Meeting a Terrific Challenge

Discussion of Larson Research Center work.

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dbundy
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Re: Meeting a Terrific Challenge

Post by dbundy » Sat Oct 13, 2018 11:13 pm

Daniel wrote:
I have read every book and paper published on the Reciprocal System and nowhere in them do I find a "Larson cube," outside of your posts. The ONLY reference Larson made with a cube was to clarify that in a 3-dimensional system, there were EIGHT possible directions, not SIX as would be presupposed (± on each axis). That's IT. There is no magic cube to solve the mysteries of the Universe.
You are right. I started calling this description of the eight 3d "directions," which Larson gave us to answer those who mistook the six directions of a 3d coordinate system as six 3d directions, Larson's cube, in honor of him, many years ago.

The reason I named it that is because of its central role in connecting the magnitudes, dimensions and "directions" of numbers and geometry. Numbers have two "directions" for each dimension as does geometry and Larson's illustration of a 2x2x2 stack of 8, 1-unit cubes, is the starting point for understanding that.

Consider a 1d scalar expansion: It expands in two opposite "directions" from a point. Numerically expressed, this length is a 21 = 2 magnitude, when the space/time expansion is 1 unit of space per unit of time. Granted, the unit of space is one unit in each "direction," but that is what makes it "magnitude only;" that is, a scalar magnitude differs from a vector magnitude in that sense.

Now, expanding this 1d scalar magnitude into the two opposite "directions" of a second dimension, the expansion produces a plane, with 22 = 4 "directions," a plane in each "direction." Finally, a subsequent expansion in the two "directions" of the third dimension produces a stack of eight cubes, one in each of 23 = 8 "directions."

Now, this is not a "magic cube to solve the mysteries of the Universe," but it does represent the mathematical and geometric truth of scalar expansion. If nature expanded by the numbers, the unit number of a 3d scalar expansion would be eight, in the RST, given the fundamental postulates of the system.

daniel wrote:
Larson's work is based on motion, not little boxes. One must learn to think in terms of motion--contours and speedometers--to understand what Larson was trying to explain. Most of your posts confuse structure with probability, which is why you cannot solve many of your issues.
I'm not sure how you arrive at the confusion that I am "confusing structure with probability." The geometry of a 3d expansion is not the structure of the eight unit stack. Nature expands/contracts in all directions simultaneously. If we analyze 3d scalar motion, we have to pick a starting point, just as we do in analyzing vector motion. The vector is not a structure of some kind, but we can construct, or graph it, mathematically to understand it better.

It's no different in our studies of scalar motion. The trouble is, in both cases, numbers don't correspond to circles and spheres, so we search for ways in which we can find correspondences between them. Most of our current science and technology is based on the correspondence between the angles of rotation and the geometry of the circle. The changing sines and cosines of rotation map the numbers of a coordinate system to the motion of rotation, but that is not the only possibility. In fact, for scalar motion we can't use them.

daniel wrote:
dbundy wrote:

"The bottom line is that the scalar system, thus defined, gives us three unit numbers, one for each non-zero dimension, by the Pythagorean theorem:

1) √(12 + 02 + 02) = √1
2) √(12 + 12 + 02) = √2
3) √(12 + 12 + 12) = √3"

First, a square root has TWO solutions, a positive and negative one, meaning you have SIX "unit numbers," not three. Second, you used the Pythagorean theorem--geometry--to obtain them. This is a COORDINATE system, not a "scalar system" (scalar = magnitude only, NO geometry). This is all based on the hypotenuse of triangles--Euclidean, not scalar. Perhaps you should study some projective geometry to get your terms correct.
Pray tell, what are the other three roots that can be calculated here? Remember (or perhaps you didn't realize), these three unit numbers are the radii of the three balls defined by Larson's cube: The radius of the inner ball, the radius of the ball just containing the 2d slice of the stack and the radius of the ball just containing the stack. It's just a physical fact that the radii of these three balls is √1, √2 and √3. Expressing that truth using the Pythagorean theorem in no way changes the expansion/contraction of the balls from scalar to "Euclidean," whatever that means. The geometry of the universe of scalar motion is postulated to be Euclidean in Larson's system, so I don't follow the logic here.

daniel wrote:
dbundy wrote:

"Adding, multiplying, subtracting & dividing these numbers is as straightforward as doing so with the so-called Reals; that is to say, the numbers are ordered and their operations are distributive, commutative and associative, regardless of their dimension."

"So-called Reals?" What do you call them?

