Sine Spiral may be an
improvement in the graphing of nonlinear and linear motion. With the
help of the recent Virtual Reality technology, most any computer can be
used to build 3D models such as Sine Spirals. We can construct and view
a Sine Spiral and have complete control over the graph, viewing it in 3D
space as if it were physically here.
There are many new math applications and theorems that may apply to
this concept. Different types of spirals are possible with the general
Sine Spiral method. Such shapes could include the Sine Tube (a sine
spiral whose period is infinitely small), the Tangent Spiral (which uses
a sine spiral whose period is infinitely small), the Tangent Spiral
(uses the equation Tan sigma = (y/r)/ (x/r) for the x and y
coordinates), and the secant spiral (uses sec sigma = 1/(x/r) for the
coordinate and csc sigma = 1/(y/r) for the y coordinate). Also, in
either two-dimensional or three-dimensional motion (when a graphing
method is available), an object can be spinning in a circle within a
circle (each level of rotation incrementally bigger than the previous),
and this will make a very special type of Sine Spiral that looks like a
spiral within a spiral within a spiral, etc., depending on how many
levels of rotation are going on. More new math applications ar sure to
be found that can apply to the Sine Spiral as it is used.
Graphing three-dimensional motion with the Sine Spiral is more
difficult to do, but can be done effectively. Graphing three-dimensional
motion using the Sine Spiral needs further refinement at this time, but
will hopefully be available for use in the near future. There are many
new avenues that open up as people figure things out in science and
math. The Sine Spiral may be another door in mathematics ready to be
opened up and entered. Through this door may be a whole new way to look
at things, a way to see objects in nonlinear motion from a standpoint
outside of time. ---------------------------------- Works Cited: Brown
R., and D. Robbins, "Advanced Mathematics: A Precalculus Course"
Boston: Houghton Mifflin Company 1987. Fleenor C., M. Shanks, and C.
Brumfiel. "The Elementary Functions".
Boston: Addison-Wesley Publishing Company, 1973. Gove, P.B., ed.
"Webster's Third New International Dictionary, Unabridged".
Springfield, MA: Miriam-Webster, 1986. Manougian, M.N. "Trigonometry
with Applications".
Tampa, FL: Mariner Publishing Co., 1980. Swokowski, E.W.
"Fundamentals of Trigonometry".
Boston: Prindle, Weber & Schmidt, Incorporated, 1982.
Conical Satellite Orbit Graphing (See Diagrams dated 10/4/2004,
3/1/2004, and 9/15/2001)
I do think the conical satellite orbit graphing idea I thought of in
winter 2001 (or the year before) could still be something valuable in
detecting and calculating collisions and for 3D space junk detection.
It's based on the hypothesis that if you compress a half-sphere into the
shape of a cone, the 180 degree arcs become straight lines, and straight
lines are easier to represent, interpret depth of, and run calculations
on than arcs. Elliptical orbits would just re straight lines at an
angle, each line representing the orbital path of an object in space.
Where two or more lines intersect, a collision is possible at the point
by either accelerating or de-accelerating the objects that the lines
represent.
Each object in a hemisphere cone is represented by a maximum and
minimum altitude, and an angle representing the direction in which the
object is traveling. There is one cone for each hemisphere. The neat
thing about the conical format is that you can see how a bunch of
objects, traveling in different directions at various altitudes, stack
up along a common line of altitude protruding through the center of the
planet, sun, moon, atom, galaxy, etc, and you can see how this line of
altitude intersects each of those objects at two points in time (one for
each hypersphere cone), along their various paths of travel.
Conical orbit graphing is a way to group a set of satellites (or
other objects in orbit) by a single line protruding through the center
of the central mass out into space (with a longitude and latitude
coordinate for each hemisphere from which the line emerges). All sorts
of nifty computer software functions can be incorporated into this as
well, such as having a 2D map of the central mass (such as a planetary
map or electron orbital map) as a clickable image map that generates a
unique pair of orbit cones for each coordinate (one for each hemisphere
of the hypersphere for objects traveling 360 degrees or more around the
planet). It would have a timing component as well and can be used as a
multi-body gravitational clock, viewable with virtual reality equipment
or a regular computer. There can also be a range component that
highlights any possible collisions within a certain proximity of the
satellites in focus (the satellites that intersect a common axis of
altitude, have one pair of cones for each axis of altitude). The user
should also be able to zoom in and out, rotate the cones, focus on
different axis' of altitude, combine complex orbits with sine- spiral
graphing techniques (see my paper on that), and watch the satellites
travel along their path lines in real-time (at an adjustable rate) using
live or recorded data collected from sensors and observational
equipment. It would help if most modern satellites were equipped to
detect space junk and satellites around them and relay it back to the
ground so that the world has a constantly updated fairly accurate map of
all of the objects and space junk in orbit around the earth, since space
flight has been compared to flying through a high-speed shooting
gallery. Ideally, some kind of Star Trek-like/Tesla
Wardencliffe-tower-like shields or something are needed for the safety
of that hazard (but not for use as a weapon), but a good 3D navigational
map can't hurt. For each satellite the computer can run a conical orbit
graphing collision detection test for each point in time along it's
projected path of travel.
