John Majka explains, according to Relativity Theory, how three observers see a charged "particle" in free space, from three different velocities. Each velocity has very different characteristics. His hypothesis concludes that "it is the distortion of space-time which appears to us as "mass" rather than "mass" causing the distortion." This is followed by justification arguments by comparing the electric permittivity and the magnetic permeability of free space, with a transmission line. He concludes, after several pages, that, "the speed of light is the resonant velocity of free space." Steady-state impedances, Non-steady state conditions and gravity are also elaborated upon, in transmission line, electrical language.
Douglas Vogt had to write a book to explain all of his multidimensional theories. Among these theories he hypothesized that the universe acts like a hologram and he presents a few applicable paragraphs to explaining a patented TV system that transmits and receives holographic information, as he points out, "just like our universe works".
Tom Van Flandern approaches creation by skipping over the 0th dimension and begins with a one "particle" discussion and a two "particle" discussion. All characteristics of both one and two particle beginnings are covered, with nothing missed. Zeno's paradox is introduced and discussed. Beginning with a bunch of real and unreal "IF" premises and tentative conditions arranged around Zeno's paradox, the discussion employs some razzle-dazzle of various concepts by pitting various paradoxial juxtapositions against each other and resolving them with two-sided explanations, under certain conditions, for example, "it is possible for substances to be unchanging at every instant, yet changed after a finite interval, ONLY if there are an infinite number of steps in the interval!"
Rick Andersen mentions Tilton and introduces the concept of "super resonance." Free space is compared to a tuned circuit. He introduces questions concerning the lowering of light velocity to manipulate a region of space.
D |
1 |
2 |
3 |
4 |
1 |
1 |
2 |
3 |
4 |
2 |
1/2 |
1 |
3/2 |
2 |
3 |
1/3 |
2/3 |
1 |
4/3 |
4 |
1/4 |
1/2 |
3/4 |
1 |
TaoJoe rearranges dimensions as an array or set of elements that interact according to an Operator. D_{1} = time, or change or motion. D_{2} is a single spatial dimension, a line; D_{3} is 2 spatial dimensions, a solid object. D_{2}, D_{3}, D_{4} = length, width, height (spatial dimensions)
D_{1/4}, D_{1/3}, D_{1/2}, D_{2/3}, D_{3/4}, D_{4/3}, D_{3/2} = fractional, composite dimensions. This scheme of notation is used to describe the permutations.
Circlemath is said to be the circular math of the inner mind, set against the straight counting line math of the outer world. Negative 1 is explored in various bases within the transitional 0-circle,1-circle and beyond.
^{Ref.taojoe } D_{4} are the three familiar spatial dimensions, each with vectors orientated 90 degrees from another.
^{Ref.taojoe } The dimensions produced by 2 D 4 and 1 D 2 are the same.
^{Ref.taojoe } D_{4} = length, width, height (spatial dimensions).
^{Ref.taojoe } 4 D 2 = 2.
^{Ref.taojoe } D_{4} = ordinary matter, always in motion, or always radiating a temperature.
^{Ref.taojoe } It was the attaining the velocity of c that split D_{4} into D_{2} + D_{2}; conversely, reducing motion below c would reunite D_{2} + D2 into D_{4}.
^{Ref.taojoe } In words again, we are taking one dimension, say the depth, away from solid matter, to get a plane in motion (once again time as a spatial dimension), but what is a plane in motion? Maybe a tube of some sort. Maybe not. The third option involves negatives. To review: D_{4}, i^{4} and +1 are all symbols we can use to represent solid matter as we know it. D_{2}, i^{2} and -1 are symbols that correlate to electromagnetic waves or photons.
^{Ref.taojoe } In words, this means that we add a new 90 degree vector to our planar electromagnetic wave, producing an abstraction we can refer to as a cubic wave. I don't know about you but I cant visualize that (for a good reason, as you will see). We can also try D_{4} - D_{1} = D_{3}.
^{Ref.taojoe } The only difference between D_{4} and D_{2} is that D_{2} = D_{4} x c (remind you of something? E = mc^{2}, perhaps?). Substituting numerals, -1 = (+1) c, and solving c = -1. In words, c has the effect of negativizing a phenomenon.
^{Ref.taojoe } D_{4} is responsible for what we call ordinary matter, both massive and detectable.
^{Ref.taojoe } Next, consider gravity's far-reaching effects. Not only matter but also even something mass-less like light, is induced to follow the shape of space. So it would be illogical to choose 2 as our denominator, since that excludes the 4 that contain matter.
^{Ref.taojoe } There are only two fractions whose denominator is 4, which would account for the matter, and everything contained in all lower dimensions. What is it then that reacts with either light or matter, to form space? What thing plays a role in the effect of gravity; appears in all the gravitational principles and equations? Distance!
^{Ref.taojoe } Therefore the only logical fraction for the dimension of gravity, or the geometry of space in one that points to the interplay of distance (D_{1}) with each of the dimensions effected by that gravity (D_{4}). The conclusion: gravity = D_{1/4}.
^{Ref.sqrtneg1 } The square root of -1 is an integer in 5-circle where -1 is 4.
^{Ref.taojoe } D_{2}, D_{3}, and D_{4} are the three familiar spatial dimensions, each with vectors orientated 90 degrees from another.
^{Ref.taojoe } D_{2}, D_{3}, D_{4} = length, width, height (spatial dimensions).
^{Ref.taojoe } D_{3} is 2 spatial dimensions, a solid object.
^{Ref.taojoe } D_{3} = before associating phenomena with D_{3} some preliminary concepts need to be discussed.
^{Ref.taojoe } D_{3} Preliminary Discussion
^{Ref.taojoe } In mathematics, equations are commonly written to describe phenomena in any of the four integral dimensions.
^{Ref.taojoe } The exponent of the variable indicates the degree of dimensionality:
^{Ref.taojoe } First degree equations have one solution and describe straight lines.
^{Ref.taojoe } Second degree equations (with x^{2}) have two solutions and describe planar geometries like circles or parabolas.
^{Ref.taojoe } Third degree equations (3 solutions) describe solid objects.
^{Ref.taojoe } For the fourth dimension, we have a choice: either write the exponent 2 or write the exponent zero. Lets do some math and play with a very simple equation. The equation is x^{4} = 1. You say x^{4} = 1- That's dumb.
^{Ref.taojoe } I can think of three ways of getting to D_{3} from something we know. We can take D_{2}, the photon and kick it up a notch, and do D_{2} + D_{1} = D_{3}.
^{Ref.taojoe } In words, this means that we add a new 90 degree vector to our planar electromagnetic wave, producing an abstraction we can refer to as a cubic wave.
^{Ref.taojoe } We can also try D_{4} - D_{1} = D_{3}.
^{Ref.taojoe } In words again, we are taking one dimension, say the depth, away from solid matter, to get a plane in motion (once again time as a spatial dimension), but what is a plane in motion? Maybe a tube of some sort. Maybe not.
^{Ref.taojoe } The third option involves negatives. To review: D_{4}, i^{4} and +1 are all symbols we can use to represent solid matter as we know it. D_{2}, i^{2} and -1 are symbols that correlate to electromagnetic waves or photons.
^{Ref.taojoe } The only difference between D_{4} and D_{2} is that D_{2} = D_{4} x c (remind you of something? E = mc^{2}, perhaps?). Substituting numerals, -1 = (+1) c, and solving c = -1. In words, c has the effect of negativizing a phenomenon.
^{Ref.taojoe } If c really does produce an opposite, then i^{1}or i , becomes its negative, i^{1}or -i , which is the value of i^{3}, which is, lo and behold, an alternate symbol for D_{3}. D_{3} then (or i,) is the realm of slower-than-light imaginary points.
^{Ref.taojoe } Since it is sub-light it must have mass, but since it is imaginary, it has no substance. And this is exactly what our plane in motion is trying to describe, a surface with no thickness and therefore no substance, enclosing on itself trying to form a sphere (or a tube) with no volume. An imaginary figure with mass. Hey, astronomy buffs, what am I describing? This is dark matter! All that unaccounted-for mass that must exist in the universe but is undetectable. Its undetectable because its not real. But because its slowed to sub-light, it has mass. And because these things are negative singularities, similar to the singularity responsible for the big bang and the subsequent universe except hollow inside (like bubbles or macaroni), they are super dense, far more dense even than black holes.
^{Ref.taojoe } D_{3} is responsible for the dark matter which is undetectable but comprises most of the mass in the universe.
^{Ref.sqrtneg1 } The square root of -1 is the integer 3 in 10-circle.
^{Ref.sqrtneg1 } In the counting line of counting circles -1 is a counting line to infinity, -1 is 2 in 3-circle and 3 in 4-circle.
^{Ref.sqrtneg1 } The problem (negative numbers) does not occur in circle math whose counting lines are closed. Positive or add means clockwise in the circle. Negative or subtract means anticlockwise. The confusion vanishes. The math remains. Thus, in 10-circle -1 is 9, which squares to 1, and has 3 and 7 as its roots. The wordiq.com statement (as above) nails the error to the masthead by defining positive numbers as greater than zero, and negative numbers by the illogical expression, "less than zero".
^{Ref.sqrtneg1 } The fact that -1 in 10-circle is 9, whose square root is 3 (and 7) is the touch of death for the mystery sequestered in the square root of -1. This one facet does not explain i, for it is only a fragment of a greater whole, but it is the key. Turn it in the lock and the door will open, for it can be readily understood. Even for those who are far from grasping the pattern of the whole, the illusion of i, the square root of -1 will begin to fade and will continue to melt away until it vanishes.
^{Ref.majka } Um = B^{2} / ( 2uo )
^{Ref.majka } where uo is the magnetic permeability of free space. The equation describing the energy density of the particle's electric field, Ue, is:
^{Ref.majka } Ue = eo E^{2}
^{Ref.majka } where eo is the electric permittivity of free space.
^{Ref.majka } The total energy, Ut, of the electric and magnetic field of a particle traveling at some speed, v, is the sum of these two equations. Converting to like terms and combining terms, the total energy equation is:
^{Ref.majka } Ut = ( eo E^{2} / 2) ( 1 + v^{2} /c^{2} )
^{Ref.majka } If we now let V = C, the equation becomes:
^{Ref.majka } Ut = eo E^{2}
^{Ref.majka } (In electrical analysis, since the symbol "i" is used to represent current flow, the symbol "j" is used to represent the square root of -1 and the symbol, w or omega, is used to represent frequency where w = 2 pi f.)
^{Ref.majka } The normalized transfer function, H(jw), of such a circuit is:
^{Ref.majka } H(jw) = 1/( w^{2} - wo^{2})
^{Ref.taojoe } D_{1} = time, or change or motion.
^{Ref.taojoe } D_{1} is time the first, or primary, or fundamental, or identity dimension, because it acknowledges change without assigning a vector to it.
^{Ref.taojoe } Lets do some math and play with a very simple equation. The equation is x^{4} = 1. You say x^{4} = 1? That's dumb.
^{Ref.taojoe } A fourth degree equation has four solutions, but here, all four solutions are the same. X can only equal 1, right? Wrong! You are forgetting how to solve an equation like this. Proceed as follows, treating the equation as if it were an ordinary quadratic:
^{Ref.taojoe } 1. Set the equation equal to zero.
2. Factor the equation into the difference of squares.
3. Let each factor independently equal zero.
4. Both these equations are quadratic. The one on the left is again factorable.
5. Repeat step 3 for this set of equations.
6. The two solutions for these equations are X = 1 (the answer we already know) or X = -1 (the answer we forgot about but which checks since (-1)^{4} does equal 1.
^{Ref.taojoe } 7. We still must solve the equation on the right side of step 3: X^{2} + 1 = 0. It is not factorable, so we will need to use the quadratic formula.
^{Ref.taojoe } When ax^{2} + bx + c = 0, then x = 0 +/- Sqrt (b^{2} - 4ac) / 2a
^{Ref.taojoe } 8. Using the formula, where a = 1, b = 0, and c = 1,
^{Ref.taojoe } x = 0 +/- Sqrt ((0)^{2} - 4 (1) (1)) / 2 = +/- Sqrt -4/2
^{Ref.taojoe } 9. Remember that the square roof of -4 is called 2i (right imaginary numbers), so the two solutions for x^{2} + 1 = 0 are i and -i.
^{Ref.taojoe } To check: Is (i)^{4} = 1? Well, i^{2} = -1, and (-1)^{2} = 1 yes.
Is (-i)^{4} = 1? (-i)^{2} = -i^{2} = -1 also yes.
^{Ref.taojoe } Therefore, the four solutions for x^{4} = 1 are +1, -1, i and -i.
Now for the interesting part. i is the first imaginary number. Lets list the first five exponents of i:
i^{0} = 1 (since anything raised to the zero power is 1)
i^{1} = i (itself)
i^{2} = -i (by definition)
i^{3} = i^{2} x i = -1 x i = -i
i^{4} = 1
^{Ref.taojoe } Observe that the exponents zero and four give the same result. When we write the imaginary number i, we are really writing i^{1}. These two symbols each have cosmological significance:
^{Ref.taojoe } The exponent portion, 1 indicates the first dimension time or motion. 1, the fundamental, imaginary constituent of reality the point. Again, time is going to behave like a spatial dimension, because a point in motion becomes a straight line. Simply by having an imaginary point in motion, we are generating the phenomenon of distance.
^{Ref.taojoe } X^{2} + 1 = 0.
^{Ref.taojoe } When ax^{2} + bx + c = 0, then x = 0 +/- Sqrt
^{Ref.taojoe } x = 0 +/- Sqrt ((0)^{2} - 4 (1) (1)) / 2 = +/- Sqrt -4/2
^{Ref.taojoe } Remember that the square root of -4 is called 2i (right imaginary numbers), so the two solutions for x^{2} + 1 = 0 are i and -i.
^{Ref.taojoe } Now for the interesting part. i is the first imaginary number. Lets list the first five exponents of i:
^{Ref.taojoe }i^{0} = 1 (since anything raised to the zero power is 1).
^{Ref.sqrtneg1 } In the counting line of counting circles -1 is a counting line to infinity. Thus it is 0 in 1-circle, because one place anticlockwise from 0 brings us back to 0.
^{Ref.sqrtneg1 } Circle..... -1 in the circle..... Square root of -1
^{Ref.sqrtneg1 } 0-circle............ Being without determination, 0-circle has no numerical elements.
^{Ref.sqrtneg1 } 1-circle............ 0............... .....0
^{Ref.sqrtneg1 } The rational/ irrational pattern is that, each integral root (0 1 2 3), is followed by double that number of non-integer results:
^{Ref.sqrtneg1 } The. 0. "sq rt. -1". in.. 1-cir. is followed by 0 irrationals, and 0+0=0
^{Ref.sqrtneg1 } We should note that 0 is the identity element for adding and subtracting.
^{Ref.sqrtneg1 } The "internalized counting line", as in the above paragraph, is given in the square roots of 0, 1, 4, 9, 16, 25
^{Ref.sqrtneg1 } Use a horizontal line instead and you have the 0 direction. This gives the answer to combinations that sum to 0.
^{Ref.sqrtneg1 } The truth it is struggling to realize is that the symbol 0 represents the mind, and the mind, as the viewpoint within which number and math exist, is above, and so indifferent to the objects it contemplates. As mentioned, 0 is not a number at all.
^{Ref.sqrtneg1 } Because at the level we teach how to spell CAT we also imply that 0 and 1 are numbers, and at the level we teach "Rover has a ball" we include the thought that zero is a number, "negative" numbers exist, and so on, mathematicians talk about i; science is unable to merge quantum and relativity theory and astronomers talk about the "size" of the universe. We search for the error in the clouds when we should be looking at our feet.
^{Ref.sqrtneg1 } The ancient Greeks pointed out that 0 is not a number because it is nothing.
^{Ref.sqrtneg1 } 0 and 1 are functional as the hinge between mind and world. 0 interfaces with the mind, 1 with the world. It is convenient to call 0 and 1 numbers, but strictly speaking they are the agents that, through counting lines and bases, create number, unraveling in the world the order that already exists in the mind.
^{Ref.sqrtneg1 } From this, expanding from the "i", a world in consciousness opens up. 0 and 1 are the active agents in generating this focal relation, and circlemath puts the whole mechanism in place for us to see.
^{Ref.majka } In electrical analysis, since the symbol "i" is used to represent current flow, the symbol "j" is used to represent the square root of -1 and the symbol, w or omega, is used to represent frequency where w = 2 pi f.
^{Ref.sqrtneg1 } The. 0. "sq rt. -1". in.. 1-cir. is followed by 0 irrationals, and 0+0=0
.... The. 1. "sq rt. -1". in.. 2-cir. is followed by 2 irrationals, and 1+1=2
.... The. 2. "sq rt. -1". in.. 5-cir. is followed by 4 irrationals, and 2+2=4
.... The. 3. "sq rt. -1". in 10-cir. is followed by 6 irrationals, and 3+3=6...and so on to infinity. What is happening here?
^{Ref.sqrtneg1 } The answer is that "i", the square root of -1, is switching the whole of mathematics from objectivity into subjectivity, translating the numerical pattern from the external or worldly domain into its internal counterpart, the mental space which we call subjectivity or time.
^{Ref.sqrtneg1 } To multiply by -1 therefore changes only the sign of the result. The "1" in the "square root of -1" is thus a dummy indicator of sign change, whose reality is an internalized counting line to infinity against a background array of fractional results, as shown in the list of -1 roots. Internal here means "in mind" and external means "in world", so that "ordinary" arithmetic, with its straight counting line to infinity belongs to the world. Circle math, with its circles rules and results belongs to the mind.
^{Ref.sqrtneg1 } We work from our knowledge of our ordinary or external math, which is objective to us. The internal circle math of the mind is however the parent and projector of the outer result. The "internalized counting line", as in the above paragraph, is given in the square roots of 0, 1, 4, 9, 16, 25. The "background array of fractional results" that seems to act as a spoiler is the tidy sequence of untidy square roots of 2 3 (two) 5 6 7 8 (four) 10 11 12 13 14 15 (six) in the -1 non-integer circle series, and the whole is the neurological setting from which developed consciousness springs.
^{Ref.sqrtneg1 } The square root problem
^{Ref.sqrtneg1 } In conventional mathematics there is no real number x, such that x^2 = -1
^{Ref.sqrtneg1 } This is because +x times +x comes to a positive result, and -x times -x also comes to a positive result, and there are no other options for x^2.
^{Ref.sqrtneg1 } Unable to assign a value to x satisfying the equation conventional mathematics stipulates that there is such a value, and calls it i for imaginary. It has no idea what "i" is, other than the square root of -1 as expressed in the equation i^2 = -1
^{Ref.sqrtneg1 } Given this, for it an impossible equation, it assigns the "no real number x" to the imagination, calling the square root of -1 "i". It then has a term for this ghost in the mind, supported not by understanding, but the fact that it integrates perfectly into mathematical operations and equations. The cost is that mathematics advances as a ritual of symbolic relations, comprehended only as the formalistic integration of signs and results.
^{Ref.sqrtneg1 } "i", as the square root of -1 is currently a blind spot in our mathematical understanding, walled off and bypassed. The task is to explain it, and in the process turn it from a block to a bond and bridge uniting in-world (linear) and in-mind (circular) math. The imaginary number serves a purpose, making an advance possible, shifting math from its ordinary domain to its higher expression. The further insight needed is to see that this "higher" math is really the internalized circular math we call the mind.