A "real" number defines an ordered set, with properties of distribution, commutation and associativity. Not the other way around. "Imaginary" numbers also have these properties because the operators are just axis designations and the magnitudes are of the real set. 2i + 3i = 3i + 2i; 2i * 3i = 3i * 2i. All it says is that you are doing a "real" operation on the "i" axis--could just as well be X, Y or Z, as well as i, j or k. You lose these properties when you exceed the single dimension that real numbers define, because we have no way to express an ordered set of planes or volumes, without reducing it to a single property such as area or volume--to stick it back on the 1D, real ordered set as a projection. Complex operations confuse people because they represent spin or twist, which cannot be represented by a displacement on a straight line--it requires a plane (Argand plane) to do it, which means you've lost the ordered set and therefore, the properties associated with the "real" line. This is the situation with straight lines, as well. A rectangle of (3,2) is not the same rectangle of (2,3), yet they are both "reals" and have the same area.
Real numbers don't exist in Larson's discrete system. In his system, motion is postulated to exist in discrete units, with two reciprocal aspects, space and time. I guess we could agree with Kronecker on this: "God made the integers. All else is the invention of man."

The trouble comes back to numbers and geometry again. There are geometric magnitudes for every number, but the converse is not true: there are geometric magnitudes for which there are no numbers. We can make symbols to represent such numbers, as we need to work with geometry algebraically, and that's a good thing, since modern science and technology would not exist without them, but the fact remains that there is this terrible incompatibility between discreet numbers and continuous geometry. Squaring the circle is impossible, which ultimately makes general relativity and quantum mechanics incompatible, theoretically. This is the biggest unsolved problem in theoretical physics.

The reason the algebraic properties are lost in the vector number system, as the dimensions of the system increase, is due to the fact that vector motion requires a starting and ending point in one "direction." There is no problem with 0d numbers, as the quantity of a collection of points (0d numbers) can be increased or decreased in one of two "directions," positive or negative "directions," to zero. However, if we extend these numbers to one dimension, we have to do so by means of rotation, and the starting and ending points we need to represent arc-lengths on a circle have to be arbitrarily chosen, because there's no way to say that one starting or ending point on the circle is greater or lessor than another.

Extending the numbers to two dimensions we not only lose the ordered property, but we also lose the commutative property because it's required to rotate the sphere of interest to point to the desired location on the surface, and this rotation now has multiple "directional" possibilities, making the outcome dependent upon the order of operations.

Of course, extending the numbers to three dimensions, really becomes problematic, because we've run out of dimensions at this point. There is no such thing as a fourth dimension, in Euclidean geometry, so it has to be invented and the operations required to manipulate these numbers is so order dependent that they lose the associative property.

Now, again these number systems correspond to vector motion; that is, change of position in one "direction" at a time. Rotation enables us to change the number of dimensions in which this motion is effective, but not without sacrificing something, in each higher dimension. This is not the case with number systems based on scalar motion, because it is motion in all possible "directions," of a given dimension, simultaneously. The binary expansion of the Tetraktys captures this dimensional truth and its relation to geometry through its correspondence with Larson's cube.

I know these scalar concepts are unfamiliar and may seem strange, but if you will start from the beginning and work your way through them, I think you will find that they are an exciting and useful way of developing Larson's new system of theory. At any rate, I do appreciate your engaging comments daniel. I hope you will continue.

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Re: Meeting a Terrific Challenge

Post by dbundy » Sun Oct 14, 2018 5:35 am

I think daniel's comment on the logic of the LRC's scalar number system is important, because it shows a need to understand Larson's cube (the 2x2x2 = 8 stack of 1-unit cubes), as fundamental. I explained it, with graphic illustrations, in the Introduction topic, but it's good to review it for those who haven't read the posts of that topic.

It really stems from the fact that, as the binary expansion of the tetraktys shows, there are no more than three non-zero dimensions in the universe of motion, and each dimension has two "directions." The fact that scalar motion is an increase or decrease of quantity, or magnitude, in the "directions" of these three dimensions, is critical to understand.

Vector motion, in a fixed reference system, has a direction that can be defined by x, y and z coordinates, but not so with scalar motion. This motion, though not recognized as such outside of Larson, is motion of magnitude (that is, size) only. However, we can express the ordered progression of these magnitudes, just as we can the set of integers.

