To add comments or start new threads please go to the full version of: MATHEMATIC/NUMBER-THEORY INSIGHTS from TOE project
PhysForum Science, Physics and Technology Discussion Forums > Relativity, Quantum Mechanics and New Theories > Physics General > Special Project: 'Complete' Cosmology Theory From Scratch
Pages: 1, 2, 3, 4, 5, 6

RealityCheck
Hello and Welcome EVERYONE, to this new thread.

As the title implies, THIS thread will run parallel with our TOE thread.

Here we will carry out any GENERAL/RELATED DISCUSSIONS of 'MATHEMATICAL' MATTERS for eventual 'plugging in' to our TOE PROJECT at the appropriate stages.

I won't give any more 'guidance' than that at this point, but expect to insert an observation or two as our TOE project's progress indicates...so let's see what develops, heh?...who knows what 'emergent' behaviour THIS thread will exhibit! hehehe.

Later guys and gals: RealityCheck.
paresh dave
QUOTE (RealityCheck+Oct 3 2005, 12:40 AM)
Hello and Welcome EVERYONE, to this new thread.

As the title implies, THIS thread will run parallel with our TOE thread.

Here we will carry out any GENERAL/RELATED DISCUSSIONS of 'MATHEMATICAL' MATTERS for eventual 'plugging in' to our TOE PROJECT at the appropriate stages.

I won't give any more 'guidance' than that at this point, but expect to insert an observation or two as our TOE project's progress indicates...so let's see what develops, heh?...who knows what 'emergent' behaviour THIS thread will exhibit! hehehe.

Later guys and gals: RealityCheck.

P-N-G I can think,
Think mass, that is charge,(what you give mass number same is charge num
Equal repulsion, opposite attraction.
G attracts both p-core and n-over.
P-G and N-G span is atomic breathing (deviation between nucleons)...E-G MOTION.-like comet –sun. Skin of heat capacity PUSSES HEAT.

Charge p (g)-n (g) is puzzle. (Equilibrium forces appear when angular shift of nucleons possible, line of forces out from vicinity) resultant effective balance force may be E-G.

Conical- cavity conical shapes.

MERCURY BRAKE ROUL (G), atomic number permits that.


Rest may be link to biospheres.
I can not able to think more than (P-N-G).

It’s my view.

We are preceding no reference stage.

can able to put real hydrogen ?
TRoc
migre,


Is the set {0,1,2,3,4,5,6,7,8,9} used in our common system of math the same system that nature uses? ("the common set")

To derive this set, a "plus one" operation is used. This is the most thorough system (covering all numbers), but sacrifices speed and Harmony. If the the function of an entity (wave, particle, energy, or mass) has the operating "code" built in, would "plus one" be used by the fastest thing in the Universe? It is my belief that it would not.

Simplified, the set could be {0,1..} N = n+1

The concepts of vibration, duality, conservation, multiplication, division, to name a few, are invalid with a quantity of zero.

So, while I'm not suggesting that we toss out the common set, I am wondering if there are other naturally produced (or self replicating) sets of quantities that could be used to describe "the way things work." (physics)

There is at least one that is well known...

Any takers?


T.Roc
paresh dave
QUOTE (TRoc+Oct 6 2005, 05:55 AM)
migre,


Is the set {0,1,2,3,4,5,6,7,8,9} used in our common system of math the same system that nature uses? ("the common set")

To derive this set, a "plus one" operation is used. This is the most thorough system (covering all numbers), but sacrifices speed and Harmony. If the the function of an entity (wave, particle, energy, or mass) has the operating "code" built in, would "plus one" be used by the fastest thing in the Universe? It is my belief that it would not.

Simplified, the set could be {0,1..} N = n+1

The concepts of vibration, duality, conservation, multiplication, division, to name a few, are invalid with a quantity of zero.

So, while I'm not suggesting that we toss out the common set, I am wondering if there are other naturally produced (or self replicating) sets of quantities that could be used to describe "the way things work." (physics)

There is at least one that is well known...

Any takers?


T.Roc

TRoc

Yes,
So many diversified replied is necessary.
We all need separate thoughts.
Way may be one of that or combination of.
TRoc
paresh dave,


Can we have Zero thoughts?

No, the best that we can do is detach from the flow, and watch them go by.

I do agree that we need a set that will produce ALL numbers though, for diversity among other reasons.



T.Roc


Layman_Steve
Hi Troc,

Isn't that set {0,1,2,3,4,5,6,7,8,9...} the set of naturals?
Don't we also need all reals, including irrationals (Pi, Phi, sqrt(2), etc)?
I am reading about some of the "physics magic" behind complex numbers (based on sqrt(-1) or "i")?

Did I miss the point?

I found it very interresting to learn that complex numbers (i + ib) while "discovered" almost 500 years ago have only recently "shown up" in nature.

Can anyone out there explain to me in simple terms, very simple terms, how dimensions are "represented" in the mathematics of cosmology and/or string theory?

Thanks
TRoc
Layman_Steve,


Yes, common set = natural set, and yes, we do need irrationals. That is my point in questioning the natural set as the basis for formulating systems of Quantity. The natural set uses "plus one" to derive itself. You cant get to Pi, phi, e, etc. from there.

This thread is a "sideline" for another thread (THEORY OF EVERYTHING BEGUN FROM ABSOLUTE CONCEPT., 'Complete' T.O.E. construction project.. We are talking about a beginning concept, from which all others can evolve. (read the first post above)

As for your question "Can anyone out there explain to me in simple terms, very simple terms, how dimensions are "represented" in the mathematics of cosmology and/or string theory?", please go here String Theory to ask that. (thanks!)


T.Roc






paresh dave
QUOTE (TRoc+Oct 6 2005, 11:56 PM)
paresh dave,


Can we have Zero thoughts?

No, the best that we can do is detach from the flow, and watch them go by.

I do agree that we need a set that will produce ALL numbers though, for diversity among other reasons.



T.Roc

TRoc,

may be set p+,p-(n),g+,g- like that.
TRoc
paresh dave,

Very nice. How will you relate the p and g dualities to n?

Can n have its' own partner too?



T.Roc



paresh dave
TRoc

Outer Conical Charge n (g) - Charge p (g) cavity conical shapes. (at Centre locketed, thermal skin here)

G equilibrium not attracts not repulsing within vicinity.

e-g:g core e is n
n-g:g outer n is e
p-g:g core

I think.

Zapper
To back up Troc in terms of 'natural' and 'common sets', I believe that the 'common set' [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] are based on a decimal system. That decimal system is man-made. A natural system could be derieved by observing nature itself without a biased number set perspective. For example the Fibonacci sequence could defined as a 'natural set'.
Instead of seeing nature in the 'common set' it could be more clearly defined and harmonious in this 'number set' [ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... ] and it would explain alot of natural structures that pass us everyday.
Pi is as important as it defines the geometry of a circle or sphere. Maybe someone else can explain to me how Pi is an important factor in organising the structure of nature.
Phi is, i believe, the 'natural set' that governs the harmony of the nature from small to big. It also defines beuty in objects if structured, according to Phi.

Cheers Zapper
Zapper
The whole concept of Phi requires a concept in which it is derieved from. That is, mathematics.
So, in order to explain something to someone who has no idea about mathematics, i would have to start from scratch and explain the concept of 'one', 'two', 'three'.
In reality we all communicate through concepts - thats how we relate to each other. Concepts give us a 'base' to work from.
TRoc
All,


A quote from Zapper:

"Another relationship found with nature that links it to the 'Golden Section' is the Fibonacci sequence. Its sequence is:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610....
This sequence is unique as it starts off with 1, adds 1 to itself to make 2.
2 adds the previous number which is one to itself to make 3.
3 adds 2, which is the previous number to make 5 and so on.
This number pattern gradually moves towards the 'Golden section' through the ratio between one number and the on either side to it. For example 144/89 = 1.6179775280898876404494382022472..., which is close to 1.6180339887498948482045868343656... "

This is the "well known" answer for a different "natural" set.

It self replicates, and doesn't include zero.
It it based on an irrational number.
It "grows" faster than "plus one".

BUT...

It doesn't cover all Quantities (even in approximation).
It doesn't include resonances.
It doesn't produce fundamental constants.

Actually, it can produce one that I know of. In a circle, the phi proportion of the circumference, in degrees, is 222.5; leaving an approximation of the fine structure constant remaining (1/137.5).


more later...


T.Roc





Zapper
T.roc

Can you explain how 222.5 is an approximation of the fine structure constant?
TRoc
Zapper,

??

360 x .618033_.. = 222.49_..

360 - 222.5 = 137.5

1 / 137 is the "approximation" of .00729_..


better?


T.Roc


Good Elf
Hi TRoc,

QUOTE (TRoc Posted on Oct 9 2005+ 06:47 AM)
Actually, it can produce one that I know of. In a circle, the phi proportion of the circumference, in degrees, is 222.5; leaving an approximation of the fine structure constant remaining (1/137.5).

Are you sure?... degrees are such an arbitrary unit to relate to such a "pure" number in the Universe. Why would "God" choose 360 degrees in a circle over some other number? The "ratio" might have some deeper purpose and that would be important.

For interest here is a "goldmine" of "fine structure constant" derivations ... you never know one may be a "natural".
Fine Structure Constant Collection
I kinda like I. Gorelik's solution because it ties into what we were discussing elsewhere.
QUOTE (Structure Constant Collection+)
n+s=1/a
n=137
s(n+s)=p^2/2
n - number of rotations in zt per one rotation in xy,
s - shift per one rotation.
Solution for this system gives:
1/a=137.036010988...
Error: 0.000000157


Cheers
TRoc
GE,


You asked "Are you sure?... degrees are such an arbitrary unit to relate to such a "pure" number in the Universe."

My answer: I'm only as sure as the accuracy of the equation above. (not very)

All I'm saying is that it CAN be approximated by using phi. I am questioning the concept that nature uses quantities that are EXACTLY represented by our numeric symbols. Man's 1 = God's 1 ?? This leads to the question: Are approximations good enough? My answer: most likely.

We all know Pi's value. It is in practical use now, and for a thousand years (or so) prior. We still don't have the EXACT answer; but it WORKS in approximating a circle when you know the radius.

We could use 355/113 to APPROXIMATE Pi, but that would be a waste of time, because we already have 3.14_.. which works! (why do the extra calculation?)

The same can be said for the above discussion, we really don't need to divide 137 point anything by 1 to get the fine structure constant, because our predecessors have defined it for us, and with an accuracy that is already bordering anal.

Let us move FORWARD in our knowledge (not in loops). I showed you an approximation for energy in electron volts. It works EASIER (with less computations) than the system shown to me by "the book." I came up with it by trial and error, not by, for example, the sq.root of 1.96468_..e-3. That type of procedure makes "the system" MORE complex; something I believe we should stay away from. (given the current state of UNDERSTANDING)

If energy transitions take place in INTEGERS, then we are allowed to "round off". One electron volt is going to do something predictable; and it won't significantly change "the system" if 1.0935_.. electron volts are used instead.