The main use of conical orbit graphing as I see it, is for detecting
collisions at points along a line of altitude, using one pair of cones
for each point in time (or as a 3D interactive video). The user should
be able to pick a time and x-y coordinate, see the satellites that
intersect that line of altitude, then zoom in on the part of the path of
the satellite that they are interested in, then click on a point in that
path, and a new set of cones will be generated using that point as an
altitude line in the center of the cones so that you can then see what
possible collisions and path intersections there are for that point in
the satellite flight path-time. All as straight lines so that it's
easier to comprehend in complex situations. The computer calculations
might even be quicker than calculating arcs. I'd assume elliptical arcs
to be the most computationally intensive using traditional methods, but
they too could probably be represented as straight lines in the software
(going diagonally across the cones from one height to another height,
and then the opposite for the other cone). It would be a 3D software
tool for visualization and collision-interception calculation (and might
be able to help protect all countries from incoming intercontinental
thermonuclear ballistic missiles by combing this visualization method
with a ground-based or space-based or airplane-based or reusable
non-offensive missile based laser/maser anti-ballistic missile defense
system. There might be many other beneficial uses for this visualization
method that I haven't thought of yet (such as charting asteroids around
Saturn or something; though hopefully it won't ever be used for, or even
be useful for offensive purposes of any kind).. I haven't written any of
the code yet or figured out much of the math yet to make it possible
yet. Scholarly help is encouraged. Copyright 8/28/2005 Justin Coslor
Applications of Conical Hyperhemisphere Graphing When Combined With Sine
Spiral Graphing (See my papers dated 5/6/1997 and 6/28/2003.)
A Sine Spiral graph can be used to depict how an object rotates in N
dimensions as it moves from point to point in time (as though it were
rotating in place through time without actually traveling forward along
a path). Then those time coordinates can be linked to a conical orbit
graph of the distance vectors that the object moves through along its
path (or use a 3D Cartesian Coordinate grid of its path if it isn't
going to travel a full orbit around the planet...or not...). This
combination of graphing techniques works regardless of whether the
object is below, on, or above the surface of the Earth, or other center
of mass in space. For instance it could be used for mapping the path of
a vessel that goes from under the ocean, up into the sky, and out into
space into an orbit around the moon or something. ***************** Each
layer of the hyperhemisphere cone is a polar grid of a different
altitude. Elliptical and circular orbits are represented as straight
lines going across a pair of cones and intersect with an axis of
altitude line that goes vertically through the center of each cone,
where the axis of altitude represents a line going through the center of
the planet and out both sides into space. Elliptical orbits go
diagonally across the cones in this fashion from one altitude to
another, and back the opposite way in the cone that represents the other
half of the hyperhemisphere. Circular orbits go straight across the
cones at whatever altitude and declination they happen to be on.
*****************
Space stations could use these mapping techniques to coordinate
their motion and to dock incoming spacecraft, and it could be useful for
spaceship navigation and satellite positioning, coordination, and
communication routing too. Navy submarines could use these sine spiral +
conical hyperhemisphere (or sine spiral + Cartesian or polar) graphs
when planning and plotting routes through the oceans of the world
through different depths and complex courses. Air-Force planes in
perpetual (or merely long distance) flight could also use it to plan or
plot their courses, so could airlines. It could simplify autonomous
agent motion through extremely complicated environments, such as space,
or for nanobots navigation in a chip or in colloidal fluid, or
autopiloted aircraft in extremely crowded skies (such as autopiloted
personal aircraft for overcrowded cities). Cross Domain Relations, for
the Mathematics of Alternative Route Exploration Aside from the first
order logic stuff, the ideas and depictions in this paper were
originally conceived of and are Copyright 5/22/2004 by Justin M. Coslor,
Rough Draft was typed on 6/9/2004 in AbiWord on an X86-Compatible
Personal Computer running GNU Sarge (a free Debian Linux Operating
System), and was encouraged by the FRDCSA.ORG project.