^{Ref.sqrtneg1 } "In Einstein's special theory of relativity there is a Lorentz transformation that leads to Minkowski's four-dimensional space. But the fourth dimension is obtained by replacing time, t, with the imaginary [square root of -1 multiplied by the speed of light, c, and by t itself]. After this, the temporal dimension becomes totally symmetrical with all other three dimensions of space."
^{Ref.sqrtneg1 } "In Quantum Mechanics, the Schrodinger's wave function also involves the same square root of -1 when the spatial locality breaks down and a conscious observer gets involved in the process of measurement."
^{Ref.sqrtneg1 } "This apparent coincidence indicates something extremely significant for understanding the universe and consciousness. It suggests that human consciousness may be a dimension of anti-space that merges into the fourth dimension of negative space so that we cannot see it as spatial any more. It appears as one dimensional time instead."
^{Ref.sqrtneg1 } Therefore, the square root -1 is the Consciousness factor.
^{Ref.sqrtneg1 } Gradually across the years the square root of -1 that divides math and maps to the schism between ideality and reality, has come forward as the kernel of the theoretical problem. Its resolution, whose completion is critical, will mark the turning point between that past and a brighter future. In this realization, the difficulty in the imaginary number, proving itself to be imaginary, will vanish, and people will wonder what it was about.
^{Ref.sqrtneg1 } The one step (switching into the circles) resolves the philosophical ground of math and the "square root of -1", which now falls into place as fully accessible at primary school level.
^{Ref.taojoe } D_{2}, i^{2} and -1 are symbols that correlate to electromagnetic waves or photons.
^{Ref.vogt } A hologram is a three-dimensional image produced by coherent light.
^{Ref.vogt } This disclosure relates to a television system that utilizes wave front reconstruction techniques to provide a real time three-dimensional image at the receiving end of the system, with the image changing in perspective as the object and/or observer moves.
^{Ref.vogt } In our existence there are no physical phase or amplitude detectors in this dimension.
^{Ref.flandern } Moreover (and this is something to note), the size of the universe would be indeterminate, even if our lone particle has "finite" dimensions.
^{Ref.flandern } 1 particle universe- Indeed, it is impossible to say whether the particle has infinite dimensions, finite dimensions, or is infinitesimal (without size), since there is no scale to measure by.
^{Ref.flandern } 2 particle universe- Now, for the first time, we have "scale" in our universe, and can measure the dimensions of the particles themselves as a fraction of the distance between them.
^{Ref.flandern } 2 particle universe- If time is treated like just another dimension (a "fourth dimension" of space), the same remarks might be extended to include the concept of a "minimum possible time unit".
^{Ref.flandern } 2 particle universe- One way to see the resolution of this paradox is by considering time to be another dimension, just like the three dimensions of space (although admittedly not exactly like a space dimension.
^{Ref.flandern } 2 particle universe- Why would they have finite space dimensions, yet infinite dimension in time?
^{Ref.flandern } 2 particle universe- If the substance of bodies always gets denser (more substance per unit volume) at smaller and smaller scales, then in the limit as dimensions approach zero, density approaches infinity and substances approaching each other must make "contact".
^{Ref.flandern } 2 particle universe- By analogy with the proposed resolution of Zeno's paradoxes for space and time, the paradox for mass is resolved, apparently necessarily, by the conclusion that substance must be infinitely divisible, and that it must approach infinite density as size decreases toward zero dimensions.
^{Ref.flandern } 2 particle universe- The upper limits to the structure of substance, the dimensions of the universe, and the extent of time, must all be as unbounded on the high side as they need to be on the small side.
^{Ref.flandern } 2 particle universe- Now introduce a second infinitesimal particle. This gives meaning to orientation, since angles can be measured from the line joining the two particles. It also provides a single measurement of length, the distance between the particles. It does not, as before, provide a scale for the empty universe, since the distance cannot be measured in units of particle diameters, which are still being assumed to have no dimensions.
^{Ref.flandern } 2 particle universe- Of course, for actual particles with finite dimensions, events of coincidence do not occur.
^{Ref.flandern } 2 particle universe- Put another way, a particle reaching one edge of a "vacuum" would skip instantaneously to the opposite edge, just as if the "vacuum" had zero dimensions, because there is no substance to mark the passage of time inside of the"vacuum", and no absolute time without substance.
^{Ref.flandern } 2 particle universe- Conversely, if substance does not start out with density which approaches infinity as dimension approaches zero, it could not assemble itself into such an infinite-density configuration in a finite time.
^{Ref.taojoe } The operator here is D, a dimensional operator that separates, or divides dimensional combinations by various permutated constituents.
^{Ref.taojoe } D_{1} is time the first, or primary, or fundamental, or identity dimension, because it acknowledges change without assigning a vector to it.
^{Ref.taojoe } D_{2}, D_{3}, and D_{4} are the three familiar spatial dimensions, each with vectors orientated 90 degrees from another.
^{Ref.taojoe } The values in the table are the complete array of dimensions that are generated in this system.
^{Ref.taojoe } Any dimension divided by the identity is equal to itself; but the identity divided by any of the spatial dimensions produces different results.
^{Ref.taojoe } In the case of 2/4, the answer, reduced to lowest terms, becomes 1/2. This means that the dimensions produced by 2 D 4 and 1 D 2 are the same.
^{Ref.taojoe } There are eleven total distinct dimensions in the system, the same as the number stated by Einstein and others as existing in our universe.
^{Ref.taojoe } This mathematical system may be the most accurate model of how dimensionality is generated in the universe.
^{Ref.taojoe } D_{1} = time. Regarding time in the context of this model.
D_{2}, D_{3}, D_{4} = length, width, height (spatial dimensions)
D_{1/4}, D_{1/3}, D_{1/2}, D_{2/3}, D_{3/4}, D_{4/3}, D_{3/2} = fractional, composite dimensions.
^{Ref.taojoe } To generate the other 7 dimensions, we will need to divide, and to perform division of dimensions we need to examine the meaning of division as it relates to the physical universe.
^{Ref.taojoe } Now lets take a simple case using dimensions: 4 D 2 = 2. In words, when we extract 2 dimensions from a solid object (in motion, of course), the result is 2 distinct electromagnetic waves (photons), whose amplitudes are perpendicular to each other (intersecting planes).
^{Ref.taojoe } And how do we induce this dimensional split? By accelerating our solid object (already in motion) to the speed of light (c).
^{Ref.taojoe } I think that each of the eleven dimensions represents one basic aspect, one constituent of the universe.
^{Ref.taojoe } In mathematics, equations are commonly written to describe phenomena in any of the four integral dimensions. The exponent of the variable indicates the degree of dimensionality.
^{Ref.taojoe } When we wanted to break our solid object into simpler dimensions, we had to increase its motion to the speed of light.
^{Ref.taojoe } When we step down a dimension, a dimension, we reduce exponents by 1, and to reduce the exponent 2 of i^{2} to 1, to get i^{1}, we can take the square root (in fact, that is the definition of i is that i equals the square root of negative one).
^{Ref.taojoe } In words again, we are taking one dimension, say the depth, away from solid matter, to get a plane in motion (once again time as a spatial dimension), but what is a plane in motion? Maybe a tube of some sort. Maybe not. The third option involves negatives.
^{Ref.taojoe } Four dimensions, the integral dimensions, have been explained. The seven fractional dimensions remain, and while each is still associated with naturally occurring phenomena in the universe, the job now becomes a lot tougher.
^{Ref.taojoe } It is the geometry of space we are observing, and like other geometries in the universe, it has a dimension associated with it. But which one?
^{Ref.taojoe } How do we decide which of the seven fractional dimensions is responsible for gravity, or rather, for the geometry of space?
^{Ref.taojoe } What do you notice as you move from D_{zero}, total nonexistence, up through the dimensions to D_{4}? I notice that there is a progression from the least substantive, most abstract manifestation; to the opposite of that the manifestation that is wholly concrete and not an abstraction at all.
^{Ref.taojoe } And if we are lucky, it may additionally be possible to identify elements being utilized by the operator D by deciding which two of the four basic dimensions are interacting to produce the appropriate fraction. In choosing, we must bear in mind that the seven available fractions all have a value less than 2, and five of them are less than 1. This means that they all represent significant abstractions. It is a certainty that none of these dimensions contain massive entities, though it is possible, as with gravity, that their effects are far-reaching enough for us to be aware, or at least suspicious, of their existence.
^{Ref.taojoe } There are only two fractions whose denominator is 4, which would account for the matter, and everything contained in all lower dimensions. What is it then that reacts with either light or matter, to form space? What thing plays a role in the effect of gravity; appears in all the gravitational principles and equations? Distance!
^{Ref.taojoe } Distance is created by the first dimension, motion.
^{Ref.taojoe } Therefore the only logical fraction for the dimension of gravity, or the geometry of space in one that points to the interplay of distance (D_{1}) with each of the dimensions effected by that gravity (D_{4}).
^{Ref.taojoe } Also, a big problem is that a dimension requires a geometry.
^{Ref.taojoe } In Einstein's special theory of relativity there is a Lorentz transformation that leads to Minkowski's four-dimensional space.
^{Ref.taojoe } But the fourth dimension is obtained by replacing time, t, with the imaginary [square root of -1 multiplied by the speed of light, c, and by t itself].
^{Ref.taojoe } After this, the temporal dimension becomes totally symmetrical with all other three dimensions of space"
^{Ref.taojoe } "That means that time is itself one more dimension of space but is perceived by our consciousness as different and uniquely temporal.
^{Ref.taojoe } "Surprisingly, in Quantum Mechanics, the Schrodinger's wave function also involves the same square root of -1 when the spatial locality breaks down and a conscious observer gets involved in the process of measurement. This apparent coincidence indicates something extremely significant for understanding the universe and consciousness. It suggests that human consciousness may be a dimension of anti-space that merges into the fourth dimension of negative space so that we cannot see it as spatial any more. It appears as one dimensional time instead."
^{Ref.majka }we may assume that the universe does not forbid speeds other than the speed of light but would provide an impedance to them.
^{Ref.majka }At light speed, the universe offers no impedance to the propagation of electromagnetic waves.
^{Ref.flandern }C. The One Particle universe
^{Ref.flandern }In the real universe there is a frame of reference to provide meaning to distance and direction.
^{Ref.flandern } The reference frame is provided both by the presence of distant matter in the universe, as well as by seas of rapidly moving "agents", such as photons and neutrinos.
^{Ref.flandern } The essential point is that the reference frame is provided by the presence of substance in the universe.
^{Ref.flandern }In the absence of other substance in the universe, our lone particle would be incapable of motion,for motion could have no meaning.
^{Ref.flandern }Moreover (and this is something to note), the size of the universe would be indeterminate, even if our lone particle has "finite" dimensions.
^{Ref.flandern }In a 1 particle universe: The number of such particles which can fit into the universe around it is infinite in any case.
^{Ref.flandern } The idea that the presence of distant matter in the universe is the origin of inertial forces is known as "Mach's Principle".
^{Ref.flandern }Our example may start to seem a little less hypothetical if we postulate a finite limit to all of the substance in the real universe, with nothing beyond.
^{Ref.flandern } Under this assumption, (finite limit) the entire substance of the universe would be like our single particle; and all remarks about its size or motion in a larger infinity of space and time would be fully applicable; i.e they would be indeterminate.
^{Ref.flandern } C. The Two Particle universe
^{Ref.flandern } Now, for the first time, we have "scale" in our universe, and can measure the dimensions of the particles themselves as a fraction of the distance between them.
^{Ref.flandern } There is no such thing as "absolute length" in this universe -- we cannot tell if the two particles are "close together" or "far apart".
^{Ref.flandern } Their separation is indeterminate relative to the universe beyond.
^{Ref.flandern } Note that the two particles cannot "see" or influence each other in any way except by collision, since our otherwise empty universe definitely contains no photons or agents to produce forces or actions at a distance, such as electromagnetism or gravitation.
^{Ref.flandern } In the real universe, a suspended pendulum would continue swinging back-and-forth in the same direction in the universe, ignoring the spin of the body.
^{Ref.flandern } But our two-particle universe can have no such properties, because there can be no framework to provide a "remembered" preferred orientation for the pendulum.
^{Ref.flandern } But our two-particle universe can have no such properties, because there can be no framework to provide a "remembered" preferred orientation for the pendulum.
^{Ref.flandern } Indeed, the pendulum could not swing at all, because there is no gravity in this imagined universe.
^{Ref.flandern } Now if the particle on which the pendulum is suspended is imagined to have local gravity only, so that the pendulum can swing; but gravity which does not reach out to influence the second particle, so that no framework is provided to the universe; then clearly the pendulum must keep its orientation with respect to the particle it resides on, since that is the only framework it has.
^{Ref.flandern } But as soon as we imagine a sort of universal gravitation, this immediately provides a framework for the pendulum.
^{Ref.flandern } By these constructions, we begin to see the origins of the what are called inertial forces, and the importance of a frame of reference to the properties of the universe we live in.
^{Ref.flandern } Then a body traveling at a uniform velocity from point A at time 1 to point B at time 2 is traveling on a straight line in this space-time universe.
^{Ref.flandern } In the real universe, the density of matter greatly increases as scale decreases.
^{Ref.flandern } Hence the ratio of mass to volume in electrons is enormously greater(about 1E10 g/cc) than the same ratio for matter in ordinary human experience (of order 1 g/cc), which in turn is enormously greater than the ratio for the entire visible universe (1E-31g/cc).
^{Ref.flandern } From the preceding considerations it seems altogether reasonable, and in a way compelling, to deduce that space, time,and substance are all infinitely divisible; because the consequences of the alternative are logically absurd.
^{Ref.flandern } But if they are infinitely divisible on the smaller scale, what about the larger scale? Recall our earlier argument that the entire visible universe would have undefined scale in space, time, and mass, unless such scale is provided by the presence of other substance in the greater universe beyond.
^{Ref.flandern } The upper limits to the structure of substance, the dimensions of the universe, and the extent of time, must all be as unbounded on the high side as they need to be on the small side
^{Ref.flandern } F. Meaning of Space and Time
^{Ref.flandern }Let us return again to our empty universe which contains no substance, and therefore no frame of reference, except for a single uniform particle of substance.
^{Ref.flandern } But as we have just seen, the particle must itself be composed of an infinitely divisible variety of sub-particles. We could have chosen a single particle at any of an infinite number of sub-levels to be our single particle. To avoid the issue of the arbitrary size of the particle we select, let us conceive of it as having zero radius. Although it does not, this conception will allow us to introduce one scale of distance at a time. As remarked earlier, motion and orientation have no meaning for a single particle in an empty universe.
^{Ref.flandern } Now introduce a second infinitesimal particle. This gives meaning to orientation, since angles can be measured from the line joining the two particles. It also provides a single measurement of length, the distance between the particles.
^{Ref.flandern } It does not, as before, provide a scale for the empty universe, since the distance cannot be measured in units of particle diameters, which are still being assumed to have no dimensions.
^{Ref.flandern } Therefore there is no way yet to determine whether our particles are separated by a microscopic or a macroscopic distance. There is as yet still no meaning to motion in this two particle universe.
^{Ref.flandern } The two particles cannot change direction, since all directions have meaning only relative to the particle-to-particle direction.
^{Ref.flandern } And the two particles cannot change distance, since all distances have meaning only relative to the particle-to-particle distance. In a very real sense, this universe without the possibility of motion or change has no time. Time can have no meaning if there cannot be events or change to mark its progress.
^{Ref.flandern } Put differently, if there were such a thing as an absolute time which existed somehow in addition to our two particles, the lapse of a microsecond or a million years would be just the same and utterly indistinguishable>. But the existence of something with substance, such as an absolute time scale, violates the assumptions of our construction, that nothing exists except our two infinitesimal particles in an empty universe. Remember, we refer to "substance" rather than "matter" to cover ANYTHING which exists.
^{Ref.flandern } An absolute scale of time, just as for a structure or framework in space, would have substance in this broad definition.
^{Ref.flandern } In an empty universe consisting of two elementary units of substance, the ordinary properties of the universe (time, space, matter) do not exist outside of the particles and between events of coincidence.
^{Ref.flandern } It can therefore be said in a logically meaningful way that space and time which are empty of particles and events DO NOT EXIST! This eliminates a logical fallacy we have been skirting around up to now about whether the empty space and time surrounding our particles exist.
^{Ref.flandern } In our construction they do not.
^{Ref.flandern } We are asserting that every point in the perceptible universe is at every moment of time filled with contiguous substance at some infinitesimal level.
^{Ref.flandern } If substance could be imagined to become absent anywhere at any time, time there would cease and the perceptible universe would collapse until the"vacuum" was filled.
^{Ref.flandern } But if there were such regions where matter density is so low that no collisional interactions between units of substance occurred,then all substance on the edge of such regions would instantly dissipate itself into the non-interacting regions, followed by substance slightly further in, and so on.
^{Ref.flandern } All substance in this universe would dissipate instantly into the void.
^{Ref.flandern } Since this does not happen, we conclude that this universe has no such regions where collisional interactions between units of substance do not occur.
^{Ref.flandern } The same reasoning applies to time. A cessation of collisional events would bring a cessation of time; but with matter existing everywhere with sufficient density for collisions, it follows that time continues forever, in both the future and the past.
^{Ref.flandern } The substances whose presence "define" space-time must be infinitesimal compared to the substances in our experience, such as baryons or photons, or even neutrinos.
^{Ref.taojoe } There are eleven total distinct dimensions in the system.
^{Ref.taojoe }... as they do in the universe, all these geometries are in motion.
^{Ref.taojoe } I think that each of the eleven dimensions represents one basic aspect, one constituent of the universe. The attempt by physics to explain phenomena by associating discrete particles with each has gotten out of hand. It may be more useful and to the point to identify eleven fundamental properties of the universe, each associated with its own dimension.
^{Ref.taojoe } What do you suppose would happen if we slowed this point to a sub-light velocity? If c really does produce an opposite, then i^{1}or i , becomes its negative, i^{1}or -i , which is the value of i^{3}, which is, lo and behold, an alternate symbol for D_{3}. D_{3} then (or i,) is the realm of slower-than-light imaginary points. Since it is sub-light it must have mass, but since it is imaginary, it has no substance. And this is exactly what our plane in motion is trying to describe, a surface with no thickness and therefore no substance, enclosing on itself trying to form a sphere (or a tube) with no volume. An imaginary figure with mass. Hey, astronomy buffs, what am I describing? This is dark matter! All that unaccounted-for mass that must exist in the universe but is undetectable. Its undetectable because its not real. But because its slowed to sub-light, it has mass. And because these things are negative singularities, similar to the singularity responsible for the big bang and the subsequent universe except hollow inside (like bubbles or macaroni), they are super dense, far more dense even than black holes, so it makes sense, as physicists claim, that this dark matter should account for most of the mass in the universe, much more than visible matter. D_{3} = dark matter, matter which has been collapsed by one dimension.
^{Ref.taojoe } D_{1} D_{3} is responsible for the dark matter which is undetectable but comprises most of the mass in the universe.