Just some ramblings on 360: It is the best number to use because it is divisible by the most integer quantities, AND cover the IMPORTANT angles of 90/270 (x axis), 0/180 (y axis), and the "45's" (to allow sine curve-like tapering off of interactions).

If we divide a circle (or a forward moving 2 phase virtual spiral) by 12 equal parts, and analyze the "colors" of the parts (frequencies), is it COINCIDENCE that there is a "separation" at 138.4165_.. degrees that divides yellow and green? Fine structure does not apply here, it can only be used to define values between cyan and yellow. This is completely analogous to there being NO harmonic between B and C in sound. This means that, from 0 degrees, values less than 138.4 (ie. 137.0360..) can be viewed as dualistic. From 180 degrees, in the same direction, you can make another separation. What you end up with is "red" and "yellow" frequencies as being a duality, and "violet" and "cyan" being another. At low frequencies, you see the same lines of separation represented by the void between E & F, and B & C. In electromagnetism, the 90 degree EMF demonstrates the same principal. Now our wave mechanic has the right tool in his box.

The virtual spiral is CONTINUOUS, and the coupled dualistic phases are QUANTUM (discreet). This is where we need to go in order to join SR/GR and QM. Gravity is not a "coupled" wave, but still is seen to "propagate" at © because of the relationship between mass and inertia. If you could subtract that relationship, gravity would be seen as "instantaneous".


T.Roc



Zapper
ohmy.gif ohmy.gif

Oh right! I didnt see it that way thanks Troc
TRoc
GE & all,


I apologize for any confusion I have caused, which, if you were following the last 2 paragraphs, surely must have happened. I left out a KEY part of the idea.

You divide the circle into 12 equal parts, and this gives the Quantum.

However, Time is Continuous; in order to make a complete turn (cycle, octave, etc.), you must ADD a step because the "zero degree" line is also zero time. That is reflective of nothing, and we are looking for symmetry.

If we (the 12 yogis) are going to share a pie that is cut into 12 pieces, and we remove them 1 by 1 (take a mental picture of the pie with 1 piece missing), when we are at the last piece, we have ANTI-symmetry. (reverse direction) If the "act of removing" the pieces is called a "step", then we will have to take 13 steps to return symmetry. Consider the mental picture of the 1st piece missing, and its' opposite, all EXCEPT the 1st piece missing.

So, in my previous post, I failed to mention that 138.41615_.. was a "time", or step derivative. At the 5th step, we have removed 5 pieces, and are at "the line" between the 5th and 6th piece. In my color analogy, this is the line between green and yellow. You can not subtract values from green to get yellow, and you cannot add values to yellow to get green. (this is current theory)

1/13 is a "step", 1/137 is a "piece".

I cannot resist mentioning the "esoteric" properties of the numbers 1-3-7

1 = the beginning; the tonic
3 = the dimensions; the function of the triad
7 = the fundamentals; the basic approach
(13) = the "secret", or hidden approach to circular symmetry

I hope that my idea is more clear to you now.


T.Roc


TRoc
All,


So, now we have two sets of numbers, each produced by an "internal code". Our standard "n+1", and the "golden set" of N=N(-1)+N(-2). (the "-" symbol representing position in the set from N)

Let me introduce another that might meet the requirements. The idea is to produce ALL possible whole integers, have fast "build up", include resonance (probably THE most important single "phenomenon" in Science), not include zero (as a quantity; we can still use exponentials), allow rational numbers, have symmetry, use circular, or spiral geometries (the sine wave/vibration), and if you could produce some constants, that would be a bonus.

The set {1,2,4,8,16,32..} certainly meets the resonance criteria, produces very large numbers quickly (ie. 3x10e8), has no zero, could produce the spiral form BUT, it lacks true symmetry (all Yin, and no Yang), leaves out the irrationals, and (AFAIK) produces no Constants.

We have the Good, The Bad, and the Ugly. Well, ok, we have the Slow (but complete), the Beautiful (but sporadic), and the Fast Harmonic (but selfish).
It's starting to sound like a family reunion! biggrin.gif It's almost as if these sets could produce personalities too. Hmm..

Is there a way to combine these 3 sets? Some set that will satisfy our need to reproduce the reality that we see? A set that would cover such profound and independent concepts such as Harmony, Resonance, Vibration Waves, The Quantum, Electromagnetism, The Speed of Light, Gravity, Attraction/Repulsion, Direction Relative to the Observer, The Spectrum of Visible Light, Thought, Sound, Intuition, Smell, Taste, Conservation, Frequency/Wavelength, Spin, Consciousness Levels, Multiple Wave Relations, Symmetry, Duality, Energy, Etheric "particles", Heat, and the things that can be produced with the above ideas, in case I left them out.

That is quite a tall order, but I believe that is what we are Seeking. An "Omni-band Harmonic Matrix of Resonant Potentials", or something to that affect.

huh.gif

TRoc
paresh dave
Hi RC,thanks.


TRok, ZAPPER,Good Elf ,esin, ZEPHIR, and all.


GREVITATION AND TEMRETURE (HEAT CAPICITY) DIRECT RELATION MAY BE.
Gravitation permits to capture heat within nucleolus, not allowed radiating or conducting, not very with thermoplastic flow as convection of heat (out coming plates cool down but
Drop mass regain heat within magma.)(Earth never going to cool down. captured heat at equilibrium states.)
Each point of material from surface of globe to centre gravitation force and heat capacity both increases.
Gravitational pressure of vicinity mass of any globe of existence (IMPOSE SHRINK EFFECT-EXITED STATE OF MATTER) towards center of body posses and capture allow holding thermal energy (CURRECTION BY EXPANSION-EQUILIBRIUM OF EXITEMENT).
Same atom heat capacity increase refers towards centre. (Increase from surface to centre)

In matter positive raise temperature (above Kelvin) from surface to centre. (Planets)

In anti matter negative raise temperature (below Kelvin) from surface to centre. (Comets)

Why gravitation permits to capture heat within nucleolus, not allowed radiating or conducting, not very with thermoplastic flow as convection of heat.
Sintering process we are aware of.
Answer is when gravitation increases pair nucleons within nucleolus (p-g and n-g)lying d closer, which is exited states, correction done by increase heat capacity, absorb and permitted to posses, so nucleons lying apart again.
This is stable states of existence

Giving example of compression of spring by gravitation, decompress by temperature.
Each globe of universe subject to combine gravitation/temperature effect. (May be some option here)

Combine effect is pulling the mass is attraction from surface up to any intermediate vicinity of globe.

Combine effect is neither attraction nor repulsion to mass is nature within intermediate vicinity of globe.

Combine effect is below intermediate vicinity of globe up to centre of globe Combine effect is mass subject to up thrust. (Negative gravitation) or repulsive gravitation force observe.
GREVITATION AND TEMRETURE NEED TO COMBINE STUDY.NEVER THEY SEPARATE EXIST.
I think.

Guest_Excal
Hello, Everyone.

I just posted over in the TOE forum, where I've asked a few basic questions. Turns out that the concept of the scalar potential (‘SCALAR’ OMNI-DIRECTIONAL POTENTIAL) is easily identified in the number system by simply considering something that, according to Hestenes, Clifford was the first to recognize the importance of, but originated with Grassmann: Two interpretations of number are possible, the "quantitative" and the "operational." The quantitative interpretation answers "how much" or "how many," but with the operational interpretation, number describes a relation between quantities.

This has a profound impact on things when we define a scalar number this way. For example, we can take the mysterious number 1 (which is what we are starting the TOE off with, when you think about it), and by operationally defining it, we can do away with zero, and form, from the natural numbers, the rational numbers. So, 1/1, a rational number, formed from the first and most important natural number, becomes the perfectly symmetrical scalar of the integer number system, though an operational interpretation.

Here's how: n/n = n+1/n+1 = n + ...n/n+...n, which is scaling in its truest and finest form. But by spontaneously breaking the symmetry to m/n, or n/m, our operationally defined scalar becomes the signed integer system, without zero! That is,

...n/m, n/n, m/n = ...-1, 0, 1...!

Thus, in one simple step, by changing our view of what a number is, we've generalized the natural number system (counting numbers) to include "direction," or a polarity.

The next step was also shown by Clifford. By expanding the dimensions of the signed integers, operationally interpreted, we can generate the reals, the complexes, the quaternions, and the octonions, which generalizes numbers to include "direction" and three dimensions + the scalar. Taking the dimensions to higher numbers only causes a repetition of the original pattern (Bott's periodicity), which should tell us something about the meaning of higher dimenisons.

To see how this works, just look at the first four rows of the Pascal triangle.

More later...

Excal






Nick
QUOTE (TRoc+Oct 6 2005, 05:55 AM)
migre,
Is the set {0,1,2,3,4,5,6,7,8,9} used in our common system of math the same system that nature uses? ("the common set")

To derive this set, a "plus one" operation is used.  This is the most thorough system (covering all numbers), but sacrifices speed and Harmony.  If the the function of an entity (wave, particle, energy, or mass) has the operating "code" built in, would "plus one" be used by the fastest thing in the Universe?  It is my belief that it would not.

Simplified, the set could be {0,1..}  N = n+1

The concepts of vibration, duality, conservation, multiplication, division, to name a few, are invalid with a quantity of zero.
T.Roc

Rock Instead of n+1 in a decimal or some other finite base why not an infinite base?

This is as close as I can get to your thought. Would God need to use a finite base in math? I think not.
TRoc

Excal,

Thank you for the differentiation. I had failed to define the opposing (dualistic) concepts of operational and quantitative. My posts in the TOE project took much longer because of that. The key difference between what I seek, and what you have posted, is that I would like to traverse the line between positive and negative without generating the problematic Zero. {-1,0,1..} This is done with the first set mentioned, using Phi/phi as the operator. {-1.618,-.618,.618,1.618..}

Nick,

No, I do not believe God (or the beginning cause) would use the finite base. Indeed, I do not believe there would be "math" at all; that would be our after the fact interpretation of the duality (quantitative vs. operational). If you have followed my TOE posts, you will see my view on a concept that is "one" being interpreted as dualistic (two) only after some "movement" has taken place (position, reflection, observation, etc.).


ALL,


I believe that Wave Theory is the answer that we seek, and that our current Wave Theory is focused entirely on one half of the duality. Namely the operational side, or more relevantly, the form of the wave. From Taylor and Euler, to Schroedinger and Einstein, our focus has been on the form of a vibrating string (including curvature of "space-time"). It is not surprising that many great minds of today are looking for "strings" in space, and still viewing empty space as being endowed with the potential for "curvature". In Socratic form, I ask "what of the flute?" The manifestation that we call "sound" is not created by the deforming of the instrument, yet the flute and violin can be "coupled" in coherent vibration, from simple to complex.

My Theory will focus on the other side of the duality, the relationship of the vibration after its formation. My "set" will reflect this, and produce consistent numeric answers to the combination of 3 or more vibrations. If you think deeply, you will see that we have only studied 1 at a time, or the relationship between 2 vibrations (that are often couched in duality such as cold/hot, +/-) I will not change any of these understandings, only go to a more fundamental level that will better explain the known, and allow us to enter a new era of understanding the complex.