1) Integer progression: n = 1, 2, 3, ...∞

2) 1d scalar progression: 2, 4, 6, ...(2n)1

3) 2d scalar progression: 4, 16, 36, ...(2n)2

4) 3d scalar progression: 8, 64, 216, ...(2n)3

Of course, just as the third non-zero dimension of the tetraktys contains both the lower non-zero dimensions and the zero dimension below it, so too does the 3d scalar expansion. We can prove this with the geometry of Larson's cube, which is the 3d geometric scalar expansion that is the equivalent of the 3d numerical expansion shown in the list above.

It would be so convenient if nature expanded numerically, but as we all know it doesn't. There is a disconnect between the numerical and the physical expansions, because nature expands in all directions, simultaneously. However, we can use the 3d numeric/geometric expansions to index, if you will, the 3d expansion of nature.

On this basis, there are three natural units of expansion isomorphic to the 3d numeric/geometric scalar expansion, which are defined by it. They are the three radii of the balls defined by the geometry of the cube. The first is the radius of the inner ball, which is always equal to n (n x √1). The second is the radius of the ball, which just contains the 2d slice of the stack. It is always equal to n x √2, and finally there is the radius of the ball, which just contains the stack. It is always n x √3.

When we consider these three radii, we can recognize that they form the basis of three scalar number systems, the progression of which is isomorphic to the dimensional progressions shown in the list above:

1) 1d scalar progression: 1√1, 2√1, 3√1, ... n√1

2) 2d scalar progression: 1√2, 2√2, 3√2, ... n√2

3) 1d scalar progression: 1√3, 2√3, 3√3, ... n√3

The fact that these three units are calculated by the use of the Pythagorean theorem does not mean there are negative versions of them, because neither circles nor balls can have a negative radius, and even if they could, it would make no difference, since:

1) √((-1)2 + 02 + 02) = √1

2) √((-1)2 + (-1)2 + 02) = √2

3) √((-1)2 + (-1)2 + (-1)2) = √3

The use of the Pythagorean theorem in the proof of these unit scalar magnitudes does not violate the fundamental postulates of the Reciprocal System in any sense. What it does do, however, is put us on a very firm foundation for describing a multi-dimensional scalar algebra, which does not exhibit the same pathology as the LST's vector-based multi-dimensional algebra, the four so-called "normed division algebras."

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Horace
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Re: Meeting a Terrific Challenge

Post by Horace » Sun Oct 14, 2018 11:47 am

dbundy wrote:
Sun Oct 14, 2018 5:35 am
...there are no more than three non-zero dimensions in the universe of motion, and each dimension has two "directions." The fact that scalar motion is an increase or decrease of quantity, or magnitude, in the "directions" of these three dimensions, is critical to understand.
IMO before considering multiple dimensions of motion, it is even more critical to understand, that one dimension of a unit of motion does not posses an intrinsic "direction", even if it can assume two possible "directions".
The concept of that "direction" arises only as a relationship between two or more units of motion, in as much as it can be stated, that one unit of motion has an opposite or the same "direction" to a second unit of motion.

This means that it is a logical error to write about one unit of motion as being intrinsically e.g. "inward", because from the perspective of a second unit of motion, that unit can appear as "outward"...even if from the perspective of a third unit, the first one appears as "inward".

In other words, those "directions" are not absolute - they are relative.

The same can be said about aspects of motion - space and time; and it is wrong to write about an aspect as e.g increasing without relating it to the increase or decrease of an aspect pair belonging to a second unit of motion.
This is because a unit of space belonging to one unit of motion can appear as increasing/expanding to the unit of space belonging a second unit of motion, while appearing as decreasing/shrinking unit of space to a unit of space belonging to a third unit of motion, when all three temporal aspects are assumed to be always e.g. increasing/expanding.

P.S.
A unit of motion is a change by one unit of space in one unit of time.

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Re: Meeting a Terrific Challenge

Post by dbundy » Tue Oct 16, 2018 12:38 pm

Hi Horace!

Good to hear from you again. You wrote:
IMO before considering multiple dimensions of motion, it is even more critical to understand, that one dimension of a unit of motion does not posses an intrinsic "direction", even if it can assume two possible "directions".
The concept of that "direction" arises only as a relationship between two or more units of motion, in as much as it can be stated, that one unit of motion has an opposite or the same "direction" to a second unit of motion.
I'm still trying to understand this point you've been making for a long time. The reason I put the word "direction" in quotes is to distinguish it from the word direction, as normally thought of in a fixed reference system. The number line has no direction in a fixed reference system, but it does have two "directions" relative to the unit datum, 1/1, or 0 displacement. We normally designate them as positive and negative relative to 0.

