Energy and Structure

[Buddy]

Imagine a weight suspended on a spring. That is a resonator. If you set it in motion, it eventually settles down due to its own internal resistance (friction). In a clock, the weight is the flywheel, which spins back and forth against a spring. However in a clock the motion doesn't settle normally, because the clock mechanism creates a negative resistance (anti-friction) which, added to the positive resistance (friction) of the resonator, cancels (adds) to a total resistance of zero. As long as this state is held, the resonator will be in constant, stable motion.

When the clock is starting, the resonator is given a negative total resistance, to build up the movement of the resonator. When the max vibration is reached, a second threshold is crossed. Positive resistance is then added, so that the total resistance on average cancels to exactly zero, so that the movement of the resonator is constant and stable, not growing or decaying.

Yes, like harmonic oscillators as damped and undamped systems, no? I get this from reviewing the Wikipedia entry on Oscillation:

_http://en.wikipedia.org/wiki/Oscillation

...where there is also an animated graphic of a spring-mass system that resembles how I conceive your "weight suspended on a spring."

Looking at that graphic, when the spring-mass system goes from motionless to in-motion, the starting point of it's ascension would be a "DO" on the musical scale in our octave analogy. It's an "affirming force" and it rises. When it gets to its highest point, it meets denying force, or gets denied any way to continue, so it now begins to descend back to its starting DO. Also in terms of our octave analogy, it will rise through six different points and will descend back through those points.

I believe I'm seeing an aligning of contexts here. I hope you're able to read and make sense of my posts because I think I'm seeing the correlations between your analogies and the ones made or inspired by Gurdjieff's octaves. Please let me know if You don't so I can make the connections as obvious as I can for your assessment.

For now, I'd like to make one more post related to the musical octave and just stop with that. It would be good to know at that point if you think we're on a track that's going somewhere or if you want the topic to go somewhere else.

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Gurdjieff says that the Law of Seven is the first fundamental law of the universe. The author of Cosmic Secrets says that the musical octave is the best way to represent this. FWIW, the Law of Seven is also referred to as the Law of Vibrations as well as the Law of Octaves.

As a musical octave, DO, RE, MI, FA, SO, LA and TI represents the Law of Seven. When G mentions this Law as a movement of forces, we need to set aside any straight-line representation and use a symbol of a line that starts at some point, continues on in a line to a point where it then angles sharply, continuing to the next point where it angles sharply again and continues until it meets back at the starting point where there there will be a total of seven "deflections."

If G were making a musical analogy of the Law of Seven, he might use the info we already covered regarding the scale of C on a piano keyboard, but we might get stuck on a model of a single straight line and probably incorrectly assume that the line of force(s) he is talking about is also linear. It's not, because he has already said in Beelzebub's Tales that the flow of forces is constantly deflecting and uniting again at its ends and that each of these deflections can also be called a "center of gravity." This is how given processes complete themselves, or how "This Law, passing through everything newly arising and everything existing, always makes its completing processes..."

Changing gears here for a minute, the Law of Three is another fundamental law of the universe and in our analogy of a musical octave, the three "things" being referred to correspond to mathematical "thirds" of a scale in question.

If you look back at my previous post, notice that in the second half of that quote, under the heading "THE TOTALITY OF 256 VIBRATIONS" there is a representation of the diatonic ratios as fractions. Notice that at FA the ratio is 1/3, at LA the ratio is 2/3 and at the ending DO or "ALL", we can think of its fractional equivalent as 3/3.

So, the points of an octave that will correspond to the Law of Three, or three forces, are FA, LA and ending DO. Eventually, we might want to try and determine what aspects of some thing we are investigating correspond to these three forces or points and how the three relate to its totality, or perhaps the outermost octave of the thing before going deeper and exploring its inner octaves. I'm not totally sure of this though, so bear with me a little longer if you think it might be worthwhile to do so.

Now, to begin to relate some of this info to some "whole phenomenon", some "totality" or a "something", we might try to put it in context with some of the things Gurdjieff has already said about Man. Man is a "something."

To begin with, we should first say that Man consists of seven centers in total: Higher Mental center, Higher Emotional Center, Intellectual center, Emotional center, Moving center, Instinctive center and Sex center. Further, he says that in his present state, Man has little or no contact with the higher mental center and higher emotional center and that the sex center was designed to activate later on in life.

In addition, Gurdjieff has depicted Man as a three-story factory; each story arising from the three primordial forces. These three forces have also been called the affirming, the denying and the neutralizing forces.