I will post this set in about a week from now.


from Cusco,
TRoc



Zephir
Try to have look to the Aether wave theory. The spacetime changes are forming the massive environment for the subsequent dimension convolution / vibration level according the geometrodynamic theory.
jal
An up and coming junction-
QUOTE
TRUE VOID ‘PRIMARY VACUUM’ BULK. This absolute frame ‘true bulk’ is thus the INFINITE AND UNBOUNDED ‘FLAT’ UNIVERSAL SOURCE AND SUBSTANCE OF BALANCED-ENERGY-LOCATION ‘absolute points’. Each and every ‘absolute point’ LOCATION within this true-bulk ‘primary vacuum energy’ matrix is naturally and logically ‘connected’ to/from every other such location via a....

JUNCTION
#1.
We should NOT eliminate the possibility that the total "bulk" did not change/evolve to become our universe. It could still be there. We could be an embedded/sprouted universe in this "bulk" that we perceive/observe to have operating instructions/rules/laws/structures/particles which could be different then the in the "bulk".
#2.
The words that we use will influence the development of our logic. I hate the word "sphere" because it automatically includes 3d.
We should eliminate the possibility that our universe could have have gone through a 2d stage or even that "a 2d spot" can be oriented/folded/rotated/spin which would make us perceive 3d/etc.
I know that I could have used a diferent word. (string/branes/membranes/wave)
jal
TRoc
All,


Some introductory thoughts:

From 0 to 1
How can something arise from nothing? I believe it to be impossible.

From 1 to 2
How is this primary resonance (& duality) performed? (a "double" value) Keep in mind that whatever function you do must be repeatable, and continue to produce harmonic values. It cannot be "plus one", as I have shown above. It cannot be "multiplied by itself", or we would never get off the ground.

Restating some requisites: Produce the digits 1-9, so our system of math is the same. Produce irrational numbers (nature doesn't seem to mind "rounding"). Use resonance and harmonics as primary functions. (also produces large numbers in the fastest way) As simple as possible (dualistic, symmetrical, rotational). Not allow, by any function, to arrive at "nothing" (zero). Perhaps most importantly, be reflective of the world we see, hear, and feel (other senses too, but these have the most empirical data accumulated; ie. "light, sound, and temperature"), even as far as outlining consciousness and communication of thought. If your brow is not "crunched up" right now, you need to re-read! There is no way to fully understand what I have just said without building the "Omni-band Harmonic Matrix of Resonant Potentials" for yourself (on MS Excel, for example) and PRINTING it (& then pasting the pages together). It is too big to appreciate on your monitor, and I will not attempt to do it for you. It will be available "at fine bookstores near you" in the next few weeks, at a price that should be less than your "wage" investment of the 1 hour or so needed to make your own copy.

(here: to 5 decimal points for brevity; for the complete story: allow for 15 per column, 13 columns, and start with 1.0267093)


{1.02670, 1.08776, 1.15244, 1.22097, 1.29357, 1.37049, 1.45198, 1.53832, 1.62979, 1.72671, 1.82938, 1.93816, 2.05341}

The "operator" is the 12th square root of 2. (1.059463094355929526456182529494)

Subsequent "rows" in the "matrix" are harmonics of this row; down from this row, all values are x2 ; up from this row, all values are /2 .

The CENTER of the matrix is vertically between column 7 and 8, and horizontally between row 14 and 15. (with the above "set" being row 1)

© 2004 Thomas Roccetta


enjoy!

TRoc

TRoc
To your questions:


Do not start this on your line (row) 1 in Excel. Start at around line 50 (or 100 if you want a larger matrix).

I do not know if the up/down direction is infinite; I have a hunch that the numbers will "merge", or "reconnect" at some magnitude, as they do from right to left.

You will need to go at least 35 row down from line 14, and 35 rows up from line 15 in order to "see the light" (in color!).

The most important constant produced is the speed of light. Rotational-symmetric movement products from the CENTER produce ©. (IE. up 1 and right 1 x down 1 and left 1; or up 35 and right 5 x down 35 and left 5)


TRoc


Confused2
I hope a newbie contribution is OK here..

My thought is that a 'proper' universe would have collapsed back into itself very rapidly. The result being that we have the 'wrong' constants which (fortunately) don't work properly. We may be looking for elegance in what is essentially a foul-up. A very nice foul-up so I'm not complaining.
-c2
jal
I'm trying to learn how to post a picture.
user posted image
jal
Can someone send me the instructions on how to post pictures, by e-mail?
jal
Guest_jal
Without any comments... I do believe that I got it!!! biggrin.gif
user posted image
jal
fivedoughnut
To start a G.U.T/T.O.E from scratch we really ought to discover inter-relationships between pi, phi and all other basic ratio's/constants etc, in context with the observable "reality"in which we exist.

I've a G.U.T feeling the answer is "staring us in the face"

However, because we as individuals are unfortunately limited to certain aspects of the"obvious" this neat idea of pooling resources together is great!

I wish I could help with the math.....fivedoughnut's skullnumbingly "blind" in this area, although if it's anything to do with "common sense" assumptions based upon inter-linking measureable phenomina... I'll be happy to assist.
Layman Steve
QUOTE (fivedoughnut+Dec 9 2005, 01:45 PM)
To start a G.U.T/T.O.E from scratch we really ought to discover inter-relationships between pi, phi and all other basic ratio's/constants etc, in context with the observable "reality"in which we exist.

I've a G.U.T feeling the answer is "staring us in the face"

I think you're on to something. smile.gif

I hope and believe a TOE will come back to this. There is something special about irrational numbers. Numbers have such a wonderful way of describing nature. Irrational numbers shouldn't be left out.

Imaginary numbers have shown themselves to have a profound place in nature. So to, will irrational numbers - I mean to imply much more than the ratio of a circumference to a diameter and such. Of course. I can't prove anything - too weak with the hard math...

I have other outrageous instincts about the cosmos. I'm not a scientist, but I appreciate what they've done and how they've done it.

<OT>
Have you ever calculated Pi in base2 (binary)? You stretch out the digits, but have fewer symbos to work with (0, 1).
</OT>

PS: I still check in once and a while - few posts obviously. Good job RC. I'm sure this can't be easy.
philip347
The theory of everything, will not surfice, as it is given.

This has to do with the passage of time and how time relates to mass as an active quantity, dealing with how mass time is realized in situ.

X Saying for today:I don't like humans.They are over controlled robots, that jump to other's demands and influences.

I don't necessarily care for you, who you are, nor what you represent.

Your first mistake, is in the thinking that all other beings off the surface of this planet, either like Earth humans, or deems them important to talk to.
jal
SYMMETRY
someone from an other thread wink.gif
QUOTE
Err~ to put that last statement more clearly, its like all these different scientific fields each have a corner of a Rather large jigsaw Puzzle put together Err~ perhaps lacking a few interlocking pieces and all that is needed is the amalgamation of these corners via these crucial missing pieces so that the bigger picture really stands out to every one who chooses to drool over it.


I would like to see a review on this page of what people understand of symmetry.
I do not mean all the different kinds of symmetry just/only "symmetry."
Most of the confusion would be resolved if we really understood and agreed on what is symmetry.
See this page,
symmetry
QUOTE (->
QUOTE
Err~ to put that last statement more clearly, its like all these different scientific fields each have a corner of a Rather large jigsaw Puzzle put together Err~ perhaps lacking a few interlocking pieces and all that is needed is the amalgamation of these corners via these crucial missing pieces so that the bigger picture really stands out to every one who chooses to drool over it.


I would like to see a review on this page of what people understand of symmetry.
I do not mean all the different kinds of symmetry just/only "symmetry."
Most of the confusion would be resolved if we really understood and agreed on what is symmetry.
See this page,
symmetry
Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations.

Let's find the hidden presumptions that are affecting out thinking.
Once an agreement is agreed, then there are few interesting questions.
1. What is the real meaning of " operations"?
2. What is really happening when the process is happening?

I seriously believe that we are not all on the same page.
Let's do a SYMMETRY 101.
JAL
I will be absent next week, to take care of bread and butter issues.
Go ahead.... start without me... cool.gif
TRoc
Everyone,


OK. If I may add GEOMETRY (based on the mathematics involved) to this thread, and use jal's last comment as the segway, let's continue this discussion.


What are the FORMS of symmetry, and broken symmetry?

What quantities do they embody?

How does RESONANCE fit into these ideas?


Just to get it rolling again.

T.Roc

jal
Hi TRoc!
I must say that from reading your posting, that your perceptions are different. As a result, I think that your are a good one to explain that perhaps, I repeat, perhaps, the information that we receive and send out to our environment may be blinding us to the "reality" of what is out there. It is causing us to make wrong presumptions and assumptions.
QUOTE
Let's find the hidden presumptions that are affecting out thinking.

In other postings I have alluded to the fact that Mrs. Susy, Mr. Slim and Mr. Slinky are up against the wall and cannot find out why particles have mass.
I think that the only way out is first , to investigate how/if there is a new way of looking at the problems. Not with different theories but by examining the way that our mind is wired.
As a re-enforcing point, our senses tell us... the earth is flat.... the sun is going around the earth...
However, our calculating/logical mind has figured out that those observations are false.
To me, that would be the first step to investigate. (We might not find anything)
smile.gif
jal
TRoc
Hi jal,


Yes, our perceptions need to be included, and by de facto, are. All "observations" are filtered through our "senses" in some way or another. All of modern Science began with explanation attempts to the colors emitted from heated, or energized bodies. This was the foundation for spectral analysis, etc. and is still used.

Technology (even "advanced") can mislead us. My "old school" 8' carpenters level says I am on flat ground, as does my "newfangled" laser level; yet both are wrong in the big picture because I am indeed on a sphere. The speedometer says I have zero velocity, yet I know that can never actually happen. Measurements are too linear to capture reality from any perspective, only the one we were just in. (point A to B, or time A to B )


T.Roc

jal
TRoc....
you are definitely to the point... smile.gif this time smile.gif
Would it be helpful/fruitfull to look for or find a way to get to a theoretical ZERO...Would it help us or reveal something new about symmetry?
jal
TRoc
jal,


Yes, and no. Getting down to the smallest size/scale possible, yes. Reducing to the quantity of zero, no. Zero is a quantity that stems from labels: when we "name" something, and separate it from everything else, we enable the concept of zero. However, since everything is interconnected, and built from the same basic elements, in the same pattern, zero is not a real quantity. Where one thing "ends", and another one "begins" is filled with space that is still defined by the terms of the 2 bodies. "Full time" (no limit to distance, or lessening by time alone) communication (ie. gravity), and conservation of mass, energy, etc., demands this.

Example: the definition of "skin" ends where those parameters stop, but skin is a part of "me", and "I" don't start, or stop there. My body heat is radiating out in infra-red quantities, and communicating past the traditional "end" of my body. Also note (pun rolleyes.gif ), if I am making sound, that will travel past the "zero point" of my skin (body end), as will "photons" reflecting off of me and DEFINING me to someone else's view. Now the "me" is in someone else's head! (yikes) huh.gif

The location "HERE" is as temporal as the time "NOW", or the action "TO BE".