When we consider the use of a vector to describe a magnitude of motion, the length of the vector represents the magnitude of the motion and, if we limit it to the same two "directions" as the number line (positive or negative), it would point either to the right or to the left of the 0 point, or origin. If we plot two vectors, one to the left and one to the right, and they are the same length, the resultant is 0, or no motion.

However, scalar motion, i.e. magnitude only motion, cannot be represented by vectors, in this manner, since the scalar expansion of a line, from 0 to 1, for instance, has to expand in both "directions" of the one dimension simultaneously. Thus, at time equal to .1, say, the motion has existed for .1 unit of time. Its magnitude is not 0, its space component is continuing to increase over time. Likewise at t =.2 and t = .3 and so on to t = 1, at which point the space component, s, equals 1 too, but 1 in each of the two opposite "directions." If the motion reverses at this point, the s component begins to decrease, while the t component proceeds to t = 2. When t reaches 2, s has collapsed to 0.

Now, if we have a second unit of oscillating scalar motion, it may be increasing out to +&- 1 or decreasing from +&- 1 to 0, in the same manner as the first unit, but I can't see how that fact depends upon what the first unit is doing, or where it is in its cycle.

You wrote:
This means that it is a logical error to write about one unit of motion as being intrinsically e.g. "inward", because from the perspective of a second unit of motion, that unit can appear as "outward"...even if from the perspective of a third unit, the first one appears as "inward". In other words, those "directions" are not absolute - they are relative.
Try as I might, I can't see how that could be true. Each unit's "direction," whether in or out at a given point in time, seems to me to be relative to 0. If it's expanding from the point of 0, it's outward motion. If it's diminishing toward the point of 0, it's inward motion. This is absolute in the sense of the postulates of the system, I do believe.

You wrote:
The same can be said about aspects of motion - space and time; and it is wrong to write about an aspect as e.g increasing without relating it to the increase or decrease of an aspect pair belonging to a second unit of motion.
This is because a unit of space belonging to one unit of motion can appear as increasing/expanding to the unit of space belonging a second unit of motion, while appearing as decreasing/shrinking unit of space to a unit of space belonging to a third unit of motion, when all three temporal aspects are assumed to be always e.g. increasing/expanding.

P.S.
A unit of motion is a change by one unit of space in one unit of time.
Again, I really don't get it. If I understand you correctly, you're asserting that, though the space (time) of a unit of motion is diminishing toward 0, at a given point in time, viewed from the perspective of the space (time) of another unit of motion, it may actually be expanding from 0? I cannot understand in what sense this could be true. Sorry.

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Horace
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Re: Meeting a Terrific Challenge

Post by Horace » Tue Oct 16, 2018 5:22 pm

dbundy wrote:
Tue Oct 16, 2018 12:38 pm
The reason I put the word "direction" in quotes is to distinguish it from the word direction, as normally thought of in a fixed reference system.
For the sake of others participating in this thread, first let's clarify your vernacular used during the consideration of only one dimension of one unit of scalar motion, in non-vectorial system:
Q1) How many "directions" can this unit assume and how do you name them?
Q2) How many "directions" can the two aspects of this unit assume collectively and how do you name them?
e.g.: left, right, increasing, decreasing, inward, outward, etc...

P.S.
If you decide, that one of these questions does not make sense in terms of your vernacular, please explain why.

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Re: Meeting a Terrific Challenge

Post by dbundy » Tue Oct 16, 2018 9:56 pm

Horace wrote:
For the sake of others participating in this thread, first let's clarify your vernacular used during the consideration of only one dimension of one unit of scalar motion, in non-vectorial system:
Q1) How many "directions" can this unit assume and how do you name them?
Q2) How many "directions" can the two aspects of this unit assume collectively and how do you name them?
e.g.: left, right, increasing, decreasing, inward, outward, etc...

P.S.
If you decide, that one of these questions does not make sense in terms of your vernacular, please explain why.
Q1: One dimension - two "directions." I don't name them.

Q2: s/t = 1/1 is unit motion and so is t/s = 1/1. The only possibility for deviation from this unity is a continuous "direction" reversal over 1 unit, in one aspect or another. If the reversals (oscillation) are in the space aspect, the unit speed is decreased toward zero. If the reversal is in the time aspect, the unit speed is increased toward infinity. When space is oscillating, time has no direction, only "direction" (increasing). When time is oscillating, space has no direction, only "direction" (increasing).