^{Ref.taojoe } Constancy in the eyes of humans, and constancy in the eyes of the universe, is not the same thing. (I have already hinted at this.) The speed of light is a kind of threshold for a change of state of various cosmic manifestations.
^{Ref.sqrtneg1 } (Schrodinger's wave function also involves the same square root of -1 when the spatial locality breaks down and a conscious observer gets involved in the process of measurement.) This apparent coincidence indicates something extremely significant for understanding the universe and consciousness. It suggests that human consciousness may be a dimension of anti-space that merges into the fourth dimension of negative space so that we cannot see it as spatial any more. It appears as one dimensional time instead."
^{Ref.majka }There is an observer which is at rest with respect to the charged particle.
^{Ref.majka }This observer "sees" the gravitational field and the electric field of this particle.
^{Ref.majka }Let us now add a second observer. The second observer is exactly like the first observer except that it is traveling at some constant speed, v, which is less than the speed of light, with respect to the first observer and the charged particle.
^{Ref.majka }This second observer also "sees" the gravitational field and the electric field of the charged particle. However, this second observer also "sees" a magnetic field surrounding the charged particle.
^{Ref.majka }At the same time, the three observers see the charged particle differently. At a relative speed of zero, the observer "sees" a mass and an electric field.
^{Ref.majka }At a relative speed other than zero but less than that of light, the second observer "sees" a mass, an electric field and a magnetic field.
^{Ref.majka }At a relative speed of light, the third observer "sees" an electromagnetic wave with no gravitational field and no electric field other than that associated with the electromagnetic wave itself.
^{Ref.majka }The hypothesis is that as the relative speed of a charged particle increases from zero to that of light, the particle appears to change to an electromagnetic wave because of the expansion of the magnetic field. This magnetic field combines with some of the static electric field, in proportion to the energy of the magnetic field, to form an electromagnetic wave.
^{Ref.majka }At the speed of light, the electric field is entirely combined with the magnetic field and the particle appears as an electromagnetic wave.
^{Ref.majka }where uo is the magnetic permeability of free space. The equation describing the energy density of the particle's electric field, Ue, is:
^{Ref.majka }Ue = eo E^{2}
^{Ref.majka }where eo is the electric permittivity of free space.
^{Ref.majka }The total energy, Ut, of the electric and magnetic field of a particle traveling at some speed, v, is the sum of these two equations. Converting to like terms and combining terms, the total energy equation is:
^{Ref.majka }Ut = ( eo E^{2} / 2) ( 1 + v^{2} /c^{2} )
^{Ref.majka }That is, when traveling at high speeds, charged particles exhibit particle characteristics and electromagnetic wave characteristics. If, as is hypothesized, the magnetic field combines with a portion of the static electric field to create an electromagnetic wave, duality is expected.
^{Ref.majka }Since the particle is only partially an electromagnetic wave, it should exhibit duality at speeds less than light.
^{Ref.majka }The accepted theory is that mass increases as speed increases. The finding by Bucherer in 1908, that the electric field to mass (e/m) ratio is less for high speed particles, has been accepted as proof of an increase in mass.
^{Ref.majka }The hypothesis proposes that the reason for this finding is not that the mass has increased but rather that the electric field and the mass have decreased.
^{Ref.majka }That part of the electric field which combines with the magnetic field to create an electromagnetic field can not participate in static charge measurements.
^{Ref.majka }The hypothesis states that the apparent mass of the particle decreases with relative speed and that the magnetic field combines with a portion of the electric field to produce an electromagnetic wave.
^{Ref.majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }We are familiar with speeds less than light. At a zero relative speed, the "stopped" electromagnetic wave appears as a "particle" and exhibits a gravitational field and an electric field.
^{Ref.majka }Similarly, as the speed of a particle increases, the effects of the static electric field are decreased.
^{Ref.majka }At frequencies greater than the resonant frequency, the impedance of the circuit is due primarily to the inductor. We may then assume that, by analogy, at speeds greater than the speed of light, the magnetic field will dominate and will appear to be as constant as the electric field at sub-light speeds.
^{Ref.majka }At these speeds, (greater than the "speed" of light) it may appear that the electric field is a function of the magnetic field.
^{Ref.majka }The impedance offered by the capacitor is analogous to the electric field of a charged particle and the impedance offered by the inductor is analogous to the magnetic field of a charged particle in motion.
^{Ref.majka }Let us assume a series LC circuit, as described above, with no applied signal. The inductor does not have an initial magnetic field nor does the capacitor have an initial electric field.
^{Ref.majka }The electric and magnetic fields of a particle have been associated with the impedances offered by the capacitor and inductor of an analogous series LC circuit.
^{Ref.majka }At speeds greater than light, the hypothesis predicts that the effects of the electric and magnetic fields will be reversed.
^{Ref.majka }At these speeds, it is likely that the magnetic fields will become polar and the electric fields will become circular, that is, a magnetic monopole will result.
^{Ref.majka }At speeds much greater than that of light, the electric field may be expected to collapse.
^{Ref.majka }This collapsed electric field may also pull or twist the fabric of space-time and form a type of field not now known.
^{Ref.andersen }There the hypothesis (majka) was set forth that the ratio of magnetic-to-electric field "seen" by an observer of a charged particle is a function of the relative motion between the particle and observer.
^{Ref.andersen }Also, that the magnetic permeability of free space and the electric permittivity thereof are analogous to the distributed inductance and capacitance, respectively, of a transmission line.
^{Ref.andersen }Tilton introduces the concept of "super resonance", in which he discusses a dual oscillator system (two pendulums). On page 67 he illustrates a theoretical electrical version of the dual pendulums-- dual LC tank circuits coupled together through back-to- back zener diodes.
^{Ref.majka }This second observer also "sees" the gravitational field and the electric field of the charged particle. However, this second observer also "sees" a magnetic field surrounding the charged particle.
^{Ref.majka }At a relative speed other than zero but less than that of light, the second observer "sees" a mass, an electric field and a magnetic field.
^{Ref.majka }The hypothesis is that as the relative speed of a charged particle increases from zero to that of light, the particle appears to change to an electromagnetic wave because of the expansion of the magnetic field. This magnetic field combines with some of the static electric field, in proportion to the energy of the magnetic field, to form an electromagnetic wave.
^{Ref.majka }At the speed of light, the electric field is entirely combined with the magnetic field and the particle appears as an electromagnetic wave.
^{Ref.majka }At speeds less than that of light, the magnetic field of the electromagnetic wave collapses. The collapsing field distorts or twists space-time which appears to us as a gravitational field.
^{Ref.majka }This hypothesis seems to be justified by equations from classical physics. The equation describing the energy density of the particle's magnetic field, Um, is:
^{Ref.majka }Um = B^{2} / ( 2uo )
^{Ref.majka }where uo is the magnetic permeability of free space. The equation describing the energy density of the particle's electric field, Ue, is:
^{Ref.majka }Ue = eo E^{2}
^{Ref.majka }The total energy, Ut, of the electric and magnetic field of a particle traveling at some speed, v, is the sum of these two equations. Converting to like terms and combining terms, the total energy equation is:
^{Ref.majka }Ut = ( eo E^{2} / 2) ( 1 + v^{2} /c^{2} )
^{Ref.majka }If we now let V = C, the equation becomes:
^{Ref.majka }Ut = eo E^{2}
^{Ref.majka }which is also the energy density equation of an electromagnetic wave. Classical physics equations also show that the direction of the magnetic field of a charged particle, traveling at some speed, is such that the Poynting Vector cross product is satisfied.
^{Ref.majka }...when traveling at high speeds, charged particles exhibit particle characteristics and electromagnetic wave characteristics. If, as is hypothesized, the magnetic field combines with a portion of the static electric field to create an electromagnetic wave, duality is expected.
^{Ref.majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }The speed of light is the square root of (1/uoeo) which are the magnetic permeability and electric permittivity of free space.
^{Ref.majka }Therefore, we may assume that the speed of light is the resonant velocity of free space.
^{Ref.majka }Similarly, we may compare the impedance of the inductor to the magnetic field of a particle in relative motion.
^{Ref.majka }At zero Hertz, there is no impedance offered by the inductor and a "particle" at zero relative speed has no magnetic field. As the frequency of the applied signal to the circuit is increased, the impedance provided by the inductor is increased.
^{Ref.majka }As the speed of the particle increases, the effects of the magnetic field are increased.
^{Ref.majka }At speeds less than that of light, the electric field is dominant and the magnetic field is a function of the electric charge.
^{Ref.majka }At frequencies greater than the resonant frequency, the impedance of the circuit is due primarily to the inductor. We may then assume that, by analogy, at speeds greater than the speed of light, the magnetic field will dominate and will appear to be as constant as the electric field at sub-light speeds.
^{Ref.majka }At frequencies greater than the resonant frequency, the impedance of the circuit is due primarily to the inductor. We may then assume that, by analogy, at speeds greater than the speed of light, the magnetic field will dominate and will appear to be as constant as the electric field at sub-light speeds.
^{Ref.majka }At these speeds, it may appear that the electric field is a function of the magnetic field.
^{Ref.majka }The impedance offered by the capacitor is analogous to the electric field of a charged particle and the impedance offered by the inductor is analogous to the magnetic field of a charged particle in motion.
^{Ref.majka }NON-STEADY-STATE CONDITIONS
^{Ref.majka }Let us assume a series LC circuit, as described above, with no applied signal. The inductor does not have an initial magnetic field nor does the capacitor have an initial electric field.
^{Ref.majka }The electric and magnetic fields of a particle have been associated with the impedances offered by the capacitor and inductor of an analogous series LC circuit.
^{Ref.majka }The hypothesis proposes that the mass of a particle is due to the collapse of the magnetic field of the particle.
^{Ref.majka }Mass is not recognized directly but a gravitational field is. A gravitational field is probably not a different form of a magnetic field.
^{Ref.majka }The gravitational field is, most likely, a result of the collapsed magnetic field.
^{Ref.majka }It is possible that the collapsed magnetic field pulls or twists the fabric of space-time in such a way as to form what we call a gravitational field.
^{Ref.majka }As the speed of the charged particle increases, the magnetic field expands and decreases its pull or twist which causes a decrease in the gravitational field.
^{Ref.majka }At speeds greater than light, the hypothesis predicts that the effects of the electric and magnetic fields will be reversed.
^{Ref.majka }At these speeds, it is likely that the magnetic fields will become polar and the electric fields will become circular, that is, a magnetic monopole will result.
^{Ref.majka }According to the Theory of Relativity, the third observer must "see" an electromagnetic wave at the location of the charged particle since their relative speed is the speed of light.
^{Ref.majka }At a relative speed of light, the third observer "sees" an electromagnetic wave with no gravitational field and no electric field other than that associated with the electromagnetic wave itself.
^{Ref.majka }The hypothesis is that as the relative speed of a charged particle increases from zero to that of light, the particle appears to change to an electromagnetic wave because of the expansion of the magnetic field. This magnetic field combines with some of the static electric field, in proportion to the energy of the magnetic field, to form an electromagnetic wave.
^{Ref.majka }At the speed of light, the electric field is entirely combined with the magnetic field and the particle appears as an electromagnetic wave.
^{Ref.majka }At speeds less than that of light, the magnetic field of the electromagnetic wave collapses. The collapsing field distorts or twists space-time which appears to us as a gravitational field.
^{Ref.majka }which is also the energy density equation of an electromagnetic wave. Classical physics equations also show that the direction of the magnetic field of a charged particle, traveling at some speed, is such that the Poynting Vector cross product is satisfied.
^{Ref.majka }That is, E x H = I.
^{Ref.majka }The hypothesis is also supported by experiments which have shown that charged particles traveling at a high speed exhibit duality.
^{Ref.majka }That is, when traveling at high speeds, charged particles exhibit particle characteristics and electromagnetic wave characteristics. If, as is hypothesized, the magnetic field combines with a portion of the static electric field to create an electromagnetic wave, duality is expected.
^{Ref.majka }Since the particle is only partially an electromagnetic wave, it should exhibit duality at speeds less than light.
^{Ref.majka }The accepted theory is that mass increases as speed increases. The finding by Bucherer in 1908, that the electric field to mass (e/m) ratio is less for high speed particles, has been accepted as proof of an increase in mass.
^{Ref.majka }The hypothesis proposes that the reason for this finding is not that the mass has increased but rather that the electric field and the mass have decreased.
^{Ref.majka }That part of the electric field which combines with the magnetic field to create an electromagnetic field can not participate in static charge measurements.
^{Ref.majka }Therefore, those experiments measuring e/m will show a lower value for high speed particles than for slower particles.
^{Ref.majka }CHARACTERISTIC VELOCITY OF SPACE
^{Ref.majka }It has been assumed that electromagnetic waves can travel only at the speed of light. The hypothesis proposes that there is an electromagnetic wave which is a characteristic of any charged particle traveling at any relative speed greater than zero and less than the speed of light.
^{Ref.majka }Since electromagnetic waves travel through transmission lines and through space, it is possible to model their propagation through space by a transmission line analogy.
^{Ref.majka }In an "ideal" transmission line, without losses, the resistance is ignored. Since it seems that an electromagnetic wave travels through space without losses, we may assume that the model for an ideal transmission line is adequate for an analysis of free space.
^{Ref.majka }electromagnetic waves, which are characteristic of charged particles, can travel at speeds other than the speed of light.
^{Ref.majka }STEADY-STATE IMPEDANCES
^{Ref.majka }The hypothesis predicts that electromagnetic waves can travel at speeds other than at the speed of light.
^{Ref.majka }At light speed, the universe offers no impedance to the propagation of electromagnetic waves.
^{Ref.majka }We are familiar with speeds less than light. At a zero relative speed, the "stopped" electromagnetic wave appears as a "particle" and exhibits a gravitational field and an electric field.
^{Ref.majka }If we assume the creation of a particle, we would see that this particle causes a disturbance which propagates as an electromagnetic wave.
^{Ref.majka }Now we change the frequency of the applied signal. Again the circuit will respond with an oscillation at it's resonant frequency.
^{Ref.majka }Similarly, if we accelerate a charged particle, an electromagnetic wave is generated. Indeed, any change in the frequency of the applied signal to a series LC circuit will generate transient oscillations just as acceleration of a charged particle will generate electromagnetic waves.
^{Ref.vogt }"There are direct analogies between this type of invention and our own existence. In Figure 4.10, Object 15 could be analogized as being the information in the diehold. Item 13 and 23 can be considered the tapehead. In this invention it is actually the two parts of the laser beam directed at the object that pick up the information that makes up the image of the object. The microwave oscillator (Item 18) and the optical oscillator (Item 16) can be considered the carrier wave and synchronizing frequencies which we have been talking about. This information is directed to Items 25 (Filter) and 26 (Device functioning as a videocon or image orthocon), which converts the light information to electromagnetic waves. This is similar to what we define as being the second dimension or the transmission dimension.
^{Ref.taojoe }.....lets take a simple case using dimensions: 4 D 2 = 2. In words, when we extract 2 dimensions from a solid object (in motion, of course), the result is 2 distinct electromagnetic waves (photons), whose amplitudes are perpendicular to each other (intersecting planes).
^{Ref.taojoe } D_{2} = wave phenomena, the electromagnetic force, or the photon
^{Ref.taojoe }We can take D_{2}, the photon and kick it up a notch, and do D_{2} + D_{1} = D_{3}. In words, this means that we add a new 90 degree vector to our planar electromagnetic wave, producing an abstraction we can refer to as a cubic wave.
^{Ref.taojoe } D_{2}, i^{2} and -1 are symbols that correlate to electromagnetic waves or photons. The only difference between D_{4} and D_{2} is that D_{2} = D_{4} x c (remind you of something? E = mc^{2}, perhaps?). Substituting numerals, -1 = (+1) c, and solving c = -1. In words, c has the effect of negativizing a phenomenon.
^{Ref.taojoe }1. The speed of light is a misleading term, for while it is the speed of electromagnetic waves.
^{Ref.majka }Since electromagnetic waves travel through transmission lines and through space, it is possible to model their propagation through space by a transmission line analogy.
^{Ref.majka }transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }Models of transmission lines are basic in the study of electricity and electronics. A model circuit diagram describing a typical, real transmission line is shown in Figure 1.
^{Ref.majka }The inductance, L, is in terms of henries per meter. The capacitance, C, is in terms of farads per meter and the resistance, R, is in terms of Ohms per meter.
^{Ref.majka }Note that the circuit diagram basically consists of one RLC circuit repeated for the length of the transmission line. The resistance, R, is responsible for losses in transmission lines.
^{Ref.majka }In an "ideal" transmission line, without losses, the resistance is ignored. Since it seems that an electromagnetic wave travels through space without losses, we may assume that the model for an ideal transmission line is adequate for an analysis of free space.
^{Ref.majka }Also, since the circuit segment is repeated for the length of the transmission line, the analysis of one segment is sufficient.
^{Ref.majka }Figure 2 shows the circuit diagram for an ideal transmission line.
^{Ref.majka }Circuit modeling involves determining the voltages and currents through the circuit. By Ohms Law (E = I x Z), the voltages and currents are related through impedances. (Note: Impedance is mathematically treated as a resistance.
^{Ref.majka }It differs from a resistance in that there are no energy losses through an impedance.)
^{Ref.majka }Figure 3 shows the same circuit with the impedances of the circuit elements.
^{Ref.majka }The values of the impedances are shown in typical electrical analysis notation. Since the impedance of an inductor varies directly with the frequency of the current through it or voltage applied to it, the impedance is in terms of the frequency, jw.
^{Ref.majka }Since the impedance of a capacitor varies inversely with the frequency of the current through it or voltage applied to it, the impedance is in terms of the inverse frequency, 1/jw. (In electrical analysis, since the symbol "i" is used to represent current flow, the symbol "j" is used to represent the square root of -1 and the symbol, w or omega, is used to represent frequency where w = 2 pi f.)
^{Ref.majka }It can be seen that this circuit is also the circuit of a series LC circuit. To go from a transmission line model to a series LC circuit model all we need do is change the terms of the parameters from henries/meter and farads/meter to henries and farads. The normalized transfer function, H(jw), of such a circuit is:
^{Ref.majka }H(jw) = 1/( w^{2} - wo^{2})
^{Ref.majka }The symbol w represents the frequency of the signal applied to the circuit. The symbol wo represents the resonant frequency of the circuit and it is numerically equal to the square root of (1/LC).
^{Ref.majka }The resonant frequency is the frequency preferred by the circuit.
^{Ref.majka }If a signal was applied to the circuit and it was not at the resonant frequency, the circuit would offer an impedance to the signal.
^{Ref.majka }If a signal at the resonant frequency was applied to the circuit, the circuit would offer no impedance. The reason for this is that the impedance of the inductor (jw) varies directly with the frequency of the applied signal.
^{Ref.majka }The impedance of the capacitor (1/jw) varies inversely with the frequency of the applied signal. At the resonant frequency, the magnitude of the impedance offered by the inductor and the capacitor are equal.
^{Ref.majka }Impedances due to inductors and capacitors are vector quantities. The direction of the inductor's impedance vector varies directly with the frequency of the applied signal in the positive direction.
^{Ref.majka }The direction of the capacitor's impedance vector also varies directly with the frequency of the applied signal but in the negative direction.