However, BE HERE NOW is still the best instructions for happiness. biggrin.gif


T.Roc
TRoc
All,


“re-routed” questions from other threads:

ktwong (from “Quantum Unreality becomes Quantum Reality”)

“So basically a Wave man with a twist in the plot to duality. So everything starts with wave... How does the plot end with particle duality?

Where's the maths for this Music Theory Guide to the Galaxy. Is it a variant to Schrodinger's. Pray tell.”


Density is the answer to “wave particle” duality. First, the term “duality” is a misnomer for something that is always EITHER/OR, but never BOTH. A simple example is a coin: when measured (flipped) it can be either heads or tails. However, when not being measured, and just being “observed” by consciousness, it is BOTH. The experiments & equations that work for the “particle” definition NEVER are solved for the wave term, and vice versa.

Waves are the true nature; when they are interacting in large enough numbers in a given area, that area is dense with vibration. That is measured as “mass”; it has intrinsic resistance to the velocity of ©, due to broken symmetry. Waves communicate with Beat Frequency (BF), the BF is, indeed, another vibration itself. Only frequencies within ~ 1.5 “octaves” of quantity to the resonant frequency of the “body” being measured will act upon it. The background radiation of the Universe is RESONANT with that of the electron, to the degree that, the interaction BF is CONTINUOUS (steady, stable “particle”).

Other than that, the different perspectives are PRODUCED by the method, not intrinsic to it. If we determine a “wave” perspective” to the photo electric effect, and QM basis, we can do away with the “particle” approach, making the demarcation of wave to particle line where mass begins to be measured in situ (not “rest mass”).


jjac (from “ENERGY/MATTER/MASS Discussions”)
“I must ask, do you see that the SGS phases or states between the plane and the vertical to the plane correspond with all possible frequencies and wavelengths. Sort of like each scale (size)of the system, from the smallest possible to the largest, is a set that corresponds with an octave?”

Absolutely! The pure, natural mathematics of resonant quantities produces “octaves” of quantities, that will line up with your geometrical model. The center lines (vertical and horizontal; x,y axis) of this “matrix”, as I have been calling it, twist in the same way as in SGS. This produces the “inverse” relationship between frequency and wavelength, and produces problems when viewed (or computed) with traditional “plus one” mathematics. From the center (x,y = 0), the product of any two rotationally symmetrical quantities = ~299,792,457.8 . You have to love that! Not ad hoc; naturally, logically PRODUCED.

This same system produces the visible spectrum of colors, as agreed upon by “us”, as well as the A-440 tuning of sound EXACTLY. It puts “red & yellow” and “cyan & violet” together, as they are experimentally shown to be, in dualistic sub-groups. Green and magenta lie on “hinge points” near the y axis “twist”. This also geometrically/numerically explains the lack of “half step” between B & C, and E & F in the musical scales. (where there is no gradient of color change; magenta & green)

Historically, the “octave” has been based on 8 steps of 7 points. The problem was, the steps were not equal. With 13 steps of 12 points, you get equal steps ie. DISCREET, QUANTA. Everybody in the “physics book” has tried this approach at one time or another, but failed to completely solve it. Balmer and Schrodinger’s integer set “n=1,2,3,4..” does not mimic the set that produces resonance. It will approximate it though (especially in limited, or simple cases). Planck’s quanta is not scalable. These things are resolved with my system.

Where your SGS ties in to this most specifically is this: the math of resonance lies between 1 and 2. The math of resonant interaction is at 3; the triad or chord, and the triangle. The math (most simple) of the right angle triangle lie at 3,4,5. This is in Pythagoras’ ratio. In spherical geometry, the triangle shifts to 4,5,6, which is the ratio of the triad. The sequence that produces (most simply) 1, 1.25, 1.5 is 12 equal steps between 1 and 2 (12th root of 2 = 1.05946..). Those are the 3 necessary quantities (in frequency or wavelength) that will produce the chord (major). Very logical, efficient, and simple use of "quantity".

Geometrically speaking, the 3 lines come together at 45 deg. from 2 right angled legs. (90 deg & 45 deg harmonic relationship)

Resonantly speaking, the 3 values come together to produce a sum (Beat Sum, BS) of the BF’s that is equal to, or resonant (simple) with the tonic, or starting point. The dominant (1.5) and the tonic (1) mediated by the mediant (it’s really called that) of 1.25, joining the harmonic relationship with a “45 deg” line.

If everything has geometry, and everything has a frequency, then the underlying, fundamental nature common to both of them, can be applied to EVERYTHING.


T.Roc

jal
Hi!
Are we ready to leave the following stages? smile.gif
perception
QUOTE
Many cognitive psychologists hold that, as we move about in the world, we create a model of how the world works. That is, we sense the objective world, but our sensations map to percepts, and these percepts are provisional, in the same sense that scientific hypotheses are provisional (cf. in the scientific method). As we acquire new information, our percepts shift. Abraham Pais' biography refers to the 'esemplastic' nature of imagination. In the case of visual perception, some people can actually see the percept shift in their mind's eye. Others who are not picture thinkers, may not necessarily perceive the 'shape-shifting' as their world changes. The 'esemplastic' nature has been shown by experiment: an ambiguous image has multiple interpretations on the perceptual level.
Just as one object can give rise to multiple percepts, so an object may fail to give rise to any percept at all: if the percept has no grounding in a person's experience, the person may literally not perceive it.


presumption
QUOTE (->
QUOTE
Many cognitive psychologists hold that, as we move about in the world, we create a model of how the world works. That is, we sense the objective world, but our sensations map to percepts, and these percepts are provisional, in the same sense that scientific hypotheses are provisional (cf. in the scientific method). As we acquire new information, our percepts shift. Abraham Pais' biography refers to the 'esemplastic' nature of imagination. In the case of visual perception, some people can actually see the percept shift in their mind's eye. Others who are not picture thinkers, may not necessarily perceive the 'shape-shifting' as their world changes. The 'esemplastic' nature has been shown by experiment: an ambiguous image has multiple interpretations on the perceptual level.
Just as one object can give rise to multiple percepts, so an object may fail to give rise to any percept at all: if the percept has no grounding in a person's experience, the person may literally not perceive it.


presumption
4  given, presumption, precondition
an assumption that is taken for granted

assumption
QUOTE
The act of taking to or upon oneself: assumption of an obligation.
The act of taking possession or asserting a claim: assumption of command.
The act of taking for granted: assumption of a false theory.
Something taken for granted or accepted as true without proof; a supposition: a valid assumption.


So far, we have not probed very deep to find the hidden presumptions that are affecting out thinking about symmetry. smile.gif
Nevertheless your post should/could become relevant at a later stage.
jal
TRoc
jal,


I'm all "ears". wink.gif

(ok, some eyes too)


What are you mathematical insights regarding symmetry, and how it will help us develop a "TOE" from tabula rasa?


T.Roc

jal
TRoc....
You want me to reveal all... biggrin.gif
It's actually very small... it's
QUOTE
Just as one object can give rise to multiple percepts, so an object may fail to give rise to any percept at all: if the percept has no grounding in a person's experience, the person may literally not perceive it.
but it has a lot of meaning.
What can be added to this is that:
The stronger you belief, the stronger will be your objection to accepting an idea that could cause a domino effect which would bring you house of belief tumbling down.
That includes everyone, including me.
If there are assumptions clouding our perceptions.... it will be a very hard nut to crack....
All of this preliminary thinking is needed before we can go forward. Since I'm not into cognitive psychology.... maybe.... there is a "helper reader" that understand mathematic who can get help for this thread.
The examples that we used were excellent in illustrating the potential problem.
Does the problem exist????
Can we make any progress in identifying it?
It's a lot to think about.
jal

Yes, I have some ideas concerning the process of symmetry. smile.gif

TRoc
jal,


Agreed on the dubious nature of individual perceptions, and "group think" pressures most of the time controlling the boundaries that individuals are willing to have, or break through.

As has been mentioned in the TOE thread, some conclusions must be held to exist a priori, just so we (as a group) don't get bogged down in an overly philosophical, endless debate.

Do we have a problem? I think so, and many others must, or we would not be having this conversation.

Where do we start? Just about anywhere seems good right now! (let's just start!)

All of "the basics", like mass, energy, velocity, frequency, charge, are already defined & accepted by most. That doesn't mean that they could not change, but that is a lot of momentum to slow down, stop, and turn around.

My own personal strategy is to start from the beginning, even conceptually, and work forward. Using empirical data, but not always "just accepting" things because that is the historical way it has been. Too many things have been proved entirely incorrect; even Nobel prizes given for ideas like the lobotomy as a useful psychological tool. The "hero's and saints", as they have become to many, are also not infallible. Even when you definitely agree with a persons "theory A", shouldn't mean that every thing that came from their mouth is above questioning. This applies to the Newtons, Planks, and Einsteins of the world.


T.Roc

jal
Hi TRoc....
We could always take a step back and re-start if we headed down the wrong path because of lack of information.
As I said, "The stronger your belief, the stronger will be your objection to accepting an idea that could cause a domino effect which would bring you house of belief tumbling down. "
Therefore, an "idea" that can cause a house of belief to come down must be presented and accepted by the non-believer. It must be perceived as being a "better house". Hopefully, an "expert" will add their comments to this. (I lack information)
Therefore, I must make some presumptions...
On Symmetry...
The language of math is very precise, yet...
QUOTE
In other postings I have alluded to the fact that Mrs. Susy, Mr. Slim and Mr. Slinky are up against the wall and cannot find out why particles have mass.
wink.gif
A) They/we are all working from the same measured particle quantities.
B ) There can only be one symmetry.
Therefore, the problem must be in the communication channels . I have seen the same thing on many web pages and even in the discussions on these forums.
People saying, "I see similarities in your work and in my work."
One of the reason that made me stop and get involved with "TOE" was because an attempt was going to be made to "get us all talking the same language".

Yes, I have some ideas concerning the process of symmetry. ( You have read them)
Everyone can look up the different symmetries that have been identified and that are used.
If you were presenting a new idea to me (Because of .... one of the previously stated reasons??) I would not be able to understand it.

