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.

----------------------------------------

This is an unfinished writing and I disclaim all liability.

----------------------------------------

---------------

Book III:

Math Ideas:

---------------

Copyright 9/13/2004 Justin Coslor

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

"Sine Spiral Graphing" applies to the discovery I made about

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

chicken in a tornado.

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.

-------------------------------- Copyright 6/28/2003 Justin Coslor

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,

ALL RIGHTS RESERVED (Please contact me for conditions of use...). This

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]

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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