^{Ref.majka }At resonance, the magnitudes of the impedances are equal but the vectors are 180 degrees out of phase with each other and thus cancel. At resonance, the circuit offers no impedance.
^{Ref.majka }The values for L and C in a series LC circuit are in terms of henries and farads. The resonant frequency, wo, is equal to the square root of (1/LC).
^{Ref.majka }The resonant frequency, then, is in terms of 1/second or Hertz.
^{Ref.majka }If we were to substitute henries per meter and farads per meter for the values of the circuit elements, then resonance would be in terms of meters per second.
^{Ref.majka }Instead of a resonant frequency we would have a resonant velocity.
^{Ref.majka }Indeed, for transmission lines, the velocity of propagation is the square root of (1/LC).
^{Ref.majka }The speed of light is the square root of (1/uoeo) which are the magnetic permeability and electric permittivity of free space.
^{Ref.majka }Therefore, we may assume that the speed of light is the resonant velocity of free space.
^{Ref.andersen } Also, that the magnetic permeability of free space and the electric permittivity thereof are analogous to the distributed inductance and capacitance, respectively, of a transmission line.
^{Ref.andersen }This leads to the suggestion that a transmission line's RESONANCE, which depends on its inductance (L) and capacitance (C), corresponds to a RESONANT VELOCITY of free space, that velocity being the speed of light (2.998E8 meters/sec). For the implications of this hypothesis, see the file GRAVITY2.ZIP.
^{Ref.majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }Models of transmission lines are basic in the study of electricity and electronics. A model circuit diagram describing a typical, real transmission line is shown in Figure 1.
^{Ref.majka }The inductance, L, is in terms of henries per meter. The capacitance, C, is in terms of farads per meter and the resistance, R, is in terms of Ohms per meter.
^{Ref.majka }Note that the circuit diagram basically consists of one RLC circuit repeated for the length of the transmission line. The resistance, R, is responsible for losses in transmission lines.
^{Ref.majka }It can be seen that this circuit is also the circuit of a series LC circuit. To go from a transmission line model to a series LC circuit model all we need do is change the terms of the parameters from henries/meter and farads/meter to henries and farads. The normalized transfer function, H(jw), of such a circuit is:
^{Ref.majka }H(jw) = 1/( w^{2} - wo^{2})
^{Ref.majka }The series LC circuit does not forbid frequencies other than the resonant frequency but it does provide an impedance to them.
^{Ref.majka }We should note that the series LC circuit does not prohibit frequencies greater than the resonant frequency.
^{Ref.majka }Since the analogy between series LC circuits and free space has held in other circumstances we may assume that space also does not prohibit speeds greater than resonant speed but will provide an impedance to them.
^{Ref.majka }In the series LC circuit, the impedance encountered by a signal with a frequency of zero Hertz is provided entirely by the capacitance. As the frequency of the signal is increased, the impedance of the capacitor is reduced.
^{Ref.majka }Let us assume a series LC circuit, as described above, with no applied signal. The inductor does not have an initial magnetic field nor does the capacitor have an initial electric field.
^{Ref.majka }Now let us apply a signal of zero Hertz and the circuit will oscillate at its resonant frequency.
^{Ref.majka }In a real circuit, resistances cause the oscillation to dampen. In an ideal circuit, the oscillation does not die out and continues forever.
^{Ref.majka }If we assume the creation of a particle, we would see that this particle causes a disturbance which propagates as an electromagnetic wave.
^{Ref.majka }Now we change the frequency of the applied signal. Again the circuit will respond with an oscillation at it's resonant frequency.
^{Ref.majka }Similarly, if we accelerate a charged particle, an electromagnetic wave is generated. Indeed, any change in the frequency of the applied signal to a series LC circuit will generate transient oscillations just as acceleration of a charged particle will generate electromagnetic waves.
^{Ref.majka }GRAVITY
^{Ref.majka }The electric and magnetic fields of a particle have been associated with the impedances offered by the capacitor and inductor of an analogous series LC circuit.
^{dictionary } Symbol Z. A measure of the total opposition to current flow in an alternating current circuit, made up of two components, ohmic resistance and reactance, and usually represented in complex notation as Z = R + iX, where R is the ohmic resistance and X is the reactance.
^{Ref.majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }Circuit modeling involves determining the voltages and currents through the circuit. By Ohms Law (E = I x Z), the voltages and currents are related through impedances. (Note: impedance is mathematically treated as a resistance.
^{Ref.majka }It differs from a resistance in that there are no energy losses through an impedance.)
^{Ref.majka }Figure 3 shows the same circuit with the impedances of the circuit elements.
^{Ref.majka }The values of the impedances are shown in typical electrical analysis notation. Since the impedance of an inductor varies directly with the frequency of the current through it or voltage applied to it, the impedance is in terms of the frequency, jw.
^{Ref.majka }Since the impedance of a capacitor varies inversely with the frequency of the current through it or voltage applied to it, the impedance is in terms of the inverse frequency, 1/jw. (In electrical analysis, since the symbol "i" is used to represent current flow, the symbol "j" is used to represent the square root of -1 and the symbol, w or omega, is used to represent frequency where w = 2 pi f.)
^{Ref.majka }If a signal was applied to the circuit and it was not at the resonant frequency, the circuit would offer an impedance to the signal.
^{Ref.majka }If a signal at the resonant frequency was applied to the circuit, the circuit would offer no impedance. The reason for this is that the impedance of the inductor (jw) varies directly with the frequency of the applied signal.
^{Ref.majka }The impedance of the capacitor (1/jw) varies inversely with the frequency of the applied signal. At the resonant frequency, the magnitude of the impedance offered by the inductor and the capacitor are equal.
^{Ref.majka }impedances due to inductors and capacitors are vector quantities. The direction of the inductor's impedance vector varies directly with the frequency of the applied signal in the positive direction.
^{Ref.majka }The direction of the capacitor's impedance vector also varies directly with the frequency of the applied signal but in the negative direction.
^{Ref.majka }At resonance, the magnitudes of the impedances are equal but the vectors are 180 degrees out of phase with each other and thus cancel. At resonance, the circuit offers no impedance.
^{Ref.majka }The series LC circuit does not forbid frequencies other than the resonant frequency but it does provide an impedance to them.
^{Ref.majka }Similarly, we may assume that the universe does not forbid speeds other than the speed of light but would provide an impedance to them.
^{Ref.majka }Since the analogy between series LC circuits and free space has held in other circumstances we may assume that space also does not prohibit speeds greater than resonant speed but will provide an impedance to them.
^{Ref.majka }STEADY-STATE impedanceS
^{Ref.majka }The hypothesis predicts that electromagnetic waves can travel at speeds other than at the speed of light.
^{Ref.majka }At light speed, the universe offers no impedance to the propagation of electromagnetic waves.
^{Ref.majka }At other than light speeds, it is expected that the universe will provide an impedance to these waves.
^{Ref.majka }We are familiar with speeds less than light. At a zero relative speed, the "stopped" electromagnetic wave appears as a "particle" and exhibits a gravitational field and an electric field.
^{Ref.majka }In the series LC circuit, the impedance encountered by a signal with a frequency of zero Hertz is provided entirely by the capacitance. As the frequency of the signal is increased, the impedance of the capacitor is reduced.
^{Ref.majka }Similarly, as the speed of a particle increases, the effects of the static electric field are decreased.
^{Ref.majka }Similarly, we may compare the impedance of the inductor to the magnetic field of a particle in relative motion.
^{Ref.majka }At zero Hertz, there is no impedance offered by the inductor and a "particle" at zero relative speed has no magnetic field. As the frequency of the applied signal to the circuit is increased, the impedance provided by the inductor is increased.
^{Ref.majka }As the speed of the particle increases, the effects of the magnetic field are increased.
^{Ref.majka }At frequencies less than the resonant frequency, the impedance of the circuit is due primarily to the capacitor.
^{Ref.majka }At speeds less than that of light, the electric field is dominant and the magnetic field is a function of the electric charge.
^{Ref.majka }At frequencies greater than the resonant frequency, the impedance of the circuit is due primarily to the inductor. We may then assume that, by analogy, at speeds greater than the speed of light, the magnetic field will dominate and will appear to be as constant as the electric field at sub-light speeds.
^{Ref.majka }The impedance offered by the capacitor is analogous to the electric field of a charged particle and the impedance offered by the inductor is analogous to the magnetic field of a charged particle in motion.
^{Ref.majka }GRAVITY
^{Ref.majka }The electric and magnetic fields of a particle have been associated with the impedances offered by the capacitor and inductor of an analogous series LC circuit.
^{Ref.andersen }An electronic circuit analogy was announced by Schelkunoff in 1938 [footnote 1] in which it was shown that the intrinsic impedance of free space is analogous to the characteristic impedance of a network, and the Q of a lossy medium is analogous to the Q of a network. That analog is in common use today in antenna design.
^{dictionary } 1. Symbol C The ratio of charge to potential on an electrically charged, isolated conductor. 2. Symbol C The ratio of the electric charge transferred from one to the other of a pair of conductors to the resulting potential difference between them. 3.a. The property of a circuit element that permits it to store charge. b. The part of the circuit exhibiting capacitance.
^{Ref.majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{Ref.majka }The inductance, L, is in terms of henries per meter. The capacitance, C, is in terms of farads per meter and the resistance, R, is in terms of Ohms per meter.
^{Ref.majka }In the series LC circuit, the impedance encountered by a signal with a frequency of zero Hertz is provided entirely by the capacitance. As the frequency of the signal is increased, the impedance of the capacitor is reduced.
^{Ref.andersen }Also, that the magnetic permeability of free space and the electric permittivity thereof are analogous to the distributed inductance and capacitance, respectively, of a transmission line.
^{Ref.andersen }This leads to the suggestion that a transmission line's RESONANCE, which depends on its inductance (L) and capacitance (C), corresponds to a RESONANT VELOCITY of free space, that velocity being the speed of light (2.998E8 meters/sec). For the implications of this hypothesis, see the file GRAVITY2.ZIP.
^{Ref.andersen }That electronic circuit analogy has been developed further by the author [Tilton] in accordance with the ideas that the permittivity and permeability of free space are analogous to the capacitance and inductance, respectively, of a tuned circuit. In this extension of the analogy, the speed of light becomes analogous to the resonant frequency of a tuned circuit.
^{majka }Let us assume that there is a charged particle in free space.
^{majka }There is an observer which is at rest with respect to the charged particle.
^{majka }This observer "sees" the gravitational field and the electric field of this particle.
^{majka }Let us now add a second observer. The second observer is exactly like the first observer except that it is traveling at some constant speed, v, which is less than the speed of light, with respect to the first observer and the charged particle.
^{majka }This second observer also "sees" the gravitational field and the electric field of the charged particle. However, this second observer also "sees" a magnetic field surrounding the charged particle.
^{majka }Now, we will add a third observer which is identical to the first two observers except that this observer is traveling at the speed of light relative to the first observer and to the charged particle.
^{majka }According to the Theory of Relativity, the third observer must "see" an electromagnetic wave at the location of the charged particle since their relative speed is the speed of light.
^{majka }At the same time, the three observers see the charged particle differently. At a relative speed of zero, the observer "sees" a mass and an electric field.
^{majka }At a relative speed other than zero but less than that of light, the second observer "sees" a mass, an electric field and a magnetic field.
^{majka }At a relative speed of light, the third observer "sees" an electromagnetic wave with no gravitational field and no electric field other than that associated with the electromagnetic wave itself.
^{majka }HYPOTHESIS
^{majka }The hypothesis is that as the relative speed of a charged particle increases from zero to that of light, the particle appears to change to an electromagnetic wave because of the expansion of the magnetic field. This magnetic field combines with some of the static electric field, in proportion to the energy of the magnetic field, to form an electromagnetic wave.
^{majka }At the speed of light, the electric field is entirely combined with the magnetic field and the particle appears as an electromagnetic wave.
^{majka }At speeds less than that of light, the magnetic field of the electromagnetic wave collapses. The collapsing field distorts or twists space-time which appears to us as a gravitational field.
^{majka }Thus, it is the distortion of space-time which appears to us as "mass" rather than "mass" causing the distortion.
^{majka }JUSTIFICATION
^{majka }Energy Density
^{majka }This hypothesis seems to be justified by equations from classical physics. The equation describing the energy density of the particle's magnetic field, Um, is:
^{majka }Um = B^{2} / ( 2uo )
^{majka }where uo is the magnetic permeability of free space. The equation describing the energy density of the particle's electric field, Ue, is:
^{majka }Ue = eo E^{2}
^{majka }where eo is the electric permittivity of free space.
^{majka }The total energy, Ut, of the electric and magnetic field of a particle traveling at some speed, v, is the sum of these two equations. Converting to like terms and combining terms, the total energy equation is:
^{majka }Ut = ( eo E^{2} / 2) ( 1 + v^{2} /c^{2} )
^{majka }If we now let V = C, the equation becomes:
^{majka }Ut = eo E^{2}
^{majka }which is also the energy density equation of an electromagnetic wave. Classical physics equations also show that the direction of the magnetic field of a charged particle, traveling at some speed, is such that the Poynting Vector cross product is satisfied.
^{majka }That is, E x H = I.
^{majka }Duality
^{majka }The hypothesis is also supported by experiments which have shown that charged particles traveling at a high speed exhibit duality.
^{majka }That is, when traveling at high speeds, charged particles exhibit particle characteristics and electromagnetic wave characteristics. If, as is hypothesized, the magnetic field combines with a portion of the static electric field to create an electromagnetic wave, duality is expected.
^{majka }Since the particle is only partially an electromagnetic wave, it should exhibit duality at speeds less than light.
^{majka }OBJECTIONS
^{majka }Mass Increase
^{majka }Bucherer Experiment
^{majka }The accepted theory is that mass increases as speed increases. The finding by Bucherer in 1908, that the electric field to mass (e/m) ratio is less for high speed particles, has been accepted as proof of an increase in mass.
^{majka }The hypothesis proposes that the reason for this finding is not that the mass has increased but rather that the electric field and the mass have decreased.
^{majka }That part of the electric field which combines with the magnetic field to create an electromagnetic field can not participate in static charge measurements.
^{majka }Therefore, those experiments measuring e/m will show a lower value for high speed particles than for slower particles.
^{majka }Momentum Selector
^{majka }Experiments with particle accelerators seem to show an increase in mass with an increase in the speed of a particle.
^{majka }After being accelerated, charged particles are passed through a velocity selector which passes only those particles which are traveling at a predetermined speed.
^{majka }Immediately, the particles are passed through a momentum selector which is a uniform magnetic field. This magnetic field produces a constant acceleration on the particle which causes the particle to travel in a circular path.
^{majka }The radius of the path is proportional to the linear momentum of the particle. Since momentum is proportional to the mass of the particle, it is assumed that the radius of the path is then proportional to the mass of the particle.
^{majka }Experiments have shown that the higher the speed of the particle, the greater the radius through the momentum selector. It has been assumed from these experiments that the greater radius is due to a greater mass.
^{majka }The hypothesis states that the apparent mass of the particle decreases with relative speed and that the magnetic field combines with a portion of the electric field to produce an electromagnetic wave.
^{majka }A decrease in apparent mass should be observed in particle accelerator experiments by a decrease in the radius of the path of the particle if mass were the determining factor.
^{majka }However, electromagnetic waves also have a linear momentum and this momentum is not affected by an external magnetic field.
^{majka }When passed through a momentum selector, an electromagnetic wave would pass straight through and not describe a circular path.
^{majka }Since the electromagnetic wave is characteristic of the particle, it's path is the same as the particle's path. The linear momentum of the electromagnetic wave adds to that of the particle and increases the radius of the path.
^{majka }CHARACTERISTIC VELOCITY OF SPACE
^{majka }It has been assumed that electromagnetic waves can travel only at the speed of light. The hypothesis proposes that there is an electromagnetic wave which is a characteristic of any charged particle traveling at any relative speed greater than zero and less than the speed of light.
^{majka }Since electromagnetic waves travel through transmission lines and through space, it is possible to model their propagation through space by a transmission line analogy.
^{majka }Transmission lines and space share common parameters. The most notable are the parameters of distributed inductance (or magnetic permeability) in henries per meter, distributed capacitance (or electric permittivity) in farads per meter, characteristic impedance in Ohms and characteristic velocity in meters per second.
^{majka }Models of transmission lines are basic in the study of electricity and electronics. A model circuit diagram describing a typical, real transmission line is shown in Figure 1.
^{majka }The inductance, L, is in terms of henries per meter. The capacitance, C, is in terms of farads per meter and the resistance, R, is in terms of Ohms per meter.
^{majka }Note that the circuit diagram basically consists of one RLC circuit repeated for the length of the transmission line. The resistance, R, is responsible for losses in transmission lines.
^{majka }In an "ideal" transmission line, without losses, the resistance is ignored. Since it seems that an electromagnetic wave travels through space without losses, we may assume that the model for an ideal transmission line is adequate for an analysis of free space.
^{majka }Also, since the circuit segment is repeated for the length of the transmission line, the analysis of one segment is sufficient.
^{majka }Figure 2 shows the circuit diagram for an ideal transmission line.
^{majka }Circuit modeling involves determining the voltages and currents through the circuit. By Ohms Law (E = I x Z), the voltages and currents are related through impedances. (Note: Impedance is mathematically treated as a resistance.
^{majka }It differs from a resistance in that there are no energy losses through an impedance.)
^{majka }Figure 3 shows the same circuit with the impedances of the circuit elements.
^{majka }The values of the impedances are shown in typical electrical analysis notation. Since the impedance of an inductor varies directly with the frequency of the current through it or voltage applied to it, the impedance is in terms of the frequency, jw.
^{majka }Since the impedance of a capacitor varies inversely with the frequency of the current through it or voltage applied to it, the impedance is in terms of the inverse frequency, 1/jw. (In electrical analysis, since the symbol "i" is used to represent current flow, the symbol "j" is used to represent the square root of -1 and the symbol, w or omega, is used to represent frequency where w = 2 pi f.)
^{majka }It can be seen that this circuit is also the circuit of a series LC circuit. To go from a transmission line model to a series LC circuit model all we need do is change the terms of the parameters from henries/meter and farads/meter to henries and farads. The normalized transfer function, H(jw), of such a circuit is:
^{majka }H(jw) = 1/( w^{2} - wo^{2})
^{majka }The symbol w represents the frequency of the signal applied to the circuit. The symbol wo represents the resonant frequency of the circuit and it is numerically equal to the square root of (1/LC).
^{majka }The resonant frequency is the frequency preferred by the circuit.
^{majka }If a signal was applied to the circuit and it was not at the resonant frequency, the circuit would offer an impedance to the signal.
^{majka }If a signal at the resonant frequency was applied to the circuit, the circuit would offer no impedance. The reason for this is that the impedance of the inductor (jw) varies directly with the frequency of the applied signal.
^{majka }The impedance of the capacitor (1/jw) varies inversely with the frequency of the applied signal. At the resonant frequency, the magnitude of the impedance offered by the inductor and the capacitor are equal.
^{majka }Impedances due to inductors and capacitors are vector quantities. The direction of the inductor's impedance vector varies directly with the frequency of the applied signal in the positive direction.