So-o-o... do you want to try to proceed?.... with the work from Mrs. Susy, Mr. Slim and Mr. Slinky, or... ktwong?.. jjac? .... good elf?...some of your presentation (which I like by the way smile.gif )
So, How do we move forward?
Maybe, tor, realitycheck, would like to give an input? smile.gif
Like you, I don't want to get bogged down in an overly philosophical, endless debate.

jal



jal
Hi!
While waiting. I would like to pass on the following simple information:
Table of the Laws of Physics and their Sources
Also, there is a power point page which should be helpfull in giving further explanations. smile.gif
by Vic Stenger
jal
RealityCheck
QUOTE (jal+Feb 19 2006, 06:10 AM)
Hi TRoc....
We could always take a step back and re-start if we headed down the wrong path because of lack of information.
As I said, "The stronger your belief, the stronger will be your objection to accepting an idea that could cause a domino effect which would bring you house of belief tumbling down. "
Therefore, an "idea" that can cause a house of belief to come down must be presented and accepted by the non-believer. It must be perceived as being a "better house". Hopefully, an "expert" will add their comments to this. (I lack information)
Therefore, I must make some presumptions...
On Symmetry...
The language of math is very precise, yet...
QUOTE
In other postings I have alluded to the fact that Mrs. Susy, Mr. Slim and Mr. Slinky are up against the wall and cannot find out why particles have mass.
wink.gif
A) They/we are all working from the same measured particle quantities.
B ) There can only be one symmetry.
Therefore, the problem must be in the communication channels . I have seen the same thing on many web pages and even in the discussions on these forums.
People saying, "I see similarities in your work and in my work."
One of the reason that made me stop and get involved with "TOE" was because an attempt was going to be made to "get us all talking the same language".

Yes, I have some ideas concerning the process of symmetry. ( You have read them)
Everyone can look up the different symmetries that have been identified and that are used.
If you were presenting a new idea to me (Because of .... one of the previously stated reasons??) I would not be able to understand it.

So-o-o... do you want to try to proceed?.... with the work from Mrs. Susy, Mr. Slim and Mr. Slinky, or... ktwong?.. jjac? .... good elf?...some of your presentation (which I like by the way smile.gif )
So, How do we move forward?
Maybe, tor, realitycheck, would like to give an input? smile.gif
Like you, I don't want to get bogged down in an overly philosophical, endless debate.

jal


Hi jal, TRoc, and all! I WILL soon contribute here and elsewhere! I have just recovered and finished tying up some loose ends. Thanks for your patience. But GEE, all you guys have been REALLY busy and original...what a LOT I've had to catch up with here, and in the other 'usual' physics forum threads! I'm truly impressed with the depth of work/thinking that has gone on. There's MUCH that will undoubtedly come in handy at the appropriate/relevant stages of the Q&A thread. Really, I'm impressed no end with all of you. It will be a pleasure to once again be back collaborating closely with such minds, in our special project.

RC.
.
jal
Hi!....
I don't want you to think that I'm dribbling out my thought one at a time just to lead you on.
Symmetry is the tool being used to understand the universe and if there was an assumption underlying it (which could be wrong) then you can see that we would/could get wrong conclusions.
I have spent time doing that examination. Here is what I am concluding...
Packing and kissing numbers make symmetry. Symmetry arized from kissing numbers and packing.
I see this as the underlying and assumed principle. I have not found anything linking the two subjects,
I ask for your help in proving (right or wrong relevant or irrelevant). Maybe your seach will be better than mine.
The web is good but it does not have everything.
The mathematicians have probably got tons of papers on it.. (that my friend is a presumption) smile.gif

Is this a good place to start an examination? smile.gif
jal

Excal
The ancient Greeks (especially Euclid) kept numbers separated from magnitudes. Numbers to them were quantities, while magnitudes were geometric measures of lengths, areas, and volumes. Numbers were useful for counting magnitudes, and just as they could be used for counting books, lumber, or people, they could be used to count lengths, areas, and volumes, but they couldn’t be lengths, areas, or volumes.

The history of mathematics in physics is largely the attempt to generalize the number concept enough to merge it with the magnitude concept. The capability to algebraically manipulate geometric magnitudes as easily as numbers is intriguing.

However, while magnitudes and numbers are both quantities, magnitudes have other properties that numbers don't have. The properties of magnitudes are:

1) Quantity

2) Dimension

3) Polarity

Hence, adding dimension and polarity properties to numbers is the long-sought goal, or, to put it another way, if we can find numbers with these properties, then we can explore the magnitudes of geometry algebraically.

There is another property of geometric magnitudes that the Greeks were fond of and that modern man has rediscovered. This is the property of symmetry. The symmetry of nature can be seen everywhere and forms the core of what we consider beauty of form and harmony. Today, it is a guiding principle of mathematics and physics. Therefore, since symmetry is so powerful, let us start with numerical symmetry. This means finding the symmetrical relation of quantities initially, since that’s all we have to work with. The most obvious mathematical operation that will do this is an operation favored by the ancient Greeks, proportion; that is, equal proportions are the ultimate expression of symmetry: this can be expressed as n:n, which is different from the more familiar identity relation, where n=n. We can characterize this difference by noting that the identity relation equates the relative value of quantities, whereas the proportional relation evaluates the relative value of quantities. Thus, n=m is the same as m=n. However, n:m is the inverse of m:n, which is the simplest mathematical expression of the symmetry property obtainable.

Amazingly enough, though, there is one, and only one, case where m:n = n:m. This occurs only when m = n. Obviously, this is the simplest and oldest mathematical relation known to man. The ancients used it in the form of a scale, or a pan balance, to measure the relative proportions of trade goods. When the weight of goods on one side equals the weight of goods on the other side, the scale is balanced. If the weight of one side is more than the weight of the other, the difference is the same regardless of which side of the balance the goods are placed. Thus, we can see how the beautiful principle of symmetry relates the relative values of two quantities and, in effect, measures them.

Now, if the quantities we want to evaluate are constantly changing, then the principle of symmetry evaluates the rate of change, rather than the number or weight of things. On this basis, equal rates of change are balanced. The numerical expression of this is: n:m = m:n = 1/1 = 1, where n and m are the change rates of two, reciprocally, related quantities. In other words, in this case, the number 1, instead of representing a quantity of one, actually represents the equality of the change rates of two dynamic quantities in equilibrium. It is a mathematical expression of the balanced, or symmetrical, condition of a dynamic system.

Now, since we want numbers that can express the quantity, dimensions, and polarity of geometric magnitudes, we should be very impressed with a number capable of expressing a dynamic symmetry, because geometric magnitudes can only be measured dynamically; that is, we have to change something to measure length, area, or volume. For instance, one way to measure length, is to move a measuring device of known length until it is parallel and coincident to the length we want to measure.

So, what can we “move” in our symmetrical, dynamic, number, n:m = m:n = 1/1 = 1, to measure length magnitude? Well, obviously, the answer is either m or n, since these are the only two rates in our number. Ok, then, let's change m. Let's double it. We get:

n:m = 1/2.

If we change n instead, we get:

n:m = 2/1,

but what does this have to do with length magnitude? Answer, everything. Think of n:m = m:n = 1/1 = 1, as a point, a balanced point. Now, the two unbalanced points n:m = 1/2 and n:m = 2/1 are unbalanced in two, opposite, “directions” from n:m = m:n = 1/1 = 1, the balance point. If we plot them on a line, we get

1/2 1/1 2/1,

where the imbalance between 1/2 and 1/1 is one unit on the line, and the imbalance between 1/1 and 2/1 is one unit on the line as well, but on the other side of 1/1. Therefore, what we have here is a numerical expression of a length magnitude; that is, this number has three properties:

1) quantity
2) dimension
3) polarity

The value of the number's quantity property is three. The value of its dimension property is one, and the value of its polarity property is two; that is, the three units of quantity (1/2, 1/1,and 2/1) are two opposed quantities measured from 1/1, like the two opposite ends of a unit length (maybe a stick or a rod), measured from its center. We can express this as a combination of integers as follows:

(1/2 + 1/1 + 2/1) = (-1 + 0 + 1),

but where it is regarded as one composite number, not a total of three separate numbers. In other words, we can think of it, as we think of complex numbers, which were invented using the "imaginary" number i and have the form:

(a + ib),

which is one composite number consisting of two different types of numbers that don’t sum to a total quantity of one type, but express the result of combining two related types of numbers. Thus, we can think of our reciprocal number as a new complex number with the form:

(aL + bM + cR),

where L, M, and R, indicate left, middle, and right respectively. Recall that these complex reciprocal numbers are numbers representing a symmetrical condition. Hence, they are numbers with three properties, only one of which is quantity. There are not just three quantities here. There are two, opposing, quantities the sum of which balance. In this type of number, the symmetrical condition can be either balanced, or unbalanced. If it is unbalanced, it can be unbalanced toward one end or the other, but not both. In the case of (1/2 + 1/1 + 2/1), the number is balanced. Therefore, the imbalance is zero, but not the number itself! We say that the zero sum of its two polarized quantities means that it is in numerical equilibrium, not that it doesn’t exist. Thus, while the integer value of

(-1 +0 + 1),

is 0, the rational sum of

(1/2 + 1/1 + 2/1),

is 4/4 = 1/1 = 1. In other words, this composite number is a one-dimensional, balanced, number, with two opposite polarities with respect to a monopole. Since ancient balances have been replaced by more modern methods of measuring proportions, we haven’t used these types of numbers much in modern times, but in building a TOE, we are looking for numerical symmetry as a starting point and this numerical symmetry has amazing powers.

For instance, we can see how its property of symmetry just keeps on giving in the binomial/trinomial expansion, where the dimension property of a reciprocal number determines the value of the other two properties, its quantity and polarity properties.

For example, recall that the dimension of our reciprocal number above is 1 and the values of its corresponding quantity and polarity properties are 3 and 2, respectively; that is, it has three quantity terms, two of which are polarized with respect to a third, non-polarized, quantity. Thus, it is a complex number composed of two types of numbers. In other words, just as the familiar complex number is a composite of two types of numbers, a real type and an imaginary type, the complex reciprocal number is also composed of two types of numbers, a unipolar type and a bipolar type. Therefore, we can say, in general, that the value of a complex, reciprocal, number consists of the values of its two properties, quantity and polarity, which are determined by the value of its dimension property. In the 1D case, the value of one of these properties, polarity, is 2^1 = 2, and the value of the other, quantity, is 3^1 = 3.

In the case of the ordinary, quantity (scalar), or non-reciprocal, numbers that we are all familiar with, increasing the dimensions of these numbers from 0 to 3, is interpreted as a general change in the type of the number; that is, the type of number goes from real to complex, from complex to quaternion, and from quaternion to octonion, etc. Each type of number has a different set of properties and algebraic rules, called normed division algebra.

However, because the reciprocal number is a numerical expression of symmetry, the result is different. As the dimensions increase, the type of number doesn't change per se, but its two properties, quantity and polarity, change value.

With the non-reciprocal numbers, the invention of the imaginary number compensates for the natural symmetry of the reciprocal number, but in the reciprocal number the three quantities, arising out of the symmetry, is a result of natural reflection. We can place signs on the two opposite quantities and call the unbalanced term on the left negative, and the unbalanced term on the right, positive, and the balanced term in the center neutral, or one bipolar term and one unipolar term. However, scalar numbers, being quantity only, don't have this capability, so the polarity property had to be invented for them.

Thus, the way you get to the opposite quantity with scalars is you just change the sign and say you did it by multiplying it by the square of an imaginary number, i. In this way, you can make two types of numbers (positive and negative) out of one type of number (positive). It seems kind of hokey now, but it has worked for two centuries and today it is regarded as arguably the greatest leap of imagination in the history of mankind. Go figure!