These ideas are intended to enhance the ability to discover and
invent new routes in any field of study, and to aid in evaluating the
relative utility of known routes, as well as to simplify some of the
problems posed by computability theory.
Figure 1.
From the foundations of relational logic, we already know that: if a
relation is xRy: X-->Y, then it is injective; or if xRy: X<-->Y, then it
is bijective; or if xRy: X-->(y1, y2, ..., yp), then it is surjective;
We also know that if a functional relation is xRy: y=f(x)=m where f(x)
represents an arbitrary function of the domain X that yields a set of
unique m's that are sub-ranges (y's) within the bounds of the range Y
(a.k.a. the Class Y), where each m corresponds to a uniquely arbitrary
domain x through the functional relation f(x). In this case, [f(x)]=R in
the equation xRy. (*Remember for later that any mathematical operator
(+, -, /, *, etc.) can be a relation. Any piece of computer software can
also be treated as a relation, since software performs operations, and
is basically a collection of algebraicly-tied operators.) But perhaps,
we can broaden the scope of the Context in order to allow for more
possibilities. This "broadening", may include metaphoric operations and
metaphoric relations between the data type(s) of the functional
relation(s) in focus and various specified number sets, orderings, and
systems of numbers (including symbolic ones). (*We'll cover more on this
later.) Let us introduce a new type of relation, that is a relation that
relates relations, and let us call it a Cross-Domain Relation, and
depict it as such: One goal of this paper is to show a system to
accurately depict the following kinds of relation: xCy: (x1, x2, ...,
xp)-->knEY (injective), or xCy: (x1, x2, ..., xp)<-->knEY (bijective),
or xCy: (x1, x2, ..., xp)-->(k1, k2, ..., kn)EY (surjective); where
every sub-range k that is an element of the range Y, has multiple
domains that relate to each k in a unique way (through a unique route).
Each sub-domain in the Domain X can come from different contexts and
each sub- domain may operate under a different relation to specific
sub-ranges in Y than other sub- domains relations to those same
sub-ranges in Y. In these relations, some of the sub-domains may be
injective, some may be bijective, and some may be surjective. In order
to label and order each set of sub-domains that is part of a unique
cross-domain relation, we introduce the ordering term "n". We can use
the "n" component here to differentiate and/or order cross-domain
relations, by combining the ordering of cross-domain-related sub-domains
(i.e. nCx) with the individual relations between those sub-domains and
any given sub-ranges (i.e. xRy), as such: nCx: N-->X (injective), or
nCx: N<-->X (bijective), or nCx: N-->{x1, x2, ..., xp} (surjective),
then nCxRy describes the cross-domain relation where n is an element in
the cross-domain N such that x=f(n), and x is an element of a particular
cross-domain subset n (as well as being a sub-domain of the domain X),
where x has a relation (a.k.a. a route) to a particular sub-range y in
Y.; (***Note: every x in X and/or every n in N can come from vastly
different contexts, yet still lead to the same y(s) in Y.) where for
each sub-range y, f(n) is every function in the cross-domain N that
leads to multiple sub-domains in X that lead to the the multilateral
result k (which is a specific singular sub-range y in the Class Y with
multiple relations leading to it from the domain X) through the route:
F[C{f(x)}]-->kEY (kEY means k is an Element of Y), where C{f(x)}=n and
y=f(x) and x=f(n), and k represents any specific unique sub-range in Y
that can be arrived at via multiple domains' functional relations, where
each of the multiple domain's relations goes from any sub-domain x in X
to the same specific sub- range y in Y; where F[N] is the set of all
routes to all k's in Y, K is the set of all k's in Y, and F[n] is the a
relation describing set of all routes to a specific k in Y. k is used to
depict sub-ranges that have multiple ways to arrive at them; that is to
say, ways that include origin variations, and intermediary relation
combinations (middle-man combinations). In short: "If some unique X's
yield some of the same unique y's through various relations, then those
X's are said to have "cross-domain relations", because those domains
have some relations whose end results have something in common."