^{majka }The direction of the capacitor's impedance vector also varies directly with the frequency of the applied signal but in the negative direction.
^{majka }At resonance, the magnitudes of the impedances are equal but the vectors are 180 degrees out of phase with each other and thus cancel. At resonance, the circuit offers no impedance.
^{majka }The values for L and C in a series LC circuit are in terms of henries and farads. The resonant frequency, wo, is equal to the square root of (1/LC).
^{majka }The resonant frequency, then, is in terms of 1/second or Hertz.
^{majka }If we were to substitute henries per meter and farads per meter for the values of the circuit elements, then resonance would be in terms of meters per second.
^{majka }Instead of a resonant frequency we would have a resonant velocity.
^{majka }Indeed, for transmission lines, the velocity of propagation is the square root of (1/LC).
^{majka }The speed of light is the square root of (1/uoeo) which are the magnetic permeability and electric permittivity of free space.
^{majka }Therefore, we may assume that the speed of light is the resonant velocity of free space.
^{majka }The series LC circuit does not forbid frequencies other than the resonant frequency but it does provide an impedance to them.
^{majka }Similarly, we may assume that the universe does not forbid speeds other than the speed of light but would provide an impedance to them.
^{majka }Electromagnetic waves, which are characteristic of charged particles, can travel at speeds other than the speed of light.
^{majka }We should note that the series LC circuit does not prohibit frequencies greater than the resonant frequency.
^{majka }Since the analogy between series LC circuits and free space has held in other circumstances we may assume that space also does not prohibit speeds greater than resonant speed but will provide an impedance to them.
^{majka }STEADY-STATE IMPEDANCES
^{majka }The hypothesis predicts that electromagnetic waves can travel at speeds other than at the speed of light.
^{majka }At light speed, the universe offers no impedance to the propagation of electromagnetic waves.
^{majka }At other than light speeds, it is expected that the universe will provide an impedance to these waves.
^{majka }We are familiar with speeds less than light. At a zero relative speed, the "stopped" electromagnetic wave appears as a "particle" and exhibits a gravitational field and an electric field.
^{majka }In the series LC circuit, the impedance encountered by a signal with a frequency of zero Hertz is provided entirely by the capacitance. As the frequency of the signal is increased, the impedance of the capacitor is reduced.
^{majka }Similarly, as the speed of a particle increases, the effects of the static electric field are decreased.
^{majka }Similarly, we may compare the impedance of the inductor to the magnetic field of a particle in relative motion.
^{majka }At zero Hertz, there is no impedance offered by the inductor and a "particle" at zero relative speed has no magnetic field. As the frequency of the applied signal to the circuit is increased, the impedance provided by the inductor is increased.
^{majka }As the speed of the particle increases, the effects of the magnetic field are increased.
^{majka }At frequencies less than the resonant frequency, the impedance of the circuit is due primarily to the capacitor.
^{majka }At speeds less than that of light, the electric field is dominant and the magnetic field is a function of the electric charge.
^{majka }At frequencies greater than the resonant frequency, the impedance of the circuit is due primarily to the inductor. We may then assume that, by analogy, at speeds greater than the speed of light, the magnetic field will dominate and will appear to be as constant as the electric field at sub-light speeds.
^{majka }At these speeds, it may appear that the electric field is a function of the magnetic field.
^{majka }To repeat for clarity:
^{majka }The impedance offered by the capacitor is analogous to the electric field of a charged particle and the impedance offered by the inductor is analogous to the magnetic field of a charged particle in motion.
^{majka }NON-STEADY-STATE CONDITIONS
^{majka }Let us assume a series LC circuit, as described above, with no applied signal. The inductor does not have an initial magnetic field nor does the capacitor have an initial electric field.
^{majka }Now let us apply a signal of zero Hertz and the circuit will oscillate at its resonant frequency.
^{majka }In a real circuit, resistances cause the oscillation to dampen. In an ideal circuit, the oscillation does not die out and continues forever.
^{majka }If we assume the creation of a particle, we would see that this particle causes a disturbance which propagates as an electromagnetic wave.
^{majka }Now we change the frequency of the applied signal. Again the circuit will respond with an oscillation at it's resonant frequency.
^{majka }Similarly, if we accelerate a charged particle, an electromagnetic wave is generated. Indeed, any change in the frequency of the applied signal to a series LC circuit will generate transient oscillations just as acceleration of a charged particle will generate electromagnetic waves.
^{majka }GRAVITY
^{majka }The electric and magnetic fields of a particle have been associated with the impedances offered by the capacitor and inductor of an analogous series LC circuit.
^{majka }The hypothesis proposes that the mass of a particle is due to the collapse of the magnetic field of the particle.
^{majka }Mass is not recognized directly but a gravitational field is. A gravitational field is probably not a different form of a magnetic field.
^{majka }The gravitational field is, most likely, a result of the collapsed magnetic field.
^{majka }It is possible that the collapsed magnetic field pulls or twists the fabric of space-time in such a way as to form what we call a gravitational field.
^{majka }As the speed of the charged particle increases, the magnetic field expands and decreases its pull or twist which causes a decrease in the gravitational field.
^{majka }At speeds greater than light, the hypothesis predicts that the effects of the electric and magnetic fields will be reversed.
^{majka }At these speeds, it is likely that the magnetic fields will become polar and the electric fields will become circular, that is, a magnetic monopole will result.
^{majka }At speeds much greater than that of light, the electric field may be expected to collapse.
^{majka }This collapsed electric field may also pull or twist the fabric of space-time and form a type of field not now known.
^{majka }Vangard Notes
^{majka }Our researches into the nature of gravity tend to support this paper. It appears that ANY FORM OF ENERGY (i.e., acoustic, electric, magnetic, motional (scalar) fields, etc...) can be properly driven to alter the energy/mass relationship to generate free energy, anti-gravity, matter transport or matter integration - disintegration - transport.
Title: Reality Revealed by Douglas Vogt and Gary Sultan
Dewey Decimal Number: BF 1031.V64 1978
Other Library Number: 001.9
Copyright Date: 1977
Vector Associates
P.O. Box 24223, San Jose,
California. 95154
Excerpt:
^{vogt }"As mentioned in the introduction of this book, there is a strong possibility that many of man's inventions (such as television, tape recorders, radio's, etc.) may be mirror images of the technology which makes up his own existence. This would mean that many of man's inventions are excellent clues and analogies to his own existence. One of the best clues to dramatically prove our theory of existence is man's invention of the hologram. A hologram is a three-dimensional image produced by coherent light. The object observed appears to have three-dimensional qualities. In fact, some advanced laser holograms produce images that make it impossible to tell the difference between the image and the actual object."
^{vogt }"As an example, Figure 4.10 is a patent by William C. Jakes, Jr. (Patent No. 3,566,021). The patent is for a real time, three-dimensional television system. Other laser television systems have been patented, we merely use this one as an example to show the parallel similarities between our own existence, as theorized by us, and the image created by a hologram television."
"This disclosure relates to a television system that utilizes wave front reconstruction techniques to provide a real time three-dimensional image at the receiving end of the system, with the image changing in perspective as the object and/or observer moves. The coherent light from a laser is first modulated at a frequency in the microwave range and one sideband of the coherent light is filtered out and used to illuminate an object scene. The light reflected from the object scene impinges upon a photodetector while a narrow reference beam of coherent light raster scans the photodetector to thereby generate a signal which is modulated in phase and amplitude in accordance with the interference pattern formed on the photodetector. The signal carrying the modulating phase and amplitude information is then transmitted to a remote receiver. At the received end, the phase and amplitude modulated information is recovered and stored, a frame at a time, in respective storage devices. At the end of a complete frame the stored information is read out and respectively applied to an array of phase and amplitude optical modulators. Also, at the end of a complete frame received information, a second laser at the receiver is pulsed with the light therefrom directed toward said array. In this manner, an image of the original object is obtained at the receiver. The described operation is continued a frame at a time." (3-p168)
^{vogt }Figure 4.10 Diagram of a laser holographic TV
^{vogt }Listed parts:
^{vogt }10. The holographic screen 25. Filter
11. Laser light source 26. Device functioning as a videocon or image orthocon
12. Light beam splitter 27. Transmission of the signal
13. Laser scanning beam 30. Coherent light source
14. 22, 29, 19 Various mirrors 31. Amplitude detector
16. Optical modulator 32. Phase detector
17. Optical filter 33. Storage device
18. Microwave oscillator 34. Storage device
^{vogt }"There are direct analogies between this type of invention and our own existence. In Figure 4.10, Object 15 could be analogized as being the information in the diehold. Item 13 and 23 can be considered the tapehead. In this invention it is actually the two parts of the laser beam directed at the object that pick up the information that makes up the image of the object. The microwave oscillator (Item 18) and the optical oscillator (Item 16) can be considered the carrier wave and synchronizing frequencies which we have been talking about. This information is directed to Items 25 and 26, which converts the light information to electromagnetic waves. This is similar to what we define as being the second dimension or the transmission dimension. Item 32 and 34 are the phase detector and frequency modulator for the vertical lines of information of the image. Items 31 and 33 are the amplitude detector and frequency modulator of the horizontal information. In our existence there are no physical phase or amplitude detectors in this dimension. The diehold somehow uses the phase angles and potentials from its eight different transmitting sides in such a manner that the signal modulates itself into existence. Item 30 can be considered a continuation of the carrier wave frequencies."
^{vogt }"It is within man's grasp to have a computer produce the images that we see without the necessity of photographing any object. This would be even closer to the way of our own existence. Sometime an analogy is so obvious and so simple that it is difficult to comprehend its application to man's own existence. Maybe this is a result of man's fear of knowing the truth of his own existence." Vogt.
[A series of eight lectures specially prepared for Compu-Serve Information Systems (CIS), for presentation in ASTROFORUM.
Copyright 1990 by Tom Van Flandern of Washington, DC [CIS ID code71107,2320].
Please seek the author's permission before reprinting more than two paragraphs.
If redistributed in electronic form, must include this statement of source and copyright.]
CHAPTER II.
THE NATURE OF SPACE, TIME, AND MATTER
We will be using Scientific Notation for large or small numbers.
For example, 1E5 = 10 to the power 5 = 10,000; 2E-7 = 2 times 10 to the power -7 = 0.000 000 2.
Ask about this, or anything else you aren't familiar with.
Use private messages, if you wish.
But your questions are probably shared by others.
A. Introduction
^{flandern }Last week we saw the value of deductive reasoning for determining the nature of reality, provided that a suitable starting point can be found.
^{flandern } This week we will reason deductively about the nature of the physical universe starting with a minimum of assumptions.
^{flandern } In fact, I propose that we start out with nothing whatever: a universe completely empty of everything which exists.
^{flandern }Is space an absolute thing, existing even without matter in it? Or does it depend upon the existence of matter to give it meaning? Let us define "substance" broadly as anything which exists, whether it takes the form of matter, energy, or "other".
^{flandern } In order to answer the question of whether space itself exists in the sense of having substance of any kind, we need to introduce some additional useful properties of substance.
^{flandern }B. The One Particle Universe
^{flandern }Let our starting universe remain empty of everything except a single infinitesimal "stationary" particle. Now imagine the same particle in motion.
^{flandern } How fast is it going, and in what direction? There is nothing for it to move relative to, and nothing to provide orientation.
^{flandern } All directions are equivalent, and all distances are equivalent.
^{flandern } The only way it can be otherwise is if space itself has a sort of "structure" to it, a framework to provide meaning to orientation and scale and motion.
^{flandern }However we have postulated an empty universe. In it, there is no matter, no energy, no substance of any kind except the single particle. How can there be "structure" without substance? In the real universe there is a frame of reference to provide meaning to distance and direction.
^{flandern } The reference frame is provided both by the presence of distant matter in the universe, as well as by seas of rapidly moving "agents", such as photons and neutrinos.
^{flandern } The essential point is that the reference frame is provided by the presence of substance in the universe. I would not insist that MATTER is needed; but I take it as self-evident that some sort of substance is required, or there can be no reference frame in space. In the absence of other substance in the universe, our lone particle would be incapable of motion,for motion could have no meaning.
^{flandern }Moreover (and this is something to note), the size of the universe would be indeterminate, even if our lone particle has "finite" dimensions.
^{flandern } Indeed, it is impossible to say whether the particle has infinite dimensions, finite dimensions, or is infinitesimal (without size), since there is no scale to measure by.
^{flandern } The number of such particles which can fit into the universe around it is infinite in any case.Our lone particle would even be incapable of spin.
^{flandern } If it had parts, they might move relative to one another.
^{flandern } But a uniform spherical lone particle cannot spin about any axis,because there is nothing outside the particle to spin relative to.
^{flandern } By extension, the particle could not be made to exhibit the properties of spin, such as centrifugal force -- a tendency to hurl objects off itself due to spin; nor would it tend to flatten from very rapid spin.
^{flandern } The origin of these "inertial forces" is surely rooted in the substance which defines the framework of space. Without a framework, without substance (except for the particle), without "agents" to produce forces, surely there could be no meaning to, nor consequences of, "spinning".
^{flandern } (The idea that the presence of distant matter in the universe is the origin of inertial forces is known as "Mach's Principle".
^{flandern })Our example may start to seem a little less hypothetical if we postulate a finite limit to all of the substance in the real universe, with nothing beyond.
^{flandern } (The "Big Bang" Theory in its simplest form is such a case, in which all substance remains inside a sphere whose surface consists of photons moving outward at the speed of light since the instant of the original explosion.
^{flandern }) Under this assumption, the entire substance of the universe would be like our single particle; and all remarks about its size or motion in a larger infinity of space and time would be fully applicable; i.e. they would be indeterminate.
^{flandern }C. The Two Particle Universe
^{flandern }Consider again our simple lone particle in an empty universe. Now let us imagine a second particle just like the first at another location, not touching.
^{flandern } Now, for the first time, we have "scale" in our universe, and can measure the dimensions of the particles themselves as a fraction of the distance between them.
^{flandern } There is no such thing as "absolute length" in this universe -- we cannot tell if the two particles are "close together" or "far apart".
^{flandern } Their separation is indeterminate relative to the universe beyond.
^{flandern } It can only be measured in terms of the number of particle diameters.
^{flandern }We have also introduced meaning to motion, since the separation measured in particle diameters can vary.
^{flandern } But with only two particles, if the separation "varied", we could not tell whether the particles had moved, or perhaps only changed diameter(shrunk or expanded) -- either would give the same result.
^{flandern } We can also now detect spinning.
^{flandern } Note that the two particles cannot "see" or influence each other in any way except by collision, since our otherwise empty universe definitely contains no photons or agents to produce forces or actions at a distance, such as electromagnetism or gravitation.
^{flandern }Consider a hypothetical pendulum suspended at the "north pole" of one of the two particles, taken as spinning.
^{flandern } In the real universe, a suspended pendulum would continue swinging back-and-forth in the same direction in the universe, ignoring the spin of the body (e.g.)
^{flandern } the Earth) underneath it (as many museum exhibits of the Foucault Pendulum demonstrate).
^{flandern } But our two-particle universe can have no such properties, because there can be no framework to provide a "remembered" preferred orientation for the pendulum.
^{flandern } Indeed, the pendulum could not swing at all, because there is no gravity in this imagined universe.Now if the particle on which the pendulum is suspended is imagined to have local gravity only, so that the pendulum can swing; but gravity which does not reach out to influence the second particle, so that no framework is provided to the universe; then clearly the pendulum must keep its orientation with respect to the particle it resides on, since that is the only framework it has.
^{flandern } But as soon as we imagine a sort of universal gravitation, this immediately provides a framework for the pendulum.
^{flandern } The proximity of the pendulum to a spinning particle is then no longer relevant, since the pendulum "senses"only the universal gravitational framework, and must maintain its orientation in that frame. Nothing about the forces acting on the pendulum would tell it the particle, above which it is suspended, is spinning.
^{flandern }By these constructions, we begin to see the origins of the what are called inertial forces, and the importance of a frame of reference to the properties of the universe we live in.
^{flandern } We also begin to see why it must be that scale and motion are relative,not absolute, in nature.We have just seen that absolute motion has no meaning without a frame of reference; and that such a reference frame must logically be provided by some sort of substance. This gives us a basis for looking at a very famous dilemma called "Zeno's Paradox".
^{flandern }D. Zeno's Paradox
^{flandern }Zeno's Paradox deals with the ultra-small structure of space and time. In its essence, the paradox notes that, if a moving body is in a specific place at every instant, then there is no instant when it is in transition from one place to another; and therefore motion is impossible. Since this contradicts everyday experience, it is called a paradox.
^{flandern }The same paradox can be expressed in a different form: to move from point A to point B one must first complete the trip to the mid-point.
^{flandern } Having reached that far, one must next reach the new mid-point of the remaining distance. But however far one has traveled, one must first travel half the remaining distance before one can travel all of it.
^{flandern } Hence one can never reach point B, because an infinite number of "half-the-distance" steps are required.
^{flandern }It might be, of course, that space is not infinitely divisible -- that there is a smallest possible increment of distance. But this leads to all sorts of conceptual problems.
^{flandern } Consider points X and Y, separated by the smallest possible increment of distance. Now consider another point Z, also separated from X by the minimum possible distance, but in a slightly different direction.
^{flandern } Then the distance between points Y and Z is less than the minimum possible distance, contradicting the starting assumption.
^{flandern } But if space were "grid-like", so that adjacent cells had no overlap, then motion in any desired direction would not be possible, unless one took a zigzag path from grid-point to grid-point! Clearly, the postulate of a"minimum possible distance" is problematical.
^{flandern }If time is treated like just another dimension (a "fourth dimension" of space), the same remarks might be extended to include the concept of a "minimum possible time unit".
^{flandern } Or we may make a separate argument about time. If there were a minimum possible time unit, then all existing substance would have one condition at one time moment, and some slightly different condition at the next time moment.
^{flandern } By hypothesis, there is no possible interval in time, nor any moment in between when anything could have happened to provide a transition from the first condition to the second.
^{flandern } It is therefore just exactly as if everything existing at the first time moment ceased to exist,and then was created from nothingness in its new condition at the next time instant.
^{flandern }We conclude then that space and time must be infinitely divisible in order to avoid these dilemmas.
^{flandern } But is this not also ruled out, by Zeno's argument? The problem is with our intuitions: while it is easy for us to imagine a whole as composed of an infinite number of parts, it is difficult for us to imagine an infinite number of components being assembled into a finite whole. As is well known in mathematics, an infinite series CAN have a finite sum.
^{flandern } For example, there are an infinite number of possible fractions or decimal numbers between zero and one, yet obviously only a finite interval.
^{flandern }In Gamow's book, "One, Two, Three ... Infinity", we learn how to count and compare things made up of an infinite number of parts, using one-to-one correspondences.
^{flandern } Such a one-to-one correspondence can be set up between points in a space interval, and decimal numbers between zero and one. Since the interval from zero to one is finite by definition, the one-to-one correspondence shows us that the space interval is finite also.
^{flandern } With another one-to-one correspondence we also conclude that it is possible to traverse the space interval in a finite time as well.
^{flandern } This is an important point, even though our intuitions do not deal with it easily.
^{flandern } The infinite mathematical series (1/2 +1/4 + 1/8 + 1/16 + ...), where each new term is half the preceding one, has a finite sum of 1.0.