Anyway, once this was done, why stop there? If you think of i^2 as a 180 degree rotation from the positive side of zero to the negative side of zero, then a rotation of i is a 90 degree rotation. So, what happens when you increase the dimension of these numbers? You increase the number of imaginary numbers! In other words, increasing the dimensions of these numbers increases the different types of numbers.

For example, increasing the dimensions of the non-reciprocal number from 0 to 1 increases the quantity of imaginary numbers from 0 to 1, creating a new type of non-reciprocal number with opposite polarity from the one with 0 polarity, but now conveniently considered as possessing positive polarity. These are the familiar complex numbers. They are a composite number with the positive, or real, type of numbers, and the negative, or imaginary, type of number. Incrementing the number of dimensions from 1 to 2 adds two more imaginary types of numbers, to the real type and the first imaginary type. In this manner, one can form complex numbers with three positive (real) and three negative (imaginary) terms.

These numbers are called quaternions. Again, the quaternions have three types of numbers, the real number type, the complex number type, with one imaginary number, and the quaternion type, with two imaginary numbers.

Finally, incrementing from 2 to 3 dimensions brings us to the octonions, but these are regarded, as a combination of two sets of quaternions, since the quaternions have all the imaginary numbers required in a three-dimensional system. This might seem complicated to explain, but we can put it all together in the first four levels of the binomial expansion known as Pascal's triangle:

0 2^0 = 1 = 1 type (1 2^0 (real))
1 2^1 = 11 = 2 types (1 2^0 (real) and 1 2^1 (complex))
2 2^2 = 121 = 3 types (1 2^0 (real), 2 2^1 (complex), 1 2^2 (quaternion))
3 2^3 = 1331 = 4 types (1 2^0 (real), 3 2^1 (complex), 3 2^2 (quaternion), 1 2^3 (octonion))

Now, clearly there is geometric information in these numbers. If you start with the 2^0 positive scalars (reals), you can regard them as geometric points that have no polarity, then comes the 2^1 complexes. Think of these as 1D lines (a line between two points). Next the quaternions are 2D planes (four lines between four points), and then the octonions are cubes (eight lines between eight points) formed from two intersecting planes (quaternions), forming the three, orthogonal, axes of a 3D volume. It's all kind of messy and unsatisfying and mysterious, but perhaps you can see why: the principle of symmetry is missing from this interpretation of numbers. The ad hoc invention of imaginary numbers enabled mathematicians to compensate for the lack of symmetry in their numbers, but, as a result, the union of number and geometric magnitude is incomplete and confused.

Ok, so let me show you the same thing now, but this time in terms of the reciprocal numbers, the numerical expression of the equilibrium stemming from the symmetry of proportions. Remember, these numbers also have two properties, quantity and polarity, the values of which are determined by the dimensional property of the number, a characteristic that emerges from the intrinsic symmetry of the reciprocal number. As the dimensions increase from 0 to 3, the value of the quantity property increases exponentially with base 3, and the value of the polarity property increases exponentially with base 2. (notice that there is 1 quantity associated with every pole of a multipole, including the 1 quantity associated with the monopole, 1/1, term).

0 2^0 = 1 = 1 polarity (balanced polarity), 3^0 = 1 quantity
1 2^1 = 11 = 2 polarities, 3^1 = 3 quantities
2 2^2 = 121 = 4 polarities, 3^2 = 9 quantities
3 2^3 = 1331 = 8 polarities, 3^3 = 27 quantities

Here, we have a binomial/trinomial expansion, as the two properties, quantity, and polarity, expand exponentially. Now, behold the magic of symmetry:

1) Line 0 is a 0D reciprocal number corresponding to a geometric point magnitude, a balanced number equivalent to the magnitude of one point, with no dimensions, and 1, one-quantity, monopole:

RN^0 = (1/1) => 2^0 = 1 = 1 => 3^0 = 1 quantity

2) Line 1 is a 1D reciprocal number corresponding to a geometric line magnitude, a balanced number equivalent to the magnitude of unit length, with one dimension and 1, one-quantity, monopole and 1, two-quantity, dipole:

RN^1 = (1/2 + 1/1 + 2/1) => 2^1 = 11 = 2 => 3^1 = 3 quantities

3) Line 2 is a 2D reciprocal number corresponding to a geometric plane magnitude, a balanced number equivalent to the magnitude of unit area, with two dimensions and 1, one-quantity, monopole, 2, two-quantity, dipoles, and 1, four-quantity, quadrapole:

RN^2 = (1/2 + 1/1 + 2/1)^2 => 2^2 = 121 = 4 => 3^2 = 9 quantities

4) Line 3 is a 3D reciprocal number corresponding to a geometric volume magnitude, a balanced number equivalent to the magnitude of unit volume, with three dimensions and 1, one-quantity, monopole, 3, two-quantity dipoles, 3, four-quantity, quadrapoles, and 1, eight-quantity, octopole:

RN^3 = (1/2 + 1/1 + 2/1)^3 => 2^3 = 1331 = 8 => 3^3 = 27 quantities

Thus, the long, elusive, goal of mathematical physics, to unify number and magnitude, is reached at last through the principle of symmetry. To fully appreciate this will take some time, but let me help you get started:

Recall that the three quantities and two polarities of the 1D reciprocal number completely define a unit line as two opposite numbers, 1/2 and 2/1, equi-distant from the center, 1/1. Now, the 2D reciprocal number must do the same for the unit plane, and the 3D reciprocal number must do it for the unit volume. If they do this, the numbers and geometric magnitudes are equivalent.

1) The RN^2 reciprocal number, corresponding to the plane unit magnitude, has four polarities (2^2 = 4), and nine associated quantities (3^2 = 9), which can be represented as a 3x3 matrix, or combination of nine quantities, each with its corresponding polarity:

|+-|+|++|
| - |0| + |
|--| - |-+|

where ‘+’ is the positive polarity of a dipole, ‘-‘ is the negative polarity of a dipole, ‘++,’ ‘--,‘ ‘+-,’ and ‘-+’ are the four polarities of a quadrapole, and 0 is the balanced, or non-polarity, of a monopole.

2) ) The RN^3 reciprocal number, corresponding to the volume unit magnitude, has eight polarities (2^3 = 8), and 27 associated quantities (3^3 = 27), which can be represented as a 3x3x3 matrix, or combination of 27 quantities, each with its corresponding polarity:

|+-|+|++|
| - |0| + |
|--| - |-+|

|+-|+|++|
| - |0| + |
|--| - |-+|

|+-|+|++|
| - |0| + |
|--| - |-+|

You have to use your imagination here a little, because I’ve separated out the three, orthogonal, dimensions of the 3x3x3 matrix for simplicity, but you should be able to see that the four, quadrapole, poles in any given plane will combine with two, orthogonal, dipole poles to form the eight poles of the octopole:

1) +++
2) ---
3) ++-
4) --+
5) -+-
6) +-+
7) +--
8) -++

The interesting and unusual feature of all these RNs is that they each contain the monopole at the center, which, of course, is the source of their symmetry, and, as such, are indispensable.

There is so much more to say about these numbers, but this is more than enough for now. The thing is, they give our TOE an enormous advantage, something undreamed of in current theories.

Excal
jal
Excal.... ohmy.gif
It seems that you have expanded from your earlier presentation of
QUOTE
Posted: Oct 23 2005, 09:27 AM
rolleyes.gif

Your post is one of the reasons that I wanted to have this discussion on "symmetry 101". I could not find your concept when I investigated symmetry.
Maybe, there should be a new definition of symmetry? smile.gif
I hope that you'll stay around to make everyone understand your points.
Have you developed your thinking to encompass what I was saying,
QUOTE (->
QUOTE
Posted: Oct 23 2005, 09:27 AM
rolleyes.gif

Your post is one of the reasons that I wanted to have this discussion on "symmetry 101". I could not find your concept when I investigated symmetry.
Maybe, there should be a new definition of symmetry? smile.gif
I hope that you'll stay around to make everyone understand your points.
Have you developed your thinking to encompass what I was saying,Packing and kissing numbers make symmetry. Symmetry arises from kissing numbers and packing.
.
Have these two concepts been explored by the mathematicall community?
For clarity, should we be exploring both concept in separate threads? How?
This problem (clarity) will certainly get overwhelming in some of the discussions in the other threads.
I was not attempting to bury your concept. (ditto to previous posters). I would like to explore it with you and others. I also, believe that you feel the same way towards my wanting to explore symmetry 101. smile.gif
jal
StevenA
I had a large post I had to delete because I keep getting reluctant to sidetrack things with too many ideas but here's something that people might find interesting.

Rational geometry/trigonometry (Divine Proportions)
http://web.maths.unsw.edu.au/~norman/papers/Chapter1.pdf
TRoc
jal, Excal, jjac, StevenA..

I had this completed, but required a rewrite after the in depth post by Excal, and the very timely and relevant inclusion of the paper on “rational trigonometry” from Steven. I think that this will be a bridge between the concept that Excal is using, and mine. A sort of “new” kind of symmetry

I think symmetry is a very important part of all this. The kind of symmetry I am introducing is “resonant symmetry.” How is ties to geometry is important too.


A point : = 1

2 points, connected by a line : = 2

This is the end of “simple symmetry". 2 points, 1 line (or step). Any additional points along this line, just continue this line.

3 points not in a line form a triangle. The "line" is important, it represents movement. An equilateral triangle may be viewed as "symmetrical", but not under a "moving" symmetrical relationship.

For a moving symmetrical relationship, the distances traveled need to be "resonantly symmetrical". 1 unit forward, 1.25 units "up", 1.5 units connecting back to the start. The first set of whole numbers to accomplish this are 4,5,6. The ratios of 1/1, 5/4, and 3/2; and the dualistic/symmetric 3/2, 7/4, and 2/1. Continuing with the quantities of 7 (4x1.75) and 8 (4x2), completing the "octave" of quantity.

Having shown that 1 and 2 have "simple" symmetry, and the ratios 3/2 are identical, this leaves the 5/4 and 7/4 relationship ... broken symmetry?

The whole numbers, and the ratios gave our ancestors a hard time with this "symmetry" thing. Much later, others (still using music as their guide) developed the differential & integral ideas, which, not too much longer, was combined with waves. However, this was a complex system, using "infinities". The simple version (which should just keep the original labels of "subtraction & addition", for simplicity) produces results that work for geometry and waves.

Some quotes from the paper subtitled “divine proportions”:

QUOTE
The key concepts of rational trigonometry are simpler, and mathematically more natural, than those of classical trigonometry. Quadrance is easier to work with than distance (as most mathematicians already know) and a spread is more elementary than an angle. The spread between two lines is a dimensionless quantity, and in the rational or decimal number fields takes on values between 0 and 1, with 0 occurring when lines are parallel and 1 occurring when lines are perpendicular. Forty-five degrees becomes a spread of 1/2, while thirty and sixty degrees become respectively spreads of 1/4 and 3/4. What could be simpler than that?