Here's a blend of a few of Gurdjieff's analogies: In Chapter Two, the author mentions that in All and Everything, Gurdjieff identifies the moving and instinctive centers (the lower story) as the denying force, the emotional center (the middle story) as the neutralizing force, and the intellectual center (the upper story) as the affirming force.

With this in mind, I think I'll pause or stop here to see what happens next.

 

[obyvatel]

Another way to look at it is that resistance dissipates energy. Whatever the antonym of dissipate is, that is what negative resistance does. Accumulate is the closest I could get.

The term negative resistance comes from electrical circuits. Electrically, negative resistance means the behavior is such that the electrical potential difference between two points is reduced as the electric current flow between the two points is increased. This is unlike materials that obey Ohm's law where an increase in electric current flow leads to an increase in electrical potential difference between these points.

Practically, one way of achieving negative resistance is by some energy input to the system. Take the example of a crystal oscillator . The crystal (eg. quartz) has its own natural frequency of oscillation along with some damping component electrically modeled as a positive resistance. In a practical oscillator, an electronic circuit which requires its own power supply (additional energy input to the system) is added to the crystal to act as a negative resistance which overcomes the damping (positive resistance) of the crystal and sustain a stable oscillating frequency.

Plasma is interesting. It displays differential negative resistance properties under certain conditions. Donald Scott's "Electric Sky" mentions this. His website on the electric sun hypothesis (_http://electric-cosmos.org/sun.htm) shows the electrical characteristics of laboratory plasma (see figure 2 on the web page). When the current density in plasma has reached a high level, called the arc mode, it exhibits negative resistance properties. This is not seen when the current density is low and the plasma is in the dark current mode where plasma displays positive resistance property. Unlike the electronic circuit example above which needs an extra energy input to the system, negative resistance in plasma is a property of the medium itself under high current density condition. The behavior is non-linear. The non-linear electrical properties of plasma can perhaps be correlated with the property of plasma to form self-organizing structures.

 

[monotonic]

Another way to look at it is that resistance dissipates energy. Whatever the antonym of dissipate is, that is what negative resistance does. Accumulate is the closest I could get.

The term negative resistance comes from electrical circuits. Electrically, negative resistance means the behavior is such that the electrical potential difference between two points is reduced as the electric current flow between the two points is increased. This is unlike materials that obey Ohm's law where an increase in electric current flow leads to an increase in electrical potential difference between these points.

Practically, one way of achieving negative resistance is by some energy input to the system. Take the example of a crystal oscillator . The crystal (eg. quartz) has its own natural frequency of oscillation along with some damping component electrically modeled as a positive resistance. In a practical oscillator, an electronic circuit which requires its own power supply (additional energy input to the system) is added to the crystal to act as a negative resistance which overcomes the damping (positive resistance) of the crystal and sustain a stable oscillating frequency.

Yes, only substances which are energized beforehand seem to exhibit negative resistance. By definition (mathematically), negative resistance accumulates energy instead of dissipating it, and since energy is neither created nor destroyed, it has to have come from somewhere to be accumulated.

BTW, I know these things because I design analog circuits as a hobby. I learned about oscillation because I have to design highly controlled circuits that aren't unstable.

Plasma is interesting. It displays differential negative resistance properties under certain conditions. Donald Scott's "Electric Sky" mentions this. His website on the electric sun hypothesis (_http://electric-cosmos.org/sun.htm) shows the electrical characteristics of laboratory plasma (see figure 2 on the web page). When the current density in plasma has reached a high level, called the arc mode, it exhibits negative resistance properties. This is not seen when the current density is low and the plasma is in the dark current mode where plasma displays positive resistance property. Unlike the electronic circuit example above which needs an extra energy input to the system, negative resistance in plasma is a property of the medium itself under high current density condition. The behavior is non-linear. The non-linear electrical properties of plasma can perhaps be correlated with the property of plasma to form self-organizing structures.

It would seem that it is the negative resistance of the "arc mode" that excites the resonances of the molecular medium to create light, a visible arc. In this case, negative resistance seems to be exhibited on a molecular scale and this results in distributed negative resistance properties within the plasma, as well as that between its electrodes, and the possibility of inner-plasma modal space emergent phenomena. This is one thing that intrigued me. The C's have remarked on what causes light many times. Here is clearly one way of causing light. Every medium has its own resonant light profile, be it air, nitrogen or salt, which glows yellow when there is an arc through it. This is how a spectrometer works. I have a plasma globe that has a special gas composition, so that the plasma streamers glow green, blue and orange in different places.

