Hi all,
Jal,
If, as you suggest, there is a pattern that can describe the progression, how will we know which one is "more fundamental".
You have described a relationship with the "circle & sphere",
WN? has talked about the "hexagon" lattice.
Do we have a "fundamental shape", or geometry?
http://en.wikipedia.org/wiki/Directional_statisticsQUOTE
Directional statistics is the subdiscipline of statistics that deals with directions (unit vectors in R^n), axes (lines through the origin in R^n) or rotations in R^n.
The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of circular data.
Other examples of data that may be regarded as directional include statistics involving days of the week, months of the year, compass directions, dihedral angles in molecules, orientations, rotations and so on.
The thing about "vibrations" is, that each movement (off axis; or from center) happens only "1/2" of the time, and the other 1/2 of the time, is counter directional. We have not reached a "shape" yet, but it is easier to model with a circle, in the initial explanation.
However, even in the realm of the circle/sphere, we are not limited to only moving from the center, out. So, the relationship of the radius to the circumference may NOT be "fundamental".
What happens when "the node splits in 2"? Basically speaking, they "co-exist" while still in the realm of "one unit", or our original shape, or boundary. Actually, there would be a "slight bulge" in our original circle, of some "minimal increase" in size, or volume. This value could only be expressed as a ratio, of the fundamental unit (1) to said increase.
One only has to watch cellular division under a microscope to understand the "fundamental nature" of the method. It is the progression from "1 to 2"
The question of which is more fundamental, between the "loci" (the center of the circle) or its boundary, calculated by the constant ratio of pi, is also worth asking.
These loci, or nodes, I propose follow the Pauli principle, and are the fundamental process of "not sharing space", which runs as the inverse of superposition. This is called "anti-bunching" sometimes. The relationship between "anti-bunching and bunching" produces the general "Brownian motion", and "zwitterbewung", which is our general "oscillating phenomenon", measured by "vibrations", or "cycles".
They can not "follow" this principle, until they are "born", or fully created, "pinched off from the bubble", and actually
measurably separate from each other, in which case, they are "thrust apart" at the speed of light. This is "aided" by the creation of our "other" fundamental component, the "anti-node", which will form AS SOON as geometrically possible.
Again, we are at the point where we ask, "what is this distance, or interval"?
In order to "volumize" the area surrounding a "node", we must give the boundary some LIMIT. It could be pi . This has worked for many different models, and each has brought understanding to us. With these "dualistic" properties of "phase", or the difference (in time and space) between nodes and anti-nodes, IF we use pi, we must admit an "anti-pi" sort of concept.
If we start from a circle, split it in half, and separate them by the diameter (the fundamental unit x2) then we arrive at the "sine wave". This is the "anti-pi", and it operates in "complex space" in order to measure the duality. The vibration is measured from -1 to zero to +1 , because, using pi, the loci has the value of "zero", because it was the "starting point" of the "radius" (1/2 diameter).
Even if we use angles, or degrees, we have to contend with "zero"; even worse, letting a position have "0, 360" shared relationship. This does nothing to measure "fundamental units", where we just need to get from "one", to "two" as our first step. The "first step" is primordial; it is "the urge" that is in all life. It is the oscillation that is in all energy, regardless of velocity, or other change in linear coordinates.
Note, that we have not even completed a "fundamental cycle", and we are already into complex mathematics, and "non-physical counting". Was that "large first step" really necessary?
Can we come up with a system, that will measure the "linear, unit (integer) progression" of the "radius becoming the diameter" AND the ratio of that radius to the circumference (pi)?
This system will have to be "compatible" with BOTH addition and multiplication. This sounds easy, and perhaps even "a bad question"; don't we always follow such a "cardinal rule"?
Can we follow that rule, if our increments are "-1, 0, +1 "? Not without "higher mathematics", which require a fantastic amount of complexity to "stay in step" with this progression. I'm not really going to comment more, and just be "negative". The point is, that we should at least explore the possibility of a different method, that stays with these "fundamental" principles.