The direction of the oscillations can be labeled in and out, increasing/decreasing, expanding/contracting, etc. but since this is not a direction as we usually understand it, I choose to label it "direction" instead, to avoid confusion with vector motion, which can be specified in a fixed reference system (up/down, left/right, forward /backward.)

Also, I differ with Larson in an important way: Since the progression is 3d, the oscillations must be also. Therefore, a 1d oscillation, such as that we are considering for purposes of illustration, does not exist separately, but only as part of the 3d oscillation. The 3d oscillations (space or time) constitute units of motion that are subject to the algebraic rules of numerical units that we know and love.

I hope that helps.

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Re: Meeting a Terrific Challenge

Post by Horace » Wed Oct 17, 2018 4:19 pm

dbundy wrote:
Tue Oct 16, 2018 9:56 pm
Q1: One dimension - two "directions." I don't name them.
Using the word "direction" in the context of both Q1 and Q2 is confusing to your readers at best.
Could you come up with a way to differentiate these two concepts without the need to write out the full phrases, like e.g.:
A "direction" of a unit of scalar motion,
vs.
A "direction" of an spatial aspect of that unit,
dbundy wrote:
Tue Oct 16, 2018 9:56 pm
Q2: s/t = 1/1 is unit motion and so is t/s = 1/1. The only possibility for deviation from this unity...
Please stay on topic. I did not ask how deviation is accomplished over multiple units. Only about"directions" involved in one unit of motion.
dbundy wrote:
Tue Oct 16, 2018 9:56 pm
The only possibility for deviation from this unity is a continuous "direction" reversal over 1 unit, in one aspect or another.
Since I asked about only one unit of motion, I expected the answer to relate to only one unit, too.
Thus, I am unsure whether you are writing about non-unity speeds over one unit of motion (if FP even allow such deviation over one unit) or non-unity speeds calculated over multiple units of motion (which is outside the scope of my question).
In case it is the former, I have to object on the basis of violation of the Fundamental Postulates and ask you what is the sense of quantizing motion, with unit boundaries and all, if the changes of "direction" are not limited to these boundaries only ?

...or the reversed question: In your interpretation of FPs, what is possible at the unit boundary, that is not possible within a unit ?
If any change is possible within a unit, then why have boundaries ...and consequently - any units at all ?

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Re: Meeting a Terrific Challenge

Post by dbundy » Thu Oct 18, 2018 7:28 am

I was trying to provide some context for my answer to your question, not go off topic, Horace. When you speak of "A 'direction' of a unit of scalar motion," the only way the phrase makes sense in my theory, is the "direction" relative to unit speed, s/t = 1/1. At s/t = 1/2, the "direction" is one unit to the low speed side. At s/t = 2/1. the "direction" is one unit to the high speed side. This is just like the integers: 1/2 equals -1, in terms of displacement, and 2/1 = +1, in terms of displacement.

With respect to the "direction" of the oscillating aspect, expanding the radius out from zero is one "direction," while collapsing it in toward zero is the opposite "direction." With respect to the "direction" of the non-oscillating aspect, the "direction" is always outward, as it continuously expands or increases uniformly.

We can also talk about the "directions" of the resolutes of the oscillation. In which case, the 1d component has two, opposite, "directions," positive and negative (+ & -), just like the number line; The 2d component has four, extending diagonally from the center point to each corner of the four squares (++, +-, -+, --), while the 3d component has eight, extending diagonally from the center to the corners of the eight cubes (+++, ++-, +--, -+-, ---, --+ -++, +-+).

Horace wrote:
Since I asked about only one unit of motion, I expected the answer to relate to only one unit, too.
Thus, I am unsure whether you are writing about non-unity speeds over one unit of motion (if FP even allow such deviation over one unit) or non-unity speeds calculated over multiple units of motion (which is outside the scope of my question).
In case it is the former, I have to object on the basis of violation of the Fundamental Postulates and ask you what is the sense of quantizing motion, with unit boundaries and all, if the changes of "direction" are not limited to these boundaries only ?

...or the reversed question: In your interpretation of FPs, what is possible at the unit boundary, that is not possible within a unit ?
If any change is possible within a unit, then why have boundaries ...and consequently - any units at all ?
I'm having trouble understanding the issue here. Imagine a unit 3d oscillation. The outside boundary is the surface of a sphere, with a radius of one unit. The inside boundary is the point, with a radius of zero. The "direction" of change takes place at the spherical boundary, from outward to inward, while it changes at the inner boundary, from inward to outward.