Here is a DIY discharge tube in action:

Notice there is not one glow region, but many. I suspect this is a modal oscillation of the tube. The oscillation has reached the maximum vibration, where at each excursion the current is high or low enough that the modal nodes have current as high as the left electrode. If so, then it is radiating EM waves at 11GHz (a wavelength of 1 inch)! It is also interesting to note that the wavelength seems roughly equal to the diameter of the tube.

 

[monotonic]

With this in mind, I think I'll pause or stop here to see what happens next.

In ISOTM Gurdjieff says that the science of the 3 forces revolves not around the specific substances, but rather which force was acting through them at the time. After trying to map the 3 forces to the variables involved in resonance and these things I've realized that any one of the forces, as they are currently defined, may act through any given variable. It only depends on which variable you're solving for, and the variables tend to arrange themselves in a triangle.

I think you could probably draw many forces in a tree-like structure with the trunk being the 3rd force. It could also be a pyramid made of triangles joined at the corners.

I've tried looking at your excerpts from many different angles but I can't quite connect it with resonance (I think I just haven't seen enough of the material). Here is an idea though. Oscillation can be caused by a feedback loop. However the control of the oscillation can also be put in a feedback loop. When the resistance is modulated above and below zero, the resulting oscillation has an envelope that resembles a sine wave, but is distorted in exponential fashion. This to me fits better with the octaves analogy because octaves are a logarithmic scale (a logarithm reverses an exponential curve).

I've attached an image of a simulation I made. It is a simple resonator connected to a varying resistance. The top graph is the resistance, which I've set to go negative and positive over time. The lower two traces are the oscillation. As you can see, while resistance is negative, oscillation grows, and while resistance is positive, it decays. The bottom graph is the actual oscillation, which responds exponentially to the resistance, resulting in the distorted sine shape. The middle graph is the same thing, run through a logarithm which reverses the exponential distortion. This graph corresponds to the layout of a piano, because perceptual frequency is logarithmic. However the bottom graph corresponds to the actual behavior of the numbers when you speak of ascending and descending octaves. For this reason elementary resonance doesn't seem to fit with those excerpts; it is not exponential. However "envelope resonance" as this might be called could fit.

 

[Buddy]

....depends on which variable you're solving for, and the variables tend to arrange themselves in a triangle.

I think you could probably draw many forces in a tree-like structure with the trunk being the 3rd force. It could also be a pyramid made of triangles joined at the corners.

The triangle whose mathematics were discovered by Pascal, maybe? FWIW, according to the author, we now realize in retrospect that Pascal's Triangle is created by the universe's diatonic structure.

Here is an idea though. Oscillation can be caused by a feedback loop. However the control of the oscillation can also be put in a feedback loop. When the resistance is modulated above and below zero, the resulting oscillation has an envelope that resembles a sine wave, but is distorted in exponential fashion. This to me fits better with the octaves analogy because octaves are a logarithmic scale (a logarithm reverses an exponential curve).

A logarithmic scale like in Fibonacci Numbers which are intimately connected with the golden ratio and whose spiral shapes can be found in Nature? The author is also saying that the mathematical configuration of the diatonic forces explains "why" Pascal's Triangle and the Fibonacci Series are so prevalent in the universe.

For this reason elementary resonance doesn't seem to fit with those excerpts; it is not exponential. However "envelope resonance" as this might be called could fit.

I'll have to think about the graph tomorrow. There's also still a lot in that book I have to grok before being able to synthesize any of it with the concept of "resonance." I think the book doesn't mention resonance, per se, and I'm wondering if that might be due to resonance generally being a dimensionless parameter? Meaning, it's not something you measure directly like intensity of vibration, but it can be accounted for like when engineers design bridges? Or maybe I'm mixing up different contexts of resonance.

There's plenty on vibration and oscillation, though, and I think I need to study a bit more of it before commenting further.

 

[John G]

http://ictwiki.iitk.ernet.in/wiki/index.php/The_harmonic_oscillator:_zero_point_energy_and_tunneling

Significance of the solutions. (Why is harmonic oscillator important?)

1. The quantum mechanical harmonic oscillator has some energy even when the quantum number is zero. (i.e., it cannot have zero energy of vibration). Classically an oscillator having zero energy does not oscillate!! A state of absolute rest is possible, however, only in classical systems.

Zero point energy (referring to the above) is a purely quantum mechanical concept and is fundamentally important for molecules and solids.