The straightforwardness of rational trigonometry is also evident from the polynomial form of the basic laws, which do not involve any transcendental functions, rely only on arithmetical operations, and are generally quadratic in any one variable. As a consequence, tables of values of trigonometric functions, or modern calculators, are not necessary to do trigonometric calculations. Computations for simple problems can be done by hand, more complicated problems can use computers more efficiently.  With the introduction of rational polar and spherical coordinates in calculus, this simplicity can be put to work in solving a wide variety of sophisticated problems.  Computations of volumes, centroids of mass, moments of inertia and surface areas of spheres, paraboloids and hyperboloids become in many cases more elementary. This simplicity extends to higher dimensional spaces, where the basic algebraic relations reduce the traditional reliance on pictures and argument by analogy with lower dimensions.

Rational trigonometry works over any field. So the difficulties inherent in the decimal
and ‘real number’ fields can be avoided. It is not necessary to have a prior model of
the continuum before one begins geometry. Furthermore many calculations become
much simpler over finite fields, which can be a significant advantage.



So, this moving, or interacting relationship between “distances” can be applied to wavelengths (and, of course, frequencies). The same simple and “fundamental” set rules the game: 1, 1.25, 1.5 or 4,5,6 expressed as their ratios. “Returning to the starting point” is a key part of the Conservation laws, as well as the sum of angles at 180 deg. In spherical models, pressure is lessened by the increasing of radius (and decreasing of frequency & energy); the “final” radius would be the Universe itself. However, as the paper also said, this complexity (spherical) is not required at the basic, fundamental level.

Simple resonance occurs with the ratios of 2:1, 1:1 , and 1:2. The harmonics forward are double the frequency, and half the wavelength; the rearward harmonics are the dualistic opposite.

Complex resonance begins with the triad (or triangle). It lays the groundwork for superposition, fourier analysis, time decay resonances (unstable particles), “inseparable thirds” (quark measurements), classical interpretation of atomic orbits, and more. It puts mathematics into the “ad hoc” systems of color mixing, and music itself.

The triad is formed with the “right angle” ratios: we can use the simplest, {1, 1.25, 1.5} as the prime example. There is not a lot of “physicalness” to vibrations, their interactions create more of the same: vibrations. Here I am using the quantity for frequency. The simple differential, is termed the beat frequency (BF). The BF is a new vibration itself (a distance). It is time and space dependent; if the sources “part ways”, the time measurement ends. If the sources themselves are time dependent, the whole group fades away, via their harmonics; this is the case for sound waves. This is not the case for orbiting bodies, or for fundamental "particles".

The lowest frequency in the “mix” is the tonic, or fundamental. It is the starting point, and with resonant symmetry, is also the ending point. (remember, these are differentials, so the positive / negative quantity is not used)

(1 – 1.25) + (1.25 – 1.5) + (1 – 1.5) = 1

Here, the 3 BF’s sum to recreate the fundamental; there is no loss in the interaction. This can be infinitely recreated by the harmonics of the fundamental ratios.

(2 – 2.5) + (2.5 – 3) + (2 – 3) = 2
(4 – 5) + (5 – 6) + (4 – 6) = 4
etc.

In music, a C chord is made from the triad of notes C + E + G = C or, by the values of the upper harmonics of 4,5,6 . This is the 0 of Excals’ -1, 0,+1 ; the point is preserved, the distance measured remains zero change. Aristotle’s claim that the natural state is at rest. (or in resonance)

The real story begins when this fundamental symmetry is broken.

Only one of two things can then happen: either the fundamental is lowered, or raised in value. This is the birth of the dualistic measurements; this is the -1 & +1 , of Excal. Movement is measured, or a dualistic change takes place; often in terms of Doppler. Mass & energy, positive & negative charge, north & south magnetic flow, frequency & wavelength, particle & anti particle, reflection & refraction… (all are 90 or 180 deg to each other in terms of definition)

Newton’s claim that inertia kept distances being measured consistently, terming the “force” gravity; disregarding that some other “force” must be responsible for moving the body in the first place.

I claim that fundamental force to be broken resonant symmetry: bodies in a resonant state maintain their state, bodies (or waves) is dissonant state are either attracted or repelled via a change in geometrical distance due to conservation of resonant state.

The mediator of this force is the BF; either filling the space between bodies (dissonance = density) causing them to push apart, or by super-positioning waves through resonance, and moving the bodies to one position (together). Velocity alters the time allowed for BF to take place; just the right velocity would cause an equilibrium state: the arc of an orbit. (Kepler) Also the increasing change in measurement itself. (Lorentz, Einstein)


T.Roc

Excal
Hi Everyone,

With jal's comments on symmetry and TRoc's comments on harmony and resonance, we see how quickly these things can get complicated. Just the terminology alone can overwhelm us. I had no idea what "kissing and packing numbers" were, and the fields of music and topology are immense.

I had read and saved the paper on ratios, to which StevenA referred us, months ago. Indeed, my files are just full of interesting and relevant ideas that I have found through the power of the Internet. What the web does for us, that is truly revolutionary, is it gives us access to information that could only be found in university libraries before. However, in a project like this, especially, too much information becomes a liability.

Consequently, we need some guiding principle that is as fundamental as we can find. We also need to have some idea of the general direction of the destination we seek, and then we need to formulate a hypothesis that we can test and follow the conclusions that our hypothesis forces us to follow.

These stringent requirements can only be successfully met every few centuries it seems, by the geniuses that are born, make their contributions, and then pass on. My ideas are based on one such individual named D. B. Larson. His hypothesis was that space and time are the reciprocal aspects of a universal motion, and that it's this motion that produces the universe, and is the source of its characteristic symmetry, harmony and mathematical truth.

I've taken his ideas to heart, trying to understand their implications, and have been simply blown away, in the last 10 years especially, with the power they have. I am convinced that we can build a TOE, from scratch, taking his simple idea as the fundamental assumption and building from there.

However, Larson was interested in the physical concepts, not the mathematical formalisms that have pervaded the world of science since the great intellectual upheaval in geometry revolutionized the field and left the Platonists bewildered. Nevertheless, the formalists are now mired in their formalisms and the misguided physics community, having relegated its responsibility to develop the mathematics of physics to the mathematicians, finds itself confused and in a state of crisis.

Interestingly enough, the amatuer innovators (regarded as "cranks" by the sophisticated, leading, professionals), now having unprecedented access to information, appear to be following the example of innovators in other fields in establishing the way forward through the newly opened frontiers of knowledge.

This is a fundamental social revolution the outcome of which is impossible to predict, but is fascinating to observe. I am aware of only a small part of it, which I will briefly share with you, since it is so appropriate to this project. The intellectual revolution in geometry that I refer to is ably outlined by Eric S. Raymond (see: The Utility of Mathematics).

QUOTE
For two centuries after Newton, phenomenal science aspired to the kind of rigor and purity that seemed to be embodied in mathematics.  The metaphysical situation seemed simple; mathematics embodied perfect a-priori knowledge, those sciences able to most mathematicize themselves were the most successful at phenomenal prediction; perfect knowledge would therefore consist of a mathematical formalism, arrived at by science and embracing all of reality, that would ground a-posteriori empirical understanding in a-priori rational logic.  It was in this spirit that Condorcet dared to imagine describing the entire universe as a mutually-solving set of partial differential equations.

The first cracks in this inspiring picture appeared in the latter half of the 19th century when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries.  Riemann's geometry was modeled on a sphere, Lobachevsky's on a hyperboloid of rotation.

The impact of this discovery has been obscured by later and greater upheavals, but at the time it broke on the intellectual world like a thunderbolt.  For the existence of mutually inconsistent axiom systems for geometry, any of which could be modeled in the phenomenal universe, called the whole relationship between mathematics and physical theory into question.


What we are attempting to do is affected by this as well. If we seek an underlying mathematical truth, such as the one we've been discussing, to found our TOE upon, we are seeking to establish "a-posteriori empirical understanding in a-priori rational logic." The definition of mathematics' role in this TOE is, therefore, in the "Aristotelian tradition of rationalism," and it is consequently subject to the intellectual impact of the possibility of the "existence of mutually inconsistent axiom systems for geometry;" that is, there can be no TOE in any absolute sense, but only in "multiverse" sense, which is the quaqmire of the so-called "anthropic principle" that is currently swallowing up the most prominent of theoretical physicists at an astounding rate.

Fortunately, for us, there is a way out: the simple concept of space and time as nothing more than the reciprocal aspects of motion allows us to reinterpret Euclid's fifth postulate in a way that closes the door opened "when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries."

It does this because, if space and time are nothing more than the reciprocal aspects of motion, then they don't have an independent existence. Space, which we normally think of as a set of points that satisfies the postulates of one of several geometries, becomes merely the space aspect of a past motion that determined the locations of those points. The space doesn't exist after the motion. In order to measure the space, we have to recreate it with motion, because it is motion that exists, not space, just as the existence of a rod creates its two ends. Without the rod, the locations of the ends of the rod are only mathematically expressable, or logically expressable, points in a formalism. Without the rod, or some other physical object to define them, the locations of space are only abstract concepts.

Thus, we see that Euclid's axiom, that only one line exists through a point not on a line parallel to it, in reality, holds true only for one-dimensional motion; that is, it is the motion that defines the line, so if the dimensions of the motion are never more than one, then Euclid's geometry holds, but, if the motion that defines the line ever changes so that there is motion in more than one dimension, then whatever motion is not in one dimension must be in another dimension, and Euclid's geometry doesn't hold.

Therefore, what Riemann and Lobachevsky actually proved was that motion must exist in more than one dimension. Clearly, then, by making a move from a concept of space as a container, or a medium, or a manifold of spacetime points, to a concept of space as the reciprocal aspects of motion, the impact of "inconsistent axioms of geometry" is allieviated, because geometry, as I showed in my previous post above, no longer has to be axiom-based.

Newton understood this anyway, when he observed that

QUOTE (->
QUOTE
For two centuries after Newton, phenomenal science aspired to the kind of rigor and purity that seemed to be embodied in mathematics.  The metaphysical situation seemed simple; mathematics embodied perfect a-priori knowledge, those sciences able to most mathematicize themselves were the most successful at phenomenal prediction; perfect knowledge would therefore consist of a mathematical formalism, arrived at by science and embracing all of reality, that would ground a-posteriori empirical understanding in a-priori rational logic.  It was in this spirit that Condorcet dared to imagine describing the entire universe as a mutually-solving set of partial differential equations.

The first cracks in this inspiring picture appeared in the latter half of the 19th century when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries.  Riemann's geometry was modeled on a sphere, Lobachevsky's on a hyperboloid of rotation.

The impact of this discovery has been obscured by later and greater upheavals, but at the time it broke on the intellectual world like a thunderbolt.  For the existence of mutually inconsistent axiom systems for geometry, any of which could be modeled in the phenomenal universe, called the whole relationship between mathematics and physical theory into question.


What we are attempting to do is affected by this as well. If we seek an underlying mathematical truth, such as the one we've been discussing, to found our TOE upon, we are seeking to establish "a-posteriori empirical understanding in a-priori rational logic." The definition of mathematics' role in this TOE is, therefore, in the "Aristotelian tradition of rationalism," and it is consequently subject to the intellectual impact of the possibility of the "existence of mutually inconsistent axiom systems for geometry;" that is, there can be no TOE in any absolute sense, but only in "multiverse" sense, which is the quaqmire of the so-called "anthropic principle" that is currently swallowing up the most prominent of theoretical physicists at an astounding rate.