^{flandern } It is clear that, in mathematics, there exist finite intervals with an infinite number of points between, and infinite series with finite sums.
^{flandern } By placing these in one-to-one correspondence with the physical concepts of space, time, and mass, we can reason by extension that FINITE intervals and masses may actually be composed of an infinite number of divisions; and conversely, that an infinite number of divisions may have a finite sum.
^{flandern }What about Zeno's objection, that if a moving body is someplace specific at EVERY instant, then at no instant is it moving, making motion impossible? One way to see the resolution of this paradox is by considering time to be another dimension, just like the three dimensions of space (although admittedly not exactly like a space dimension; e.g.
^{flandern } we cannot travel both ways in time).
^{flandern } Then a body traveling at a uniform velocity from point A at time 1 to point B at time 2 is traveling on a straight line in this space-time universe. To clarify this picture, suppose the body is at rest in space. It nonetheless takes a straight "line" in space-time to connect its position at one time with the same position at a later time -- the "line"representing an interval in time, instead of space. Viewed in this way, it may be seen that the body is at every instant at some specific point on a space-time line. And once again the points in the interval can be put into a one-to-one correspondence with numbers between zero and one. So even though the distance traveled by the body in zero time is zero, it is nonetheless possible to traverse a finite distance in a finite time, each interval consisting of an infinite number of time instants and space points.
^{flandern }To put this conclusion more strongly, it is possible for substances to be unchanging at every instant, yet changed after a finite interval, ONLY if there are an infinite number of steps in the interval!
^{flandern }This "one-to-one correspondence" may be the toughest concept to understand in this entire course. Does anyone have any ideas on how to better explain it?
^{flandern }E. Zeno-like Paradox for Matter
^{flandern }There is another form of Zeno's Paradox which applies to masses: if bodies are infinitely divisible, then contact is impossible. For example, when macroscopic bodies seem to touch,they actually consist of mostly empty space at the atomic level;so it must be their atoms which actually touch.
^{flandern } But atoms are themselves composed of smaller particles and mostly empty space,so it must be these smaller constituents which actually touch.
^{flandern } But if matter is infinitely divisible, this argument can be prolonged indefinitely, and nothing can ever actually touch.
^{flandern }One might use this argument to conclude that there is a smallest possible unit of matter or substance. Imagine such a"unit particle".
^{flandern } It must be utterly uncomposed.
^{flandern } It therefore cannot be broken or divided, nor even deformed by spin or collision -- since these are properties of bodies composed of yet smaller particles.
^{flandern } What then are we to assume will happen when two such unit particles collide? What density will the unit particle have? Indeed, will there be anything inside it at all? What would the unit particle's "surface" be like? Could it be hollow? With what thickness of shell? Would two colliding unit particles have to stick, since they can't rebound elastically? If they rebounded, with what resultant velocity? Would the unit particles be spherical in shape? Why would they have finite space dimensions, yet infinite dimension in time? Or do they come into and go out of existence constantly? Where and when would they appear and disappear? It should be apparent from these considerations that postulating a "minimum possible unit of substance" is no more logically palatable than a "minimum possible unit of space or time".
^{flandern } Substance must be infinitely divisible, as must space and time; or else the paradoxes quickly lead to unresolvable logical dilemmas.
^{flandern } But how then can matter ever experience "contact", if everything which might experience contact is itself composed of smaller substances? The resolution of this paradox would seem to be analogous to that for space-time. If the substance of bodies always gets denser (more substance per unit volume) at smaller and smaller scales, then in the limit as dimensions approach zero, density approaches infinity and substances approaching each other must make "contact" (i.e., at infinite density, they cannot be "transparent" to other substance).
^{flandern } In the real universe, the density of matter greatly increases as scale decreases.
^{flandern } Hence the ratio of mass to volume in electrons is enormously greater(about 1E10 g/cc) than the same ratio for matter in ordinary human experience (of order 1 g/cc), which in turn is enormously greater than the ratio for the entire visible universe (1E-31g/cc).
^{flandern } "Contact" is therefore possible for infinitely divisible matter, as long as the smaller and smaller particles continue to increase in density with sufficient rapidity, without limit.
^{flandern }I appreciate that it is very difficult for the intuition to grasp this concept.
^{flandern } Consider the approach of one minute particle of substance to another.
^{flandern } As the outer surfaces approach, the lesser particles (call them "second level" particles) of which each is composed begin to approach each other.
^{flandern } After the original particles traverse only a very small distance, the third level particles of which the second level particles are composed begin to approach each other.
^{flandern } After an even more minute traverse of distance, and after an ever smaller lapse of time, the fourth level particles begin to interact.
^{flandern } Although this continues without limit, as we have already seen, the process takes place in a FINITE time and a FINITE distance. The penetration of each level of particle into its counterparts in the approaching particle continues until the density of matter in the approaching particle is too great for it to penetrate deeper.
^{flandern } Then the smaller particles at the next level penetrate until the density becomes too great for them to make further progress, and so on.
^{flandern } By one-to-one correspondence with terms in our infinite series with a finite sum, we see that the depth of penetration has a finite limit, and requires a finite time, after which the original particles react with resistance to the intrusion of new substance into their ranks JUST EXACTLY AS IF THERE HAD BEEN A COLLISION! By analogy with the proposed resolution of Zeno's paradoxes for space and time, the paradox for mass is resolved, apparently necessarily, by the conclusion that substance must be infinitely divisible, and that it must approach infinite density as size decreases toward zero dimensions.
^{flandern } This conclusion is reached by reasoning alone; it is reinforced by the observation that matter does in fact increase rapidly in density as scale becomes smaller over a range of 40 orders of magnitude in the observable universe.From the preceding considerations it seems altogether reasonable, and in a way compelling, to deduce that space, time,and substance are all infinitely divisible; because the consequences of the alternative are logically absurd.
^{flandern } But if they are infinitely divisible on the smaller scale, what about the larger scale? Recall our earlier argument that the entire visible universe would have undefined scale in space, time, and mass, unless such scale is provided by the presence of other substance in the greater universe beyond.
^{flandern } That argument must remain true without limit.
^{flandern } The upper limits to the structure of substance, the dimensions of the universe, and the extent of time, must all be as unbounded on the high side as they need to be on the small side. This will become even clearer as we further examine the nature of substances.
^{flandern }F. Meaning of Space and Time
^{flandern }Let us return again to our empty universe which contains no substance, and therefore no frame of reference, except for a single uniform particle of substance. But as we have just seen, the particle must itself be composed of an infinitely divisible variety of sub-particles.
^{flandern } We could have chosen a single particle at any of an infinite number of sub-levels to be our single particle. To avoid the issue of the arbitrary size of the particle we select, let us conceive of it as having zero radius.
^{flandern } Although it does not, this conception will allow us to introduce one scale of distance at a time.As remarked earlier, motion and orientation have no meaning for a single particle in an empty universe. Now introduce a second infinitesimal particle. This gives meaning to orientation, since angles can be measured from the line joining the two particles.
^{flandern } It also provides a single measurement of length, the distance between the particles.
^{flandern } It does not, as before, provide a scale for the empty universe, since the distance cannot be measured in units of particle diameters, which are still being assumed to have no dimensions.
^{flandern } Therefore there is no way yet to determine whether our particles are separated by a microscopic or a macroscopic distance. There is as yet still no meaning to motion in this two particle universe. The two particles cannot change direction, since all directions have meaning only relative to the particle-to-particle direction.
^{flandern } And the two particles cannot change distance, since all distances have meaning only relative to the particle-to-particle distance.In a very real sense, this universe without the possibility of motion or change has no time. Time can have no meaning if there cannot be events or change to mark its progress.
^{flandern } Put differently, if there were such a thing as an absolute time which existed somehow in addition to our two particles, the lapse of a microsecond or a million years would be just the same and utterly indistinguishable. But the existence of something with substance, such as an absolute time scale, violates the assumptions of our construction, that nothing exists except our two infinitesimal particles in an empty universe. Remember, we refer to "substance" rather than "matter" to cover ANYTHING which exists.
^{flandern } An absolute scale of time, just as for a structure or framework in space, would have substance in this broad definition.
^{flandern }Perhaps you have thought about one possible event or change which might occur in our two-particle universe up to this point.
^{flandern } We might imagine that the two particles coincide, which is a distinguishable condition from non-coincidence. It might be fair to say that the first coincidence of the two particles marks the beginning of time; and that the interval between any two coincidences marks an interval of time. This interval still has arbitrary and indeterminate length.
^{flandern } We cannot tell if the interval to the next coincidence is longer or shorter than the last (that implies an absolute scale of time to measure against).
^{flandern } We can merely mark the progression of time by counting coincidences.
^{flandern }This brings us to an important point of our mental construction.
^{flandern } In an empty universe consisting of two elementary units of substance, the ordinary properties of the universe (time, space, matter) do not exist outside of the particles and between events of coincidence. It can therefore be said in a logically meaningful way that space and time which are empty of particles and events DO NOT EXIST! This eliminates a logical fallacy we have been skirting around up to now about whether the empty space and time surrounding our particles exist.
^{flandern } In our construction they do not.
^{flandern } Therefore our use of "substance" to mean "anything which exists" is logically correct, since a true void would not exist (in either space or time), in the operationally-defined meaning of the word "exist" as used here.Of course, for actual particles with finite dimensions,events of coincidence do not occur.
^{flandern } Instead we have what may be operationally described as "collisions", in the sense already discussed.
^{flandern } Two particles interact "collisionally" when their sub-particles at all levels approach the infinite density limitation and are forced to retreat.
^{flandern } Notice, however, that if we were to imagine an infinitesimal volume of space IN OUR REAL UNIVERSE within which there were only two uncomposed infinitesimal particles and nothing else (including forces), then all that we have concluded about distance and time not existing between events of coincidence would still be true. No time or time interval would exist until an event occurred, with the only possible events being collisions with other elementary particles of substance.Therefore, on the most microscopic levels, time must proceed"instantly" from one collision event to the next.
^{flandern } Reflection on this construction, which implies the non-existence of space and time between events in a region, begins to provide some insight into why the universe seems to behave as if space and time were relative, not absolute. We have reasoned to the conclusion that they must be. To emphasize the point that true vacuum implies non-existence, we are asserting that every point in the perceptible universe is at every moment of time filled with contiguous substance at some infinitesimal level.
^{flandern } If substance could be imagined to become absent anywhere at any time, time there would cease and the perceptible universe would collapse until the"vacuum" was filled.
^{flandern } Put another way, a particle reaching one edge of a "vacuum" would skip instantaneously to the opposite edge, just as if the "vacuum" had zero dimensions, because there is no substance to mark the passage of time inside of the"vacuum", and no absolute time without substance.
^{flandern }G. Implications and Discussion
^{flandern }Pausing for a moment to digest some implications of our reasoning, it would be fair to conclude that the only logically imaginable way in which substance can come into, or pass out of existence (in this model) is for it to "enter" or "leave" the region of collisional interactions with other substance. But if there were such regions where matter density is so low that no collisional interactions between units of substance occurred,then all substance on the edge of such regions would instantly dissipate itself into the non-interacting regions, followed by substance slightly further in, and so on.
^{flandern } All substance in this universe would dissipate instantly into the void.
^{flandern } We suppose that even solid bodies are held together by the action of agents which would disperse if not continually held together by the presence of other substances, so that even solids would dissolve. Since this does not happen, we conclude that this universe has no such regions where collisional interactions between units of substance do not occur.
^{flandern }The same reasoning applies to time. A cessation of collisional events would bring a cessation of time; but with matter existing everywhere with sufficient density for collisions, it follows that time continues forever, in both the future and the past.
^{flandern } But couldn't substance redistribute itself so that densities no longer approach infinity anywhere, thereby ending collisional events? By analogy with the dissipation of substance in space, if it could so dissipate (for example, if the amount of substance in the universe were finite), it would have already happened, virtually instantaneously.
^{flandern } Conversely, if substance does not start out with density which approaches infinity as dimension approaches zero, it could not assemble itself into such an infinite-density configuration in a finite time. We may therefore be reasonably certain that the "universe"(in our model) is infinite in space, time, and mass or scale.It must be the case that every bit of space is occupied at all times by a continuum of substance; and that wherever substance is not, existence of time, space, and matter is not.
^{flandern } The substances whose presence "define" space-time must be infinitesimal compared to the substances in our experience, such as baryons or photons, or even neutrinos.
^{flandern }Distance scales must be purely relative, with no absolute meaning to "large" or "small".
^{flandern } Likewise we should not be surprised by very large velocities.
^{flandern } If distance and time scales are unlimited on the large side, then velocities must be also.
^{flandern } We will discuss in future weeks how this can be reconciled with Special Relativity, which postulates that the speed of light is a maximum speed in the universe.Our conclusions are deductive, not inductive. So they can be invalidated only by faulty reasoning or an incorrect starting point or assumptions.
^{flandern } They do not at first glance appear to lead to descriptions of the REAL universe; for example, they do not easily reconcile with the sort of universe inferred using the Big Bang as a starting point.
^{flandern } But we will see in coming weeks that,although they do imply changes in some of our theories, the descriptions from this new starting point seem entirely reconcilable with REALITY.
^{flandern } If they add understanding and make successful predictions, I argue that is sufficient for them to be worthy of consideration AS HYPOTHESES in the field of astronomy.
^{flandern }If this model is unclear, ask questions to help clarify it.
^{flandern } If the model is clear, it must lead to a description of the real universe and make successful predictions to be of value. It is still too early to compare this model with observed reality.
^{flandern } But let's see some discussion of the implications of the model to this point, and perhaps anticipating the next step by guessing the nature of FORCE using this model.
^{andersen } A Keelynet listing called GRAVITY2.ZIP (10/17/90) presented an unpublished paper by John R. Majka called "A Different Point of View".
^{andersen } There the hypothesis was set forth that the ratio of magnetic-to- electric field "seen" by an observer of a charged particle is a function of the relative motion between the particle and observer.
^{andersen } Also, that the magnetic permeability of free space and the electric permittivity thereof are analogous to the distributed inductance and capacitance, respectively, of a transmission line.
^{andersen } This leads to the suggestion that a transmission line's RESONANCE, which depends on its inductance (L) and capacitance (C), corresponds to a RESONANT VELOCITY of free space, that velocity being the speed of light (2.998E8 meters/sec). For the implications of this hypothesis, see the file GRAVITY2.ZIP.
^{andersen } This file serves as an addendum to J. Majka's paper. Evidently Majka is not the first to seriously consider the "resonant velocity" idea.
^{andersen } In 1986, Prentice-Hall, Inc. published a college-level textbook entitled "Waveforms: a modern guide to non sinusoidal waves and nonlinear processes", by Homer B. Tilton (ISBN 0-13-946096-9).
^{andersen } On page 66 Tilton introduces the concept of "super resonance", in which he discusses a dual oscillator system (two pendulums). On page 67 he illustrates a theoretical electrical version of the dual pendulums-- dual LC tank circuits coupled together through back-to- back zener diodes.
^{andersen } Tilton claims that, if the resonant circuits are excited in their linear regions, below a certain threshold, each LC tank (or pendulum) will oscillate at its own independent resonant frequency.
^{andersen } But when they are OVERDRIVEN into the nonlinear region, they will act as if they had been tuned to a higher frequency simply as a result of the NEW MODE OF EXCITATION. I now quote directly from Tilton, pp. 67,68:
The constancy of the speed of light in free space leads to the now-classic equation of special relativity, gamma=(1-beta^{2}) ^(-1/2), which describes relativistic effects, with beta being 'normalized velocity'. Among these effects are 'space contraction', 'time dilation', and 'mass increase with velocity'.
That equation for gamma can be 'explained' by likening the speed of light in free space to a resonance condition.
An electronic circuit analogy was announced by Schelkunoff in 1938 [footnote 1] in which it was shown that the intrinsic impedance of free space is analogous to the characteristic impedance of a network, and the Q of a lossy medium is analogous to the Q of a network. That analog is in common use today in antenna design.
That electronic circuit analogy has been developed further by the author [Tilton] in accordance with the ideas that the permittivity and permeability of free space are analogous to the capacitance and inductance, respectively, of a tuned circuit. In this extension of the analogy, the speed of light becomes analogous to the resonant frequency of a tuned circuit [footnote 2].
While apparently it is not possible to decrease the value of either the permittivity or permeability of free space and thereby 'tune' it to a higher 'resonant speed', the following question nevertheless presents itself: Can a region of free space be EFFECTIVELY tuned to a higher resonant speed by exciting it in the manner of the binary systems described above? This question remains unanswered.
^{andersen }So we see that John Majka seems to be in good company with other scientists who have been pondering this idea at least as far back as the 1930's.
^{andersen } As food for thought, I'd like to pose the question: Is it possible that the notorious Philadelphia Experiment utilized these principles to warp space-time in 1943?
^{andersen } Tilton speaks of possibly "raising" the resonant speed of space -- but what about LOWERING it?
^{andersen } If Einstein was correct in saying that a mass cannot be accelerated all the way up to light speed, can we perhaps "cheat" by bringing DOWN the effective speed of light in a given area of space?
^{andersen } It would seem that by manipulating electromagnetics properly we might be able to alter the properties of a region of space and thus be able to perform all sorts of strange magic tricks.
^{andersen }Any comments relating to these points are welcome!
^{andersen }Footnote 1-- S. A. Schelkunoff, "The Impedance Concept and its Application to Problems of Reflection, Refraction, Shielding, and Power Absorption", The Bell System Technical Journal, XVII, no. 1, (January 1938), pp. 17-48.
^{andersen }Footnote 2-- H. B. Tilton, "An Electronic Analog of Relativistic Space", Electron and Ion Beam Science and Technology (New York: The Electrochemical Society, Inc., 1968), pp. 3-14.
^{taojoe }The following text describes a model of the universe. Fundamental to it is that cosmic phenomena, such as time, electromagnetic waves or particles, gravity and space itself are manifestations of dimensions or geometries. These dimensions are discussed here as the result of being extracted from the traditional set of spatial vectors: length, width, height and the vector usually described as the 4th dimension, time.
^{taojoe }Before going any further it is necessary to reassign the traditional numbers so they can be manipulated within this model. The re-assignment is not arbitrary but key and follows basic mathematical procedure.
^{taojoe } Array of Dimensions:
D |
1 |
2 |
3 |
4 |
1 |
1 |
2 |
3 |
4 |
2 |
1/2 |
1 |
3/2 |
2 |
3 |
1/3 |
2/3 |
1 |
4/3 |
4 |
1/4 |
1/2 |
3/4 |
1 |
^{taojoe }Mathematical systems are sets of elements that interact according to the rules of an operator. The operator here is D, a dimensional operator that separates, or divides dimensional combinations by various permutated constituents. D_{1} is time the first, or primary, or fundamental, or identity dimension, because it acknowledges change without assigning a vector to it. D_{2}, D_{3}, and D_{4} are the three familiar spatial dimensions, each with vectors orientated 90 degrees from another.