2. The boundary for an oscillator is endless. What does the potential look like? It is a parabola and the classical oscillator can ONLY oscillate between the two ends of the parabola. The total energy of the oscillator is all potential energy at the turning points and all kinetic energy midway between the two points. However, in quantum mechanics the wave function outside of this classical parabola can be nonzero for all values of x, though very small. Thus, the square of the wave function which represents probability density is nonzero everywhere, except for a finite number of points. It goes to zero as x tends towards infinity. This is often stated as that quantum mechanical systems can be found in regions where they are strictly forbidden if classical mechanics were applicable. This is also known as quantum mechanical tunneling. Though tunneling can often be explained by solution of the Schrödinger equation for particles in one dimensional finite potential barrier models in physics, chemistry students are introduced to tunneling from harmonic oscillator model. It is a finite potential barrier model anyway. Experimentally tunneling phenomena have been amply demonstrated. The invention of scanning tunneling electron microscope which was credited with a Nobel Prize in physics amply demonstrates the idea.

So you have some quantum harmonic oscillator math that describes how a tunneling diode does classically illegal things like negative resistance without added energy. Quantum mechanics in general kind of only obeys energy conservation via "averages". Plasma is a quantum physics thing which I think self organizes charges in ways that are illegal classically too.

To relate to the Pascal triangle and Fibonacci numbers (which are sums of diagonals on the Pascal Triangles) you first have to note the rows of the Pascal triangle are the graded dimensions of Clifford Algebra and then relate Clifford Algebra to quantum harmonic oscillator math.

Via Tony Smith:

Pierre Ramond said in hep-th/0112261: "... exceptional algebras relate tensor and
spinor representations of their orthogonal subgroups,
while Spin-Statistics requires them to be treated differently ...
all representations of the exceptional group F4 are generated by three sets
of oscillators transforming as 26. We label each copy of 26 oscillators as
Ak_0 , Ak_i , i = 1, ... , 9, Bk_a , a = 1, ... , 16,
and their hermitian conjugates, and where k = 1, 2, 3.
Under SO(9), the Ak_i transform as 9, Bk_a transform as 16, and Ak_0 is a scalar.
They satisfy the commutation relations of ordinary harmonic oscillators ...
Note that the SO(9) spinor operators satisfy Bose-like commutation relations ...
both A_0 and B_a ... obey Bose commutation relations ...
Curiously,
if both ... A_0 and B_a ... are anticommuting, the F4 algebra is still satisfied ...
One can just as easily use a coordinate representation of the oscillators
by introducing real coordinates
...[ for A_i ]... which transform as transverse space vectors,
...[ for A_0 ]... which transform ... as scalars,
and ...[ for B_a ]... which transform ... as space spinors
which satisfy Bose commutation rules ...".
Since the commuting/anticommuting F4 lives in Cl(8) and
since Cl(8) by periodicity is the fundamantal factor of all large Clifford algebras,
the commuting/anticommuting property goes to all large Clifford algebras
and in particular goes to the tensor product Cl(8) x Cl(8) = Cl(16)
in which 248-dim E8 lives naturally as 120-dim bivector Spin(16) commutators plus
128-dim half-spinor of Spin(16) anticommutators... The Commutator part of F4 is V8 + BV28
The AntiCommutator part of F4 is S8+ + S8-
V8 and S8+ and S8- are related by Triality.

 

[monotonic]

I've heard of tunnel diodes, and IIRC they are not manufactured anymore, giving oscilloscope repairers quite a headache. However I never saw anything to suggest they output more energy than put in. The V-I curve always stays in the upper right quadrant, which is all that is necessary to ensure only positive power dissipation takes place. So, The negative resistance exhibited by a tunnel diode does not in the ordinary sense seem to be related to its electrical properties when operated in the usual way.

I found a paper describing the use of tunnel diodes for harvesting zero-point energy but that has to do with thermal noise, which I cannot directly connect with negative resistance, and the only reason to use them seemed to be as a detector diode because they had the most useful characteristics as detector diodes go.

That said, I only know basic algebra, have probably forgotten half of it, and have never had a calculus class. I tried to look up Clifford algebras, thinking I might learn something by studying them, but couldn't find anything that made sense.

 

[Buddy]

That said, I only know basic algebra, have probably forgotten half of it, and have never had a calculus class. I tried to look up Clifford algebras, thinking I might learn something by studying them, but couldn't find anything that made sense.

The author of Cosmic Secrets provides a visual triangle composed of the actual numbers and arrows to show which way they add up. You'll even see the Fibonacci numbers right there in plain sight. I'm on the way to work now, but I'll post the relevant excerpt this afternoon.