I don't know how to make it any clearer than that,

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Horace
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Re: Meeting a Terrific Challenge

Post by Horace » Thu Oct 18, 2018 10:32 am

dbundy wrote:
Thu Oct 18, 2018 7:28 am
When you speak of "A 'direction' of a unit of scalar motion," the only way the phrase makes sense in my theory, is the "direction" relative to unit speed, s/t = 1/1. At s/t = 1/2, the "direction" is one unit to the low speed side. At s/t = 2/1. the "direction" is one unit to the high speed side. This is just like the integers: 1/2 equals -1, in terms of displacement, and 2/1 = +1, in terms of displacement.
And that usage of "direction" is clear to me when considering more than one unit of motion.
It is simply the direction of deviation or the operational interpretation of the ratio listed below this number line, that serves as a visual aid to depict the sign of that deviation.
Image

I don't have an issue with the concept of "direction" of deviation, nor the operational interpretation of the ratio nor graphing this interpretation on a number line, but I do have an issue with the conceptual ambiguity of the word "direction" as contrasted to the other concept, namely the "direction" of aspects of motion, about which you wrote below.

Over the tears, your readers and I, have tripped up on this ambiguity repeatedly making communication and progress difficult. I employ you to coin a different word that will apply only to the sign of deviation accounted over multiple units of motion.
dbundy wrote:
Thu Oct 18, 2018 7:28 am
We can also talk about the "directions" of the resolutes of the oscillation. In which case, the 1d component has two, opposite, "directions," positive and negative (+ & -), just like the number line; The 2d component has four, extending diagonally from the center point to each corner of the four squares (++, +-, -+, --), while the 3d component has eight, extending diagonally from the center to the corners of the eight cubes (+++, ++-, +--, -+-, ---, --+ -++, +-+).
I don't have an issue with that but let's hold off on consideration of more than 1D components, because more dimensions are not necessary convey my point and they only complicate things at this stage of the discussion. It shall be picked up later, though...
dbundy wrote:
Thu Oct 18, 2018 7:28 am
I'm having trouble understanding the issue here. Imagine a unit 3d oscillation. The outside boundary is the surface of a sphere, with a radius of one unit. The inside boundary is the point, with a radius of zero. The "direction" of change takes place at the spherical boundary, from outward to inward, while it changes at the inner boundary, from inward to outward.
If that means that the change of "direction" are limited only to these boundaries, then I don't have an issue with it.
dbundy wrote:
Thu Oct 18, 2018 7:28 am
With respect to the "direction" of the oscillating aspect, expanding the radius out from zero is one "direction," while collapsing it in toward zero is the opposite "direction."
OK, I would like to concentrate on the issue of relativity of "directions" in this context.
dbundy wrote:
Thu Oct 18, 2018 7:28 am
With respect to the "direction" of the non-oscillating aspect, the "direction" is always outward, as it continuously expands or increases uniformly.
Let's begin by elaborating on this statement above...

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Re: Meeting a Terrific Challenge

Post by dbundy » Thu Oct 18, 2018 1:59 pm

The easiest way I've found to think about it is to plot it, as what I call a world line chart. In the graphic below, I've plotted both the space oscillation and the time oscillation, for the two reciprocal sectors:

Image

Let's consider the Material sector plot. Time is progressing uniformly from bottom to top. Space is progressing uniformly from left to right. Consequently, a one-to-one progression is equivalent to unit motion, represented by the heavy black arrows pointing diagonally up and to the right. A continuous "direction" reversal in the space aspect, where the space/time ratio equals 1/2, is shown by the alternating green arrows zig-zagging upwards, as time progresses uniformly. A continuous "direction" reversal in the time aspect, where the space/time ratio equals 2/1, is shown by the alternating green arrows zig-zagging to the right, as space progresses uniformly.

As you can see, the green arrows are all plotted along the diagonals of the aspect of the unit that is oscillating, because its time (red arrow) or its space (blue arrow) is "uni-directional," while the "direction" of its reciprocal aspect is oscillating, producing the diagonal plot.

The Cosmic sector plot is the same, only the "directions" are reversed. Time is progressing uniformly from top to bottom, while space is progressing from right to left. This gives us the equivalent of a Cosmic sector number line, where, again the heavy black arrows along the diagonal represent unit motion, with no deviation, where t/s = 1/1, and a unit space oscillation produces the deviation, t/s = 1/2, while a unit time oscillation produces t/s = 2/1.

I hope this is helpful.

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