^{taojoe }The values in the table are the complete array of dimensions that are generated in this system. Any dimension divided by the identity is equal to itself; but the identity divided by any of the spatial dimensions produces different results. In the case of 2/4, the answer, reduced to lowest terms, becomes 1/2. This means that the dimensions produced by 2 D 4 and 1 D 2 are the same. There are eleven total distinct dimensions in the system, the same as the number stated by Einstein and others as existing in our universe. The universe does not know numbers but it knows ratios, and how to produc(e) results with its own elements. This mathematical system may be the most accurate model of how dimensionality is generated in the universe.
^{taojoe }Dimension Summary - I
^{taojoe }D_{1} = time - regarding time in the context of this model.
D_{2}, D_{3}, D_{4} = length, width, height (spatial dimensions)
D_{1/4}, D_{1/3}, D_{1/2}, D_{2/3}, D_{3/4}, D_{4/3}, D_{3/2} = fractional, composite dimensions which we still need to describe. Physicists talk about these dimensions being very small, curled up, as it were, inside the others. What does that mean?
^{taojoe }Lets start with what we know. D_{1} is time, or change or motion. Lets agree to use the word motion for this investigation. D_{2} is a single spatial dimension, a line; D_{3} is 2 spatial dimensions, a solid object. Of course, if they exist with D_{1}, as they do in the universe, all these geometries are in motion. To generate the other 7 dimensions, we will need to divide, and to perform division of dimensions we need to examine the meaning of division as it relates to the physical universe.
^{taojoe }It will be useful to think of division as a separation, or extraction. For instance, using numbers to illustrate, if we take the quantity 10 and we would like to extract the fives from this ten, we will end up with two sets of them. Now lets take a simple case using dimensions: 4 D 2 = 2. In words, when we extract 2 dimensions from a solid object (in motion, of course), the result is 2 distinct electromagnetic waves (photons), whose amplitudes are perpendicular to each other (intersecting planes). And how do we induce this dimensional split? By accelerating our solid object (already in motion) to the speed of light (c). This is clearly a catalyst in the process, adding enough instability to the system to cause the bound dimensions to break apart.
^{taojoe }I think that each of the eleven dimensions represents one basic aspect, one constituent of the universe. The attempt by physics to explain phenomena by associating discrete particles with each has gotten out of hand. It may be more useful and to the point to identify eleven fundamental properties of the universe, each associated with its own dimension. Thus far we have identified three (four if you choose to count the zeroeth dimension, the singularity, which exists out of time, and arguably belongs to the realm of nonexistence, rather than existence).
^{taojoe }Please consider this a work in progress. Many ideas still need refinement and clarification. Despite that, the concepts are so exciting that I could not resist posting it.
^{taojoe }So, the reader is requested to take the view that this is presented to stimulate thought and provoke discussion rather than as a formal presentation of a finished work.
^{taojoe }Dimension Summary II
^{taojoe }D_{1} = the identity dimension, time, or motion
D_{2} = wave phenomena, the electromagnetic force, or the photon
D_{4} = ordinary matter, always in motion, or always radiating a temperature.
D_{3} = before associating phenomena with D_{3} some preliminary concepts need to be discussed.
^{taojoe }D_{3} Preliminary Discussion
^{taojoe }In mathematics, equations are commonly written to describe phenomena in any of the four integral dimensions. The exponent of the variable indicates the degree of dimensionality: first degree equations have one solution and describe straight lines, second degree equations (with x^{2}) have two solutions and describe planar geometries like circles or parabolas, third degree equations (3 solutions) describe solid objects. For the fourth dimension, we have a choice: either write the exponent 2 or write the exponent zero. Lets do some math and play with a very simple equation. The equation is x^{4} = 1. You say x^{4} = 1- That's dumb. A fourth degree equation has four solutions, but here, all four solutions are the same. X can only equal 1, right? Wrong! You are forgetting how to solve an equation like this. Proceed as follows, treating the equation as if it were an ordinary quadratic:
^{taojoe }1. Set the equation equal to zero.
2. Factor the equation into the difference of squares.
3. Let each factor independently equal zero.
4. Both these equations are quadratic. The one on the left is again factorable.
5. Repeat step 3 for this set of equations.
6. The two solutions for these equations are X = 1 (the answer we already know) or X = -1 (the answer we forgot about but which checks since (-1)^{4} does equal 1.
7. We still must solve the equation on the right side of step 3: X^{2} + 1 = 0. It is not factorable, so we will need to use the quadratic formula.
^{taojoe }When ax^{2} + bx + c = 0, then x = 0 +/- Sqrt (b^{2} - 4ac) / 2a
^{taojoe }8. Using the formula, where a = 1, b = 0, and c = 1,
^{taojoe }x = 0 +/- Sqrt ((0)^{2} - 4 (1) (1)) / 2 = +/- Sqrt -4/2
^{taojoe }9. Remember that the square roof of -4 is called 2i (right imaginary numbers), so the two solutions for x^{2} + 1 = 0 are i and -i.
To check: Is (i)^{4} = 1- Well, i^{2} = -1, and (-1)^{2} = 1 yes.
Is (-i)^{4} = 1- (-i)^{2} = -i^{2} = -1 also yes.
^{taojoe }Therefore, the four solutions for x^{4} = 1 are +1, -1, i and -i.
Now for the interesting part. i is the first imaginary number. Lets list the first five exponents of i:
i^{0} = 1 (since anything raised to the zero power is 1)
i^{1} = i (itself)
i^{2} = -i (by definition)
i^{3} = i^{2} x i = -1 x i = -i
i^{4} = 1
^{taojoe }Observe that the exponents zero and four give the same result. When we write the imaginary number i, we are really writing i^{1}. These two symbols each have cosmological significance:
^{taojoe }The exponent portion, 1 indicates the first dimension time or motion. 1, the fundamental, imaginary constituent of reality the point. Again, time is going to behave like a spatial dimension, because a point in motion becomes a straight line. Simply by having an imaginary point in motion, we are generating the phenomenon of distance.
^{taojoe }When we wanted to break our solid object into simpler dimensions, we had to increase its motion to the speed of light. And yet, nothing can move faster than light. So our D_{2} entity, the photon (which we can represent mathematically as i^{2} or 1) is real, but mass-less, and has a velocity of c. When we step down a dimension, a dimension, we reduce exponents by 1, and to reduce the exponent 2 of i^{2} to 1, to get i^{1}, we can take the square root (in fact, that is the definition of i is that i equals the square root of negative one). It was the attaining the velocity of c that split D_{4} into D_{2} + D_{2}; conversely, reducing motion below c would reunite D_{2} + D2 into D_{4}. So if we wish to split D_{2} into D_{1} + D_{1}, we certainly cant lower its velocity (this doesn't split, it joins), but we cant increase it either, since c is the maximum velocity in the universe. Therefore, it stands to reason that the velocity of the point in motion, D_{1}, or i^{1}, is also c.
^{taojoe }Ready to begin a discussion of D_{3}? By using my ideas, we can make some sense of this very abstract notion, and link it to one of the real mysteries of astrophysics. I can think of three ways of getting to D_{3} from something we know. We can take D_{2}, the photon and kick it up a notch, and do D_{2} + D_{1} = D_{3}. In words, this means that we add a new 90 degree vector to our planar electromagnetic wave, producing an abstraction we can refer to as a cubic wave. I don't know about you but I cant visualize that (for a good reason, as you will see). We can also try D_{4} - D_{1} = D_{3}. In words again, we are taking one dimension, say the depth, away from solid matter, to get a plane in motion (once again time as a spatial dimension), but what is a plane in motion? Maybe a tube of some sort. Maybe not. The third option involves negatives. To review: D_{4}, i^{4} and +1 are all symbols we can use to represent solid matter as we know it. D_{2}, i^{2} and -1 are symbols that correlate to electromagnetic waves or photons. The only difference between D_{4} and D_{2} is that D_{2} = D_{4} x c (remind you of something? E = mc^{2}, perhaps?). Substituting numerals, -1 = (+1) c, and solving c = -1. In words, c has the effect of negativizing a phenomenon.
^{taojoe }We have already referred to cs splitting or destabilizing properties; how it is useful to concentrate on this anti-polarity feature.) So we can say that the photon is the negative state of matter. A little while ago, we stated that the point in D_{1}or i^{1}, has a velocity of c. What do you suppose would happen if we slowed this point to a sub-light velocity? If c really does produce an opposite, then i^{1}or i , becomes its negative, i^{1}or -i , which is the value of i^{3}, which is, lo and behold, an alternate symbol for D_{3}. D_{3} then (or i,) is the realm of slower-than-light imaginary points. Since it is sub-light it must have mass, but since it is imaginary, it has no substance. And this is exactly what our plane in motion is trying to describe, a surface with no thickness and therefore no substance, enclosing on itself trying to form a sphere (or a tube) with no volume. An imaginary figure with mass. Hey, astronomy buffs, what am I describing? This is dark matter! All that unaccounted-for mass that must exist in the universe but is undetectable. Its undetectable because its not real. But because its slowed to sub-light, it has mass. And because these things are negative singularities, similar to the singularity responsible for the big bang and the subsequent universe except hollow inside (like bubbles or macaroni), they are super dense, far more dense even than black holes, so it makes sense, as physicists claim, that this dark matter should account for most of the mass in the universe, much more than visible matter. D_{3} = dark matter, matter which has been collapsed by one dimension.
^{taojoe }Four dimensions, the integral dimensions, have been explained. The seven fractional dimensions remain, and while each is still associated with naturally occurring phenomena in the universe, the job now becomes a lot tougher.
^{taojoe }We are going to wade right into the thick of things, but first, a little stage setting. Lets revisit that old trickster dimension, D_{1}, time, which seems so elusive, yet has so many properties, and influences just about everything. To review, time is motion, specifically the motion of an imaginary point at velocity c. In theory, there was a period of dimensionality (duration is necessarily meaningless here) when a motionless singularity constituted a potential big-bang to begin the universe. In that instant, when the point acquired motion, the first dimension was born. But something else was created too, because when the point began to move, it needed a medium to move through, and this did not yet exist. So as consequences of D_{1}, we also got something called space, and something else called distance; things respectively, for the creation to move through, and for it to displace as it went on its journey. Up to now we have assumed that this distance was a straight line, since this is usually how distance is measured, and since there was nothing (apparently) to deflect a projectile from its course. Or was there? In that moment in which the singularity burst from non-existence into being, lots of things happened. One of those things was that the singularity's ambiguous state as a mass-less/infinitely massive thing became unambiguous the instant the new universe existed it was incredibly massive, so that all of a sudden there was this super-massive something sitting the center of a region of space, expanding outward into that region even as it was being created. Imagine that space as a huge net, stretched out evenly, nice and flat. Then into the net we drop a bowling ball. What happens? The weight of the ball causes the net to stretch, most noticeably where the ball sits, and less as you get further from it. But as far as the net extends there is a greater or lesser slope downward toward the center. If a marble had been sitting on the net before the bowling ball was dropped, it would have been perfectly stationary on the flat net. But with the bowling ball causing the net to have a different shape, the marble begins to roll towards the center. The greater the slope the faster the marble rolls. If you cant see the net, it looks like the marble is actually being pulled toward the bowling ball by some mysterious attractive force. But we know about the net understand that there is no force; the marble is simply rolling down a curved surface caused by the massive object at the center. And the distance the marble rolls is the length of a curved line, not a straight one.
^{taojoe }The situation is analogous to the way massive entities shape the space around them, that is, give space a geometry that is not flat, but curved here, stretched there, endlessly twisted and varied. This geometry of space caused by mass is how Einstein described gravity. Why at this stage there are physicists still maintaining that gravity is one of the four forces of nature when it is fully explainable as a geometry of space is a bit confusing. One would think that the failure either to unify it theoretically with the other forces, or to find its associated particle, the graviton, would convince them otherwise, but so far it has not done so. They continue to try to crack the nut of the gravitational force, where no force exists. It is the geometry of space we are observing, and like other geometries in the universe, it has a dimension associated with it. But which one? How do we decide which of the seven fractional dimensions is responsible for gravity, or rather, for the geometry of space? Lets try an approach that looks for patterns or trends in the qualities we attribute to the various dimensions.
Dimension Summary III
^{taojoe }D_{1} is responsible for time, motion of imaginary points and the distance they travel along gravity influenced lines.
D_{2} is responsible for wave phenomena, specifically the electromagnetic force.
D_{3} is responsible for the dark matter which is undetectable but comprises most of the mass in the universe.
D_{4} is responsible for what we call ordinary matter, both massive and detectable.
^{taojoe }What do you notice as you move from D_{zero}, total nonexistence, up through the dimensions to D_{4}? I notice that there is a progression from the least substantive, most abstract manifestation; to the opposite of that the manifestation that is wholly concrete and not an abstraction at all. Maybe by assessing the balance between the abstract and the concrete nature of a phenomenon, it is possible to order it, that is, place it in its proper position in the progression of dimensions, almost as if we were creating a size order. And if we are lucky, it may additionally be possible to identify elements being utilized by the operator D by deciding which two of the four basic dimensions are interacting to produce the appropriate fraction. In choosing, we must bear in mind that the seven available fractions all have a value less than 2, and five of them are less than 1. This means that they all represent significant abstractions. It is a certainty that none of these dimensions contain massive entities, though it is possible, as with gravity, that their effects are far-reaching enough for us to be aware, or at least suspicious, of their existence.
^{taojoe }The geometry of space then. I dare say this is a very foreign, abstract concept for most of us. The natural tendency is to view space as nothingness. To learn that it is both a vacuum, and has a definite geometry is a little hard to grasp. Yet most of get a pretty good mental picture of D_{1}, even though the point in motion is completely imaginary. Is space then a greater abstraction than time? I imagine that it is, and that were looking for a fraction less than 1. Next, consider gravity's far-reaching effects. Not only matter but also even something mass-less like light, is induced to follow the shape of space. So it would be illogical to choose 2 as our denominator, since that excludes the 4 that contain matter.
^{taojoe }There are only two fractions whose denominator is 4, which would account for the matter, and everything contained in all lower dimensions. What is it then that reacts with either light or matter, to form space? What thing plays a role in the effect of gravity; appears in all the gravitational principles and equations? Distance! It is the distance from the distortion causing mass that determines the degree of the curvature of space, or in traditional terms, the strength of the gravitational attraction. Distance is created by the first dimension, motion. Therefore the only logical fraction for the dimension of gravity, or the geometry of space in one that points to the interplay of distance (D_{1}) with each of the dimensions effected by that gravity (D_{4}). The conclusion: gravity = D_{1/4}.
^{taojoe }The concern of the next topic will surprise no one, it is that familiar demon come back to haunt us, as it has done time and time again: the speed of light. My conclusions about how the speed of light relates to dimensionality are nearly ready, but thoughts about how to set down the logical progression and justification of my argument needs more work. So all I can give you at this point is a preview. In it, the salient highlights are as follows:
^{taojoe }1. The speed of light is a misleading term, for while it is the speed of electromagnetic waves; it is also the speed of other cosmic phenomena, including things that have dimensionality less than 2.
2. The speed of light, or c, is indeed constant, but not in the way everyone believes. Constancy in the eyes of humans, and constancy in the eyes of the universe, is not the same thing. (I have already hinted at this.)
3. The speed of light is a kind of threshold for a change of state of various cosmic manifestations, just as 320 F. is a threshold for the change of state of water. (Matter can be considered to be frozen energy.) Things at light-speed are mass-less; sub-light things acquire a mass. Dimensionally speaking, D2 corresponds to this dividing line.
4. The dimension of c is 1/2. I cannot at present provide the coherent explanation that I would like. Also, a big problem is that a dimension requires a geometry. That is something I have insisted on. I do not yet have a clear picture of a geometric model for c, nor do I immediately see how to derive one. This, of course, is the challenge. I am working on it.
Source:
The Square Root of -1
Mathematics and Neurology
To Homepage: http://www.users.bigpond.net.au/a-theory-of-mind
^{sqrtneg1 }The square root of -1 is the integer 3 in 10-circle, but it is not an integer in 9-circle, where -1 is 8. It is an integer in 5-circle where -1 is 4, and we could go through the numbers, choosing an array of circles that have integral -1 square roots.
^{sqrtneg1 }Figure 1
^{sqrtneg1 }In the counting line of counting circles -1 is a counting line to infinity. Thus it is 0 in 1-circle, because one place anticlockwise from 0 brings us back to 0. It is 1 in 2-circle, 2 in 3-circle, 3 in 4-circle, 4 in 5-circle etc., as shown below.
^{sqrtneg1 }Figure 2
^{sqrtneg1 }The square roots of these -1 numbers again constitute a counting line to infinity, but one interrupted by non-integer square roots as shown in the following sequence:
^{sqrtneg1 }Circle................... -1 in the circle..... Square root of -1
0-circle............ Being without determination, 0-circle has no numerical elements.
1-circle............ 0............... .....0
2-circle............ 1.................... 1
3-circle............ 2.................... 1.4142135623730950488016887242096980785696
4-circle............ 3.................... 1.7320508075688772935274463415058723669428
5-circle............ 4....... .............2
6-circle............ 5.................... 2.2360679774997896964091736687312767354406
7-circle............ 6.................... 2.4494897427831780981972840747058913919659
8-circle............ 7.................... 2.6457513110645905905016157536392604257102
9-circle............ 8.................... 2.8284271247461900976033774484193961571393
10-circle.......... 9.................... 3
11-circle......... 10[i].................. 3.162227.... Six, or 3*2 more irrationals follow here
........ ...............................................................before we reach 4, the next perfect
square.
^{sqrtneg1 }List of -1 Square Roots in Sequential Circles
^{sqrtneg1 }The irrational square roots, which shade off in what we could call infinite imbalance, are as precise in achieving their purpose as any rational configuration. They constitute the background against which the rational stands out. Taken in conjunction with the neural organization of the brain, the contrast between the perfect and irrational results represents the foundation of difference, aligned by functioning nervous systems to create the wealth of meaningful identities that confront life. On this contrast the intelligence in living creatures builds.
^{sqrtneg1 }The integer counting line, from which, in the square root of -1 the rational/ irrational pattern arises, read into the depth of all bases, is the mathematical foundation upon which the neural determination of known entities, so our knowledge of the world arises. The irrationals provide the supporting background in consciousness for this to occur. From this cloth the known world is fashioned. We usually think that the brain is only interested in the pursuit of order in a rational world, but without the contrasting irrationals there would be no canvas to take the picture.
^{sqrtneg1 }The rational/ irrational pattern is that, each integral root (0 1 2 3), is followed by double that number of non-integer results:
^{sqrtneg1 }.... The. 0. "sq rt. -1". in.. 1-cir. is followed by 0 irrationals, and 0+0=0
.... The. 1. "sq rt. -1". in.. 2-cir. is followed by 2 irrationals, and 1+1=2
.... The. 2. "sq rt. -1". in.. 5-cir. is followed by 4 irrationals, and 2+2=4
.... The. 3. "sq rt. -1". in 10-cir. is followed by 6 irrationals, and 3+3=6...
^{sqrtneg1 }and so on to infinity. What is happening here?
^{sqrtneg1 }The answer is that "i", the square root of -1, is switching the whole of mathematics from objectivity into subjectivity, translating the numerical pattern from the external or worldly domain into its internal counterpart, the mental space which we call subjectivity or time. We have that relation before us here. The generative pattern is genetic. Its root taps a chromosomal stem. It is not the expression of brain-dependent learning. Projected into the world, and returned as an external "sense", it becomes the comprehension of mathematics, which we again invest in externality in symbolic form, the "math" we know.