 

[monotonic]

I'll eagerly wait for those excerpts.

I just read chapter 7 of ISOTM today, and it deals directly with Buddy's octave stuff, seemed to be more to the point and less hypothetical in nature. That could just be because I am only now reading about it.

Some observations on negative resistance:

1: Ordinarily damped systems become undamped with the introduction of negative resistance. Resonators vibrate when exposed to it.
2: Growing oscillations are caused by negative resistance. Decaying oscillations are caused by positive resistance.
3: Negative resistance is always to blame when a small disturbance has a large result.

An interesting question is, based on these examples, what is the reaction most people have when they encounter negative resistance in life? How do they feel, and what are things that have made you feel this way? This way we may come up with concrete examples of negative resistance acting in our lives. We may then identify the structures and forces involved in these circumstances.

PS: I've transposed "see" and "use" twice, once when writing, and once when reading. What gives!? I'm pretty tired though, without which I don't expect either case would have happened. It could be a sort of dream...

 

[Buddy]

I'll eagerly wait for those excerpts.

I just read chapter 7 of ISOTM today, and it deals directly with Buddy's octave stuff, seemed to be more to the point and less hypothetical in nature. That could just be because I am only now reading about it.

Some observations on negative resistance:

1: Ordinarily damped systems become undamped with the introduction of negative resistance. Resonators vibrate when exposed to it.
2: Growing oscillations are caused by negative resistance. Decaying oscillations are caused by positive resistance.
3: Negative resistance is always to blame when a small disturbance has a large result.

An interesting question is, based on these examples, what is the reaction most people have when they encounter negative resistance in life? How do they feel, and what are things that have made you feel this way? This way we may come up with concrete examples of negative resistance acting in our lives. We may then identify the structures and forces involved in these circumstances.

Interesting indeed. I'd like to address them when I'm done with those promised excerpts.


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Here are the excerpts I mentioned. I decided OCR was inefficient for providing the info due to the drawings, varying font sizes and positionings the author uses in the book. Image format works better. So I'll make a series of posts with image attachments that show the triangle and a few more pages that explain what you're seeing.

There are probably 70 or more pages between the excerpt previously provided and what I'm posting now, so if you feel there is too much context missing, then a specific question might help me zero in on anything extra I might need to provide.

So, here is the first of the page images with the rest to follow (must be logged in to see):
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[Buddy]

The next page:
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[Buddy]

Next:
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[Buddy]

Next to last page:
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[Buddy]

Finally:
.
.
Note: I left the bottom of this page intact, although my intention was to end this excerpt series on the phrase "Fibonacci series" at the end of the next-to-last paragraph.
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[John G]

However I never saw anything to suggest they output more energy than put in. The V-I curve always stays in the upper right quadrant, which is all that is necessary to ensure only positive power dissipation takes place. So, The negative resistance exhibited by a tunnel diode does not in the ordinary sense seem to be related to its electrical properties when operated in the usual way.

I found a paper describing the use of tunnel diodes for harvesting zero-point energy but that has to do with thermal noise, which I cannot directly connect with negative resistance, and the only reason to use them seemed to be as a detector diode because they had the most useful characteristics as detector diodes go.

That said, I only know basic algebra, have probably forgotten half of it, and have never had a calculus class. I tried to look up Clifford algebras, thinking I might learn something by studying them, but couldn't find anything that made sense.

The "averages" for quantum mechanics kicks in fast enough that you aren't going to violate energy conservation on a large scale hence the tunnel diode requires a bias voltage for its negative resistance region. But each actual tunneling event can't be described in a normal energy conservation way. Each event is described as borrowing energy from the vacuum. The bias voltage in effect raises the "zero-point" around which the classically illegal jumps around the average are done. Basically you have a bunch of charges making classically illegal jumps.

Yeah we certainly aren't going to use Clifford Algebra the way Ark and Tony Smith do. There are some easy things to see like the rows of the Pascal triangle. It's even easy to use a tensor product to say combine the Cl(8) and Cl(2) rows to get the Cl(10) row. There's an 8-fold periodicity to Clifford Algebra which means combining the Cl(7) and Cl(3) rows will not give you the Cl(10) row.

From an ISOTM point of view the idea might be to show how that three octave via Law of 3 120 degree rotation formation of the Enneagram near the end of ISOTM relates to rows of the Pascal triangle which involves math-wise (for me) relating Triality to Clifford Algebra.