some thoughts I considered for book 3 of possibility thinking explorations in logic and thought 
[Dec. 3rd, 200709:32 am]
justincoslor

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.  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 redefinition 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 NPCompleteness 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. Nonpolynomial 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{n1}X^n1 + . . . + 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 parallelprocessing 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 sinespiral graphing (Sinespiral graphing is something of my own invention, and is a 3D resentation of circular motion, where the sinewaves 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 sideview, 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 sinespiral graphing methods (I came up with the math for it when I got way behind in 10th grade MathAnalysis 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 twodimensional 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, threedimensionally. 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 twodimensional (flat) motion through space over a period of time that more clearly shows the progression of travel. At present, mapping threedimensional 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 twodimensional 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 threedimensionally 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 threedimensional 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 twodimensional 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 twodimensional 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 colinear 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 Zaxis 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 threedimensional 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 twodimensional 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 threedimensional fashion. This new dimension can be called the "Zaxis". Now that there is an X, Y, and Z dimension available, a threedimensional 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 trispiral (a spiral or shape that represents 3D motion over time). One can continue plotting the points with several objects and where the trispirals intersect, the objects intersect. One can break down the trispiral 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 Zaxis 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 Ztag, 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 twodimensional 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 Zaxis). 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 Zaxis around which the graph spirals would instantaneously bend at a ninety degree angle. A computer can easily generate this threedimensional 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 twodimensional motion (motion that moves in any direction on a flat plane), or rotation in threedimensions 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 twodimensional 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 Zaxis appropriately to accommodate changes is axis orientation). Using the Sine Spiral, graphing motion in the Zaxis, or time, requires one to employ a means to mark or reference the Sine Spiral in order to distinguish how deep down the Zaxis the motion has traveled. Without a Sine Spiral, one can only pick threedimensions 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 Zaxis 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 twodimensional or threedimensional 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 threedimensional motion with the Sine Spiral is more difficult to do, but can be done effectively. Graphing threedimensional 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: AddisonWesley Publishing Company, 1973. Gove, P.B., ed. "Webster's Third New International Dictionary, Unabridged". Springfield, MA: MiriamWebster, 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 halfsphere 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 deaccelerating 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 multibody 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 realtime (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 highspeed shooting gallery. Ideally, some kind of Star Treklike/Tesla Wardencliffetowerlike 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 xy 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 pathtime. 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 collisioninterception calculation (and might be able to help protect all countries from incoming intercontinental thermonuclear ballistic missiles by combing this visualization method with a groundbased or spacebased or airplanebased or reusable nonoffensive missile based laser/maser antiballistic 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. AirForce 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 X86Compatible 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 subranges (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 algebraiclytied 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 CrossDomain 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 subrange 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 subdomain in the Domain X can come from different contexts and each sub domain may operate under a different relation to specific subranges in Y than other sub domains relations to those same subranges in Y. In these relations, some of the subdomains may be injective, some may be bijective, and some may be surjective. In order to label and order each set of subdomains that is part of a unique crossdomain relation, we introduce the ordering term "n". We can use the "n" component here to differentiate and/or order crossdomain relations, by combining the ordering of crossdomainrelated subdomains (i.e. nCx) with the individual relations between those subdomains and any given subranges (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 crossdomain relation where n is an element in the crossdomain N such that x=f(n), and x is an element of a particular crossdomain subset n (as well as being a subdomain of the domain X), where x has a relation (a.k.a. a route) to a particular subrange 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 subrange y, f(n) is every function in the crossdomain N that leads to multiple subdomains in X that lead to the the multilateral result k (which is a specific singular subrange 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 subrange in Y that can be arrived at via multiple domains' functional relations, where each of the multiple domain's relations goes from any subdomain 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 subranges that have multiple ways to arrive at them; that is to say, ways that include origin variations, and intermediary relation combinations (middleman 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 "crossdomain 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 crossdomain 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 crossdomain relations. In practice, the user ends up with a concise pack of crossdomain 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: Crossdomain 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.     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 subrange, so that the patterns of the context of that subrange can all be packaged into one continuous string. Such a string can then be parsed easily and sorted according to factors such as: routescale (number of computable degrees or nodes v.s. potentially infinite possibilities), route category, routesize, relative route location, etc.) Example 1: In Figure 1., the domain x2 has an alternative route to y1 through the crossdomain 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 crossdomain relations. Or we can simply reference that group of routes by the metaname k1_k2. Figure 1.    The following are Graph Theory Examples of CrossDomain 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$C=a1b2d2 => Routes ACCk4 through ACCk11 (****Many more complex routes beginning at A and terminating at C exist, and can be very explicitly depicted in this manner.)     Example 3: In Figure 2, by entering each line's node relationship into a computer in a format such as: [ACB,BCD,DCC,BRC,ARC,ARF,FCE,FCC], (<This is the Context.) (Next I'll describe the Patterns in that Context...) the computer can generate on the fly all of the possible routes from any given node to any other given node, including curtailed potentially infinite loop structures (by representing loop structures via the "$" operator, as noted earlier), and it can explicitly represent the optimal routes and rank the suboptimal routes using relation and crossdomain 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 closedcircuit 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.  This is an unfinished writing and I disclaim all liability.  