^{sqrtneg1 }This "math we know", to which sensible things like chairs and tables belong, is thus the externalization of an internal (mathematical) pattern that underpins mind. Conventional mathematics, as taught in schools is the externalized reflection of this internal order. We call the internal domain subjectivity, when we wish to refer to our sense of a world around, and time when our reference is to its nature (as distinct from the outer domain of space); i.e., when only the "inner" or "outer" of the domain is in reference. Time, with its present past and future is the stem. Space is its externalized expression.
^{sqrtneg1 }Some further observations
^{sqrtneg1 }We are using decimal to show the pattern in the list of -1 Square Roots because it is the familiar base we know, but it occurs in all bases and we could show it in base 2, base 3, base 4 notation and so on. In circle math we can work in any base, though we will usually choose decimal as our base of expression. The circle algorithms then call on other, we could say, parallel bases, to recruit their results. The circular method is thus properly described as transbase in operation, coming in this a step closer to the function of the brain, which as deep to mind is uniformly transbase.
^{sqrtneg1 }We should note that 0 is the identity element for adding and subtracting. 1 is the same for multiplying and dividing (adding 0, or multiplying by 1 does not change a number). To multiply by -1 therefore changes only the sign of the result. The "1" in the "square root of -1" is thus a dummy indicator of sign change, whose reality is an internalized counting line to infinity against a background array of fractional results, as shown in the list of -1 roots. Internal here means "in mind" and external means "in world", so that "ordinary" arithmetic, with its straight counting line to infinity belongs to the world. Circle math, with its circles rules and results belongs to the mind.
^{sqrtneg1 }We work from our knowledge of our ordinary or external math, which is objective to us. The internal circle math of the mind is however the parent and projector of the outer result. The "internalized counting line", as in the above paragraph, is given in the square roots of 0, 1, 4, 9, 16, 25 The "background array of fractional results" that seems to act as a spoiler is the tidy sequence of untidy square roots of 2 3 (two) 5 6 7 8 (four) 10 11 12 13 14 15 (six) in the -1 non-integer circle series, and the whole is the neurological setting from which developed consciousness springs.
^{sqrtneg1 }To think, the mind needs continuity and contrast and here it is, continuity in the mathematical sequences and contrast in these very same sequences. The circular system then comes forward as a coherent fine-grain mathematical approach to the subject matter, "how do we think, what is the mind's mechanism and operational principle?
^{sqrtneg1 }The square root problem
^{sqrtneg1 }In conventional mathematics there is no real number x, such that
^{sqrtneg1 }x^2 = -1
^{sqrtneg1 }This is because +x times +x comes to a positive result, and -x times -x also comes to a positive result, and there are no other options for x^2.
^{sqrtneg1 }Unable to assign a value to x satisfying the equation conventional mathematics stipulates that there is such a value, and calls it i for imaginary. It has no idea what "i" is, other than the square root of -1 as expressed in the equation
^{sqrtneg1 }i^2 = -1
^{sqrtneg1 }Given this, for it an impossible equation, it assigns the "no real number x" to the imagination, calling the square root of -1 "i". It then has a term for this ghost in the mind, supported not by understanding, but the fact that it integrates perfectly into mathematical operations and equations. The cost is that mathematics advances as a ritual of symbolic relations, comprehended only as the formalistic integration of signs and results.
^{sqrtneg1 }."i", as the square root of -1 is currently a blind spot in our mathematical understanding, walled off and bypassed. The task is to explain it, and in the process turn it from a block to a bond and bridge uniting in-world (linear) and in-mind (circular) math. The imaginary number serves a purpose, making an advance possible, shifting math from its ordinary domain to its higher expression. The further insight needed is to see that this "higher" math is really the internalized circular math we call the mind.
^{sqrtneg1 }Interlude
^{sqrtneg1 }In case this needs further explanation, the contrast between perfect and irrational results as set out above, is easy enough to understand. It is there before us and it stands alongside the "ordinary" counting line to infinity, when these numbers, turned into circles (figure 2) yield another counting line of whole numbers, padded out with irrationals.
^{sqrtneg1 }Now, the point is that these irrationals are rational in their irrationality. Every single digit in every irrational number, and there are countless billions of them, is exactly determined and cannot be other than what it is. Hence we can say, "The irrationals are rational in their irrationality."
^{sqrtneg1 }The jump to neurology rests in the fact that this overall pattern maps to the understanding in mind. To give the idea, when we say of two different sense-mediated patterns, "this is a window and this is a goldfish bowl," that comprehended distinction is carried in the mind by the mathematical distinction we have just considered, between the whole numbers and the irrationals. In other words, the rational/ irrational relation, as found in mathematics is the foundation of difference as found in the sensible world, between this object and that. Making this observation takes us across the math-to-mind boundary. Interpreting "i" within a mathematically circular expression will complete the picture.
^{sqrtneg1 }"i" as a square peg in a round hole
^{sqrtneg1 }"i" is impossible by the rules that have governed math for 500 years, yet "higher math" is adamant that it needs it. As imaginary it stands as a real unreal, as enigmatic as a unicorn or phoenix. To understand this situation we need to look at the unyielding sign laws. Children are taught that "plus times plus" is plus. "Minus times minus" is plus, but "plus times minus" (or minus times plus), gives minus.[ii]
^{sqrtneg1 }There are only two signs, + (plus) and - (minus). An asterisk * means times or multiply. Positive numbers may or may not be signed. Negative numbers are always signed. The sign rules work and they have order, but they are given without reason. They have to be learned.
^{sqrtneg1 }A.................... + * + = +................. so........................ +1 * +1 = +1
B.................... + * - = -................. so........................ +1 * - 1 = -1
C.... ................- * + = -................. so......................... -1 * +1 = -1
D.................... - * - = +................. so......................... -1 * - 1 = +1
^{sqrtneg1 }a and d, whose roots are identical in sign and number yield perfect squares. Only b and c give the -1 answer that "i" requires, but the results are not squares because each utilizes non-identical signs (one +, one -). The sign laws make no provision for negative squares. The square root of -1, devoid of actuality, is therefore said to be imaginary (i).
^{sqrtneg1 }This brings ordinary mathematics to its Waterloo. Workers in complex fields saw that the concept of a function, which they took to be a numerical magnitude equivalent to the square root of -1, was useful and orderly, but it was impossible to reconcile it with the sign laws. They therefore consigned it to the imagination.
^{sqrtneg1 }A new protagonist, circle math, now takes the field. It is strictly mathematical and just as precise as ordinary math, but its rules have been nurtured and born not in mathematics but philosophy. In scope it embraces both the straight math of the world and the circular math of the mind. We see immediately (fig. 1) that -1 in 10-circle is 9, whose square roots satisfy the problem equation, and this is only a beginning.
^{sqrtneg1 }In the shallows
^{sqrtneg1 }The observation belongs to the circular system, founded on a counting line, not of numbers but number circles. Children delight in this form of math because its rules are simple and its operations without fault. It comes to them as a number game, but "number" without its intimidating aspect. 0 gives the answer, so for the children we will draw it with a smile (figure 3 below).
^{sqrtneg1 }Figure 3
^{sqrtneg1 }For example, if the sum is 2 + 3 = ? the child has but to locate the 2, locate the 3 and join them with a line. A parallel from 0 gives the answer. It also establishes the 5 direction.
^{sqrtneg1 }There is more power in this construction than meets the eye. For instance it covers every numerical calculation that sums to 5: 1+4, 2+3, 8+7, 9+6, 0+5, 4+1, 3+2, 7+8, 6+9 and 5+0.
^{sqrtneg1 }Use a horizontal line instead and you have the 0 direction. This gives the answer to combinations that sum to 0. Do not forget that 9+6=5 in ten circle is the digital stepping-stone to 9+6=15, 9+16=25, and so on. Algorithms combine the sub-steps to final answers.
^{sqrtneg1 }Notice too, that if you join the numbers on an angle you get all the other directions as well. The one figure, easy to visualize and work with, yields every possible cipher adding combination, while similar constructions cover subtraction, multiplication and division.
^{sqrtneg1 }Each sum has its distinguishing direction. Turning arithmetic into a game children learn their adding subtracting multiplying and dividing tables without noticing that they are doing so. It is not "arithmetic without tears", but arithmetic with joy and delight.
^{sqrtneg1 }For the child, the circle is a tiny arithmetic pool in preparation for oceanic journeys to come. Nor is the method confined to base ten. It works in every base, for instance on the clock face. Drawing two lines shows that 11+8=7 (11 o'clock + 8 hours comes to 7, easily done, easily remembered).
^{sqrtneg1 }Applying it to other bases, it becomes a recreation and pursuit as mathematics becomes an interest in its own right. For example:
^{sqrtneg1 }Every number in a circle squares to the same result in that circle as the square of its complement. For example, the complement of 5 in 12-circle is 7.. 5 squares to 25, which counts into 12-circle as 1 (twice round plus 1).. 7, its complement squares to 49. This also counts into 12-circle as 1 (four times round plus 1).
^{sqrtneg1 }This double squaring shows in smaller circles; say 3 in 5-circle (fig. 1), which is +3 clockwise and -2 anticlockwise.
^{sqrtneg1 }Square the 2. It comes to 4. Square the 3. It comes to 9, which counts into 5-circle as 4.. 3 squared is the same operation as -2 squared, and gives the same result as 2 squared. Children learn their sign laws hands on, and see for themselves why "- * -" comes to the same result as "+ * +".. The rote-learned tables do not help in the least in this respect, but we must now resume our pursuit of the square root of -1 at a deeper level.
^{sqrtneg1 }In the depths
^{sqrtneg1 }Typing "square root of minus 1" into a search engine can find Zhenming Zhia, who writes under "Square Root of -1 as a Consciousness factor", July14 1996, that "In Einstein's special theory of relativity there is a Lorentz transformation that leads to Minkowski's four-dimensional space. But the fourth dimension is obtained by replacing time, t, with the imaginary [square root of -1 multiplied by the speed of light, c, and by t itself]. After this, the temporal dimension becomes totally symmetrical with all other three dimensions of space"
^{sqrtneg1 }Zhia continues: "That means that time is itself one more dimension of space but is perceived by our consciousness as different and uniquely temporal. The modification with the imaginary square root of -1 therefore corresponds to the unique involvement of consciousness in the process."
^{sqrtneg1 }He adds that, "Surprisingly, in Quantum Mechanics, the Schrodinger's wave function also involves the same square root of -1 when the spatial locality breaks down and a conscious observer gets involved in the process of measurement."
^{sqrtneg1 }"This apparent coincidence indicates something extremely significant for understanding the universe and consciousness. It suggests that human consciousness may be a dimension of anti-space that merges into the fourth dimension of negative space so that we cannot see it as spatial any more. It appears as one dimensional time instead."
^{sqrtneg1 }"Therefore, the square root -1 is the Consciousness factor. Working out its implications might lead to a true revolution in human understanding.". Quotation ends.
^{sqrtneg1 }In circle terms factoring i into t imports the equation into subjective time, where subjectivity has not the meaning of opinion, but the inner mathematical realm that in circularity grounds consciousness, objective in its very subjectivity.
^{sqrtneg1 }Two mathematical systems
^{sqrtneg1 }For centuries mathematics on a stormy path, fractured internally by dispute has played an autocratic role in education in a civilization torn by dissent and war. In its circular form, now being presented, it gains a subjective level, complementary to its familiar objective expression. We now see that in its developed form it is accessible to children as well as having an application in higher math and physics.
^{sqrtneg1 }Gradually across the years the square root of -1 that divides math and maps to the schism between ideality and reality, has come forward as the kernel of the theoretical problem. Its resolution, whose completion is critical, will mark the turning point between that past and a brighter future. In this realization, the difficulty in the imaginary number, proving itself to be imaginary, will vanish, and people will wonder what it was about. Validating Zhenming Zhia's prediction, mathematics, as an internalized discipline, is the logos upon which consciousness stands and the sensed world resides. The square root of -1 is the logical switch between that sensed world and our internal world of understanding.
^{sqrtneg1 }Definition
^{sqrtneg1 }In a circular setting "i" is a function (like Narcissus looking into a pool), that reflects mathematics from its external presence into the depth of its in-mind source, but we must examine some existing definitions, starting with the fact that "negative number" is still taught as a "quantity less that zero".[iii]
^{sqrtneg1 }In "wordiq.com" we can read, "a negative number is a number that is less than zero, such as -3. A positive number is a number that is greater than zero, such as 3. Zero is itself neither negative nor positive. The non-negative numbers are the positive numbers together with zero. Note that some numbers are neither negative nor non-negative, for example the imaginary unit i.
^{sqrtneg1 }The following discussion will make a frontal assault upon this position in order to create a sense of neutrality in the matter, upon which a new system of expression can arise. Mathematics is an integral part of all knowledge. It is not a castle to itself, and it cannot have a different logos from that of language. To be a discipline it must accept discipline. We now look at this in more detail.
^{sqrtneg1 }Ordinary math relates number directly to quantity. This is logical where quantity is amount, but it enters another thought alongside, that negative number is a quantity less than zero. This is illogical because quantity is a specified amount or number. If we start with a quantity of apples and remove them one by one, the quantity diminishes. When it reaches zero we can remove no more. The concept of a number less than zero is therefore false, and false in the world, false in mathematics.
^{sqrtneg1 }The problem does not occur in circle math whose counting lines are closed. Positive or add means clockwise in the circle. Negative or subtract means anticlockwise. The confusion vanishes. The math remains. Thus, in 10-circle -1 is 9, which squares to 1, and has 3 and 7 as its roots. The wordiq.com statement (as above) nails the error to the masthead by defining positive numbers as greater than zero, and negative numbers by the illogical expression, "less than zero".
^{sqrtneg1 }It then compounds the error by creating a new category; zero as a number that is "neither negative nor positive". The truth it is struggling to realize is that the symbol 0 represents the mind, and the mind, as the viewpoint within which number and math exist, is above, and so indifferent to the objects it contemplates. As mentioned, 0 is not a number at all.
^{sqrtneg1 }Conventional math has led us so far astray that it is painful to have to point out its absurdities, but it has to be done, for imbued in our consciousness since early childhood they prevent us from seeing the world revealed in science intelligently within the context of reality. Because at the level we teach how to spell CAT we also imply that 0 and 1 are numbers, and at the level we teach "Rover has a ball" we include the thought that zero is a number, "negative" numbers exist, and so on, mathematicians talk about i; science is unable to merge quantum and relativity theory and astronomers talk about the "size" of the universe. We search for the error in the clouds when we should be looking at our feet.
^{sqrtneg1 }The ancient Greeks pointed out that 0 is not a number because it is nothing, and 1 is not a number because number means more than one. The idea of 1, as opposed to 0, is derived from entity, originally creatures and things. Singularity is the "one", through which we identify kind, and plurality is "greater than one." Plurality is thus the general term for number, which latter is the specification of plurality by count or estimation.
^{sqrtneg1 }0 and 1 are functional as the hinge between mind and world. 0 interfaces with the mind, 1 with the world. It is convenient to call 0 and 1 numbers, but strictly speaking they are the agents that, through counting lines and bases, create number, unraveling in the world the order that already exists in the mind.
^{sqrtneg1 }Once the bases as the gathering ground of the mind are formed, we have to consider "i", the square root of -1 as the focus of the mathematical connection between mind and world. To begin its resolution, take the straight counting line into the circle. In our example -1 is 9 in 10-circle, with the relations of square and root that then apply. Mathematics comes to a point in "i". From this, expanding from the "i", a world in consciousness opens up. 0 and 1 are the active agents in generating this focal relation, and circlemath puts the whole mechanism in place for us to see.
^{sqrtneg1 }The one step (switching into the circles) resolves the philosophical ground of math and the "square root of -1", which now falls into place as fully accessible at primary school level. In the circular and straight we now have two systems, one subjective inherent, the other projected and objective or worldly. They map to each other through i, which is not a number but the confluence of mind and world mapped within a mathematical continuum.
^{sqrtneg1 }We are done with the fallacy of negative numbers, ghosts of the imagination, born with Fibonacci in the 12th century and matured in the 16th. The positive sign simply means add, and the negative subtract. This, with the rules of circle math are all we need to carry all mathematical computation, clockwise and anticlockwise across the sea of number and the ocean of bases. The imagination (read mind) is just as mathematical and just as real as any castle of brick and stone.
^{sqrtneg1 }The fact that -1 in 10-circle is 9, whose square root is 3 (and 7) is the touch of death for the mystery sequestered in the square root of -1. This one facet does not explain i, for it is only a fragment of a greater whole, but it is the key. Turn it in the lock and the door will open, for it can be readily understood. Even for those who are far from grasping the pattern of the whole, the illusion of i, the square root of -1 will begin to fade and will continue to melt away until it vanishes.
^{sqrtneg1 }Circle math emerged from the shadows of ordinary math by systematically eliminating every non-universal relation and mending each gap left behind.[iv] Eventually the entire circular system stood forth, universal and complete. The task was to map mathematics to the pattern of inherent mental function. We now have "i" unraveled, and with it a whole new field of mathematical relations, rich in algorithms accessible to children who take to it like ducklings to water.
^{sqrtneg1 }Education Departments were not so happy with it. They saw it rather as a threat, much as a hen that has hatched out ducklings will run along the bank in fright as its "chicks" swim and dive in the water. They were not geared from the massive change in learning patterns that it represents.
^{sqrtneg1 }The circular mathematical system depicts mathematics as it exists in the subjectivity of mind. It arises directly from the neural mechanism that responds to the world, and heralds the end of the defect that medicine knows how every organ in the body works, with but one exception, namely that which makes sense of the whole, the brain; that finally we may know what thinking is, upon which our human being and understanding rest.
^{sqrtneg1 }Copyright © Dr. Stephen W. Taylor M.B., Ch.B 08-03-2004 Brisbane Australia
Response comment and inquiry welcome.
You can contact me by email:
stetay @ bigpond.net.au
^{sqrtneg1 }[i]. Lock ten.. A lockbar is a vinculum above selected digits "locking" them into a single expression, so that circles greater than 10-circle can be created "to infinity" without descending into the confusion of substituting alphabet letters (A B C etc.). The underline (10) used here serves as a makeshift vinculum.
^{sqrtneg1 }[ii] +x is identical in successive circles. ?x is a counting line in the same successive circles, thus 2, 2, 2 (for 2's in-world expression) is -6, -7, -8 (for 2's in-mind expression) in circles 8, 9, 10. Sign change thus indicates a subjective/ objective switch within the thinking process.
^{sqrtneg1 }[iii] For the phrase, "less than zero," see Collins English Dictionary Third Edition, OR Mathematics Illustrated Dictionary, Bendick Levin and Simon, McGraw Hill, OR Internet: open wordiq.com (encyclopedia), OR any similar source and search for "negative and non-negative numbers". A "quantity less than zero," is just another Carrollean "impossible thing to be believed before breakfast."
^{sqrtneg1 }[iv] "Mending each gap" means that circlemath arose from the piecemeal correction of every fault discovered in conventional math theory, where "fault" means relying upon assumption and/or a non-universal progression for its coherence and sense.
.
© Copyright. Robert Grace. 2004