^^^^^^^^^^^^^^^^This is what I was trying to draw and put into an
equation format. I'm not sure if I succeeded, but probably. It seemed
like there needed to be a formal word for what I was trying to depict as
relations that relate different contexts' functions' domains by a
representable equivalence or similarity in their ranges (when there is
exists such a representable equivalence or similarity), so that's how I
came up with the name "cross- domain relations". Computer software can
essentially be treated as such functions, for which cross-domain
relations that lead to alternative routes may exist for any given set or
class of software functions. It's basically all about alternative
routes. Such a mapping can be quite useful for exploring alternative, or
previously unconsidered, or unknown possibilities and modalities. In
Figure 1., X is a class that contains domains that lead to ranges within
the class Y. There may be other classes that lead to those ranges, even
if they do so indirectly through other classes by broadening the
applicable context. By saying "lead to them" I mean "relate to them" in
any "chosen" way(s). The route equations can get very complex the more
classes and destinations you're analyzing when looking for and mapping
cross-domain relations. In practice, the user ends up with a concise
pack of cross-domain relation equations that summarizes the entire
complexity of the known patterns in the contexts of any situation or
model. The equation packs can also be used to represent the possible
outlets to explore for new patterns based on perceived priority of their
beginning class of categories, and perceived
attainability/computability. . . mark off potentially infinite patterns
and recursive loops accordingly, after exploring the first few layers
only. Conclusion: Cross-domain relations can be used when depicting,
predicting, finding, manipulating, creating, using, analyzing,
backtracking, tracing, comparing, and reverse engineering alternative
routes to anything, in any field of application.
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Examples: (*In the following examples I have defined the "underscore"
character "_" to be the equivalent of the logical statement "OR", which
is equivalent of the English language statement "and/or". I use the "_"
character to link multiple routes to a sub-range, so that the patterns
of the context of that sub-range can all be packaged into one continuous
string. Such a string can then be parsed easily and sorted according to
factors such as: route-scale (number of computable degrees or nodes v.s.
potentially infinite possibilities), route category, route-size, relative
route location, etc.)
Example 1:
In Figure 1., the domain x2 has an alternative route to y1 through
the cross-domain relation n1Cx2Ry1_n1Cx1Cy1 where n1Cx2Ry1=Route k1, and
n1Cx1Cy1=Route k2. ((In my examples I like to use C to represent
bijective relations, and R to represent surjective or injective
relations.) *Note that x1Cy1Cn1=n1Cx1Cy1 ) So in this example k1 and k2
are the known routes to y1, and since we know about more than one route
to y1, we can call k1 and k2 cross-domain relations. Or we can simply
reference that group of routes by the meta-name k1_k2.
Figure 1.
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The following are Graph Theory Examples of Cross-Domain Relations:
(**In the following examples, I use R to represent an injective or
surjective route, and use C to represent the continuous directional flow
of a bijective route. I use the symbol "\$" to indicate that the routes
on each side of the "\$" have a bijective relationship. The "\$" symbol is
used when comparing two or more routes. The "=>" symbol means "directly
implies".) Example 2: First have a look at Figure 2. (***Note: in ACB,
(a1Cb1)\$(b1Ca1), because BCA exists.)
Figure 2.
Alternative routes to A from C:
ARC=a2Ra2 => a2 = ACCk1
ACBRC=a1Cb1Rb3_ a1Cb1Ca1Cb1Rb3 => ACCk2_ACCk3
ACBCDCC=a1Cb1Cb2Cd1Cd2_A\$B\$D\$C=a1b2d2 => Routes ACCk4 through
ACCk11
(****Many more complex routes beginning at A and terminating at C
exist, and can be very explicitly depicted in this manner.)
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Example 3: In Figure 2, by entering each line's node relationship
into a computer in a format such as: [ACB,BCD,DCC,BRC,ARC,ARF,FCE,FCC],
(<----This is the Context.) (Next I'll describe the Patterns in that
Context...) the computer can generate on the fly all of the possible
routes from any given node to any other given node, including curtailed
potentially infinite loop structures (by representing loop structures
via the "\$" operator, as noted earlier), and it can explicitly represent
the optimal routes and rank the suboptimal routes using relation and
cross-domain relation notation. Perhaps in some situations, one might
even order the routes by largest perimeter of closed polygonal circuit
region to smallest polygonal circuit perimeter, followed by largest open
leg to smallest open leg, when declaring a context. (*****Where ";" is
the character that indicates the parsing of each closed-circuit
polygonal region or open leg in this notation variation.) This might
look something like: [ARC_CCF_ARF;ARC_ACB_BRC;BCD_DCC_BRC;FCE] ...if the
proportions were correctly represented in my diagram, that is...
Figure 2.
These are some thoughts that I considered for book 3 of possibility
thinking explorations in logic and thought and many of them are
probably flawed so the burden of understanding lies entirely on the
reader and gossip is not allowed.
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This is an unfinished writing and I disclaim all liability.
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