If we say that we MUST use multiplication, as part of the "tool kit" that we are assembling, because we want to use the "circle" as a model shape, and we are going to have to multiply r by 2pi , then we can NOT even use "one" as our starting point", because every # multiplied by 1, stays the same. Not a good way to measure change.
So, we set one of our initial parameters at >1 .
Next, we still need a "fundamental unit", and that needs to be "one", by definition. Sparing the full explanation of "why", let's continue from jal's example, and divide the fundamental unit up into 12 parts, separated by a common "multiplier", or "rate". If we ask the question, "what number, divided by a constant rate, in 12 steps or parts, will get us to "one"? , we are asking for the "12th sq rt of 2". This is the ONLY option. (a unique solution)
Parameter 2 becomes 1.05946..
Why would this work ( measure the "linear, unit (integer) progression" of the "radius becoming the diameter" AND the ratio of that radius to the circumference) ?
Because the "radius becoming the diameter" is "1 becoming 2", and it is never more than 1. However, from the previous discussion above, we remember that this is still happening WITHIN the circle, and is actually in the opposite phase space. This is a "simple" equivalent to measuring the complex space of -1.
This is because we decided that the radius is "first", then the other values (diameter, circumference), in order of importance. So, we are not going to move in the direction of expanding the circle, until we have completed the circle, from dualistic phase space.
Why am I doing this? Because we know that a wave has a rate that fluctuates. Whatever value we start with, has a peak on one side of a fundamental area, and then, in a separate moment in time, a peak on the "other side". The method I am using says that the "other side" is not "really imaginary", but also inverse or opposite.
The "opposite" of the radius would be a "counter radius". This simply completes the definition of the "diameter", which is the combination of the 2 values, "radius in our chosen direction", and a "radius in the opposite direction". However, we measure these with the same method; that is, the radius is "one" unit, and as we increase this value, in 12 steps, to get to the diameter, which is always 2x the radius.
The end result is that when we get to 2x our original radius, we still have to multiply by 2 pi. Now we have progressed one fundamental unit, and taken as a whole, have expanded our circle by 2. [2r x 2pi = 1r x 4pi = 4r x 1pi]
This means that if our nodes are separated by one unit, and that unit is our radius, then that unit has a constant ratio to the circumference (2pi). You are describing one circle, by this fundamental unit.
When this fundamental value is doubled to 2 units, you have the equivalent of 2 circles, of the original circumference, or 1 circle of double the circumference. This embodies Huygens' Principle. Drawing circles implies choices as to HOW MANY nodes are you counting? As long as the nodes are equally spaced, you can approximate the wavefront by finding the center of all of the nodes, and drawing a "greater" circle.
The wavefront is this expanding circumference then, and can be gaged with a constant rate, that also measure the linear expansion of the radius, by the same rate. If we increase the radius by 1/4, this will give a circumference that is 1/4 larger than the original circle, created from the fundamental unit.
When we reach a distance (in one linear direction) that is 2r , we have the option of interpreting this as a "new circle" of double size, and/or as 2 circles, of the original size.
Part of the decision as to how to interpret this comes from whether or not our "source" is moving. That is further along in the discussion, where we can discuss Doppler shift, which measures the "asymmetric" version that results from "not making the choice". In other words, keep our original circle, and move (and measure) the change in position of the loci, as if it was "on the way" to becoming "2 circles" (but never getting there).
The catch here, is that I have not expressed the interpretation of QM, which says that there is a minimum based on "harmonics", or a finite integer stack of nested curves, that will match our energy measurement. Nor have I expressed the limitation placed on this scale, that GR says exists due to gravitational redshift.
However, both of these interpretations say that at the center of the circle, we have "duality", or some bit of change in position (an average), that will effect our ultimate measurement, by a change in "expectation".
I just point these out, so that when making your decision, you are asking "from where (which parameter) do we derive a fundamental unit"? Something that is "ad hoc", and added to the "system", or something that can create the system on its own?
regards, (and sorry for the length of this post)
T.Roc
heheh you cannot get half of the minimum length. hehehe 