Fortunately, for us, there is a way out: the simple concept of space and time as nothing more than the reciprocal aspects of motion allows us to reinterpret Euclid's fifth postulate in a way that closes the door opened "when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries."

It does this because, if space and time are nothing more than the reciprocal aspects of motion, then they don't have an independent existence. Space, which we normally think of as a set of points that satisfies the postulates of one of several geometries, becomes merely the space aspect of a past motion that determined the locations of those points. The space doesn't exist after the motion. In order to measure the space, we have to recreate it with motion, because it is motion that exists, not space, just as the existence of a rod creates its two ends. Without the rod, the locations of the ends of the rod are only mathematically expressable, or logically expressable, points in a formalism. Without the rod, or some other physical object to define them, the locations of space are only abstract concepts.

Thus, we see that Euclid's axiom, that only one line exists through a point not on a line parallel to it, in reality, holds true only for one-dimensional motion; that is, it is the motion that defines the line, so if the dimensions of the motion are never more than one, then Euclid's geometry holds, but, if the motion that defines the line ever changes so that there is motion in more than one dimension, then whatever motion is not in one dimension must be in another dimension, and Euclid's geometry doesn't hold.

Therefore, what Riemann and Lobachevsky actually proved was that motion must exist in more than one dimension. Clearly, then, by making a move from a concept of space as a container, or a medium, or a manifold of spacetime points, to a concept of space as the reciprocal aspects of motion, the impact of "inconsistent axioms of geometry" is allieviated, because geometry, as I showed in my previous post above, no longer has to be axiom-based.

Newton understood this anyway, when he observed that

the description of right lines and circles, upon which geometry is founded, belongs to mechanics...it is the glory of geometry that, from those few principles, brought from without, it is able to produce so many things.  Therefore, geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.


Not that he understood that numbers lead to geometry, without the need of axioms, but he understood that geometry is entirely dependent upon motion. When that motion is one-dimensional, as the motion of all objects must be at any given moment in time, geometry has some wonderful things to say about the "right lines and circles" given to it from "principles brought from without." However, it has NOTHING to say about the motion that creates them. Therefore, the "inconsistent axioms of geometry" are irrelevant to a system based on space/time as motion, not spacetime as geometry.

This is huge.

Excal



StevenA
Hi there, Excal.

I saw someone else on this site gave this link:

Food for Thought > Unified Theory > Standard units / ST conversion
http://www.blazelabs.com/f-u-suconv.asp

Though I agree with your view that space and time seem the same in many ways.

A ToE model that seems to describe everything I know about at least is a 2-D spacial plane with a thin time dimension (I've given examples of standing sandwhiched between a couple mirrors before). Google "holographic universe" if you're curious.

If you look at human perceptions, we're 2-D over time. Some people even have trouble visualizing a 3-D cube, with 8-lines, yet they can instantly recognize the surface of a face composed of effectively millions of such points.

Anyway, we tend to go off and talk about 10D universes etc. but these are largely mathematical extrapolations. The only thing we can really verify firsthand is 2-D + time. Tap something with your finger. Now we might assume the waves propogating through this must have gone through a 3-D volume over time, but consider that if that 3-D dimension is time already then 3-D already includes time and we still only sensed a 2-D surface. Even if you get out 50 scanning electron microscopes and watch while you tap things, you're simply seeing 50 2-D + time representations. Do we really have a way of knowing how "thick" the time dimension is?

Now if you crinkled up this relatively 2-D plane into a ball, it would give a chaotic view of things on a small scale, and this may not seem intuitive but from what we see, small scales aren't intuitive and consider that what's provided intuitivly to us has been preprocessed by senses and a brain that are more complex than any super computers he have and what isn't intuive we simply don't even understand or recognize anyway. Now this thin crinkled 2-D + time (I separate time, even though it's technically a 3-D dimension because I don't believe we visualize correctly how it operates) would appear on a macroscopic scale to be rather uniform (truly the other 2 spacial dimensions could operate in non-uniform ways also and correlate to natural forces like charge and mass/energy). Imagine a drop of water soaking into a sponge. The sponge is 3-D and though on a small scale water diffuses unevenly, on a larger scale it appears rather uniform. We don't see brownian type effects from distant light sources, but consider that a if these surfaces generated exponential scales, any brownian effects wouldn't be along the lines of 1/sqrt(time or distance) but instead be a log(time or distance) function, which would probably simply appear as a relatively planck scale phenomenon (We do see differences in light that's travelled a long way anyway - the red shift).

Now after you crinkle up (or wrap it into a toroid or whatever) this thin 2-D + time view, you have a true 3-D object much smaller than its original 2-D spacial diameter. From inside, it still appears massive though in reality things like faster than light speed gravity are possible, wierd quantum phenomenon and fractal characteristics to the universe etc. (I've already posted a lot on these ideas so I'll skip them here).

Now because these dimensions are tightly coupled on a small scale, macroscopically they would appear uniform. All these 3 dimensions would appear uniform and interchangeable from a wider perspective, though they wouldn't truly need to operate in a uniform manner. For example, imagine 3 different delay lines transmitting signals being quickly rotated very quickly through 3 distinct signal operations (maybe different filter structures, reverberation, non-linearities etc.) Though to determine individual samples, you'd need to track the phase of these operations "through the pipe", from a higher level view of larger samples, all 3 lines would appear to operate similarly and perform those 3 operation on each channel.

Now from that point of view. If all energies propogate at light speed, then time equates to a distance. In many ways time is the distance between two points. (I admit I still have a tough time working with these ideas biggrin.gif) From a stationary viewpoint, all you need to know is the time delay or alternately phase between interactions to triangular them. Truly you could even relax that requirement and simply require that only the chronology of observed events is corrent or alternately that the ratio of observed events is correct. (But that's something I'll try to ignore, though modelling things from a stationary viewpoint and simply using techniques like phase or delay to determine spacial locations doesn't seem like a bad idea)

So it seems time can be compressed into the 3 spacial dimensions with the benefit of describing many naturally observed phenomenon as well, and time might be viewed as the spacial distance measure as well. The other 2 dimensions could follow along your list of:

1) quantity
2) dimension
3) polarity

Though I wonder if somehow these couldn't be rewritten as simply operations on volumes, or along the lines of point, area, volume? Polarity and quantity seem similar and I can't help thinking there may be only one polarity needed - neutral. If my view of a small resonant universe is correct, then polarity would seem to correlate to a phase difference (things in phase attract, things out of phase could be repulsive). If you view forces as rates of rotation between real and imaginary axises, there might be a way to have polarity or velocity be the imaginary components of quantity and location/dimension?

(I deleted a bit here. It does seem best to leave frequency and phase as derived formulas and not the base units. Describing events in terms of periodic functions can might make some mathematical transformations easier but they might cloud the picture a bit. When you say dimension, are you thinking of the dimensionality of the property as in point, line, area or volume ... or are you thinking of distance in spacetime between things?)

I'm beginning to wade too deep, though there's a ton of ideas that seem tantalizing but I don't want to lead you guys in the wrong direction, so I'll stop there and see where you guys head.
StevenA
I think I talked a lot and didn't emphasize the importance of the curled up space.

The three distinct dimensions sound great. We know how interactions between them appear on macroscopic scales (long term stochastic aggregates) but the critical part seems to be determining what possible operations are performed on the quantum level.

Some macroscopic values we know of are

orbital period/observed time at that scale = distance^3/2.

Now if time is more of an aggregate macroscopic rate of interaction, then if detection of it correlated to detecting rather random brownian motions, then we could have a 1/(distance or scale) squared relationship. This isn't too hard to envision - we see smaller scales as appearing to operate faster than larger scales.

So if observed time was at a rate of (distance^1/2) and then if we divided these two to get a more absolute quantum time frame for orbitals, we'd have:

planetary orbital period from a scale invariant time reference = distance^3/2 / distance^1/2 = distance

and of course this would agree with the macroscopic observations of frequency = 1/distance. So from a larger scale, a solar system might appear like an atom or a photon with some randomized/brownian components.

Funky? biggrin.gif (Maybe I did something wrong and that's just a mathematical parlor trick?)

We consciously observe things from a relatively constant scale ... maybe not all particles perceive and/or react in time in a similar manner that depends on what scale they're interacting at? Could this be a continuum or relfections at discrete levels with some particles being "real" and some mirrored at a different scale of time and space?)

You can do many complex mathematical operations on stochastic bits streams with simple logic operations (I can post some examples of these functions if you guys want)

But I think the relationship above is what ties gravity to EMF forces. They are the same force but observed at different scales of time/space due to macroscopic approximation of time .... and space?! (Maybe we aren't calculating distances correctly and they are non-linear)

Maybe there's a single exponential scaling to space and time that makes things appear more linear and less fractalized. Like warping all distances and times by the power of 2/3rds or something like that. That would agree with entrophy calculations for black holes which supposedly follow the surface area not the volume, implying they are in the limit only 2-D, not 3-D.
jal
StevenA...Excal...TRoc
LET'S ALL STOP FOR A MINUTE! huh.gif
QUOTE
there's a ton of ideas that seem tantalizing but I don't want to lead you guys in the wrong direction, so I'll stop there.

I think that what we are all saying is important and that moreover that there is a link. WE ARE NOT USING THE SAME LANGUAGE.
Even worst, we cannot even connect/communicate with the accepted definitions of symmetry (standard theory). (Which is the language spoken by everyone else)
Look at any wikipedia.org page and you will see the references at the bottom.
Look at,mathworld.wolfram.com page and you will see the references.
I previously said,
QUOTE (->
QUOTE
there's a ton of ideas that seem tantalizing but I don't want to lead you guys in the wrong direction, so I'll stop there.

I think that what we are all saying is important and that moreover that there is a link. WE ARE NOT USING THE SAME LANGUAGE.
Even worst, we cannot even connect/communicate with the accepted definitions of symmetry (standard theory). (Which is the language spoken by everyone else)
Look at any wikipedia.org page and you will see the references at the bottom.
Look at,mathworld.wolfram.com page and you will see the references.
I previously said,
Therefore, an "idea" that can cause a house of belief to come down must be presented and accepted by the non-believer. It must be perceived as being a "better house". Therefore, I must make some presumptions...
On Symmetry...
The language of math is very precise, yet...
QUOTE 
In other postings I have alluded to the fact that Mrs. Susy, Mr. Slim and Mr. Slinky are up against the wall and cannot find out why particles have mass.
A) They/we are all working from the same measured particle quantities.
B ) There can only be one symmetry.
Therefore, the problem must be in the communication channels

"They" have a common language. (symmetry) "They" have communication problems. (SUSY, LQG, BRANES, ETC.)
What have we got to communicate and compare? How can we link/communicate what we are saying with "they"?
Just because I have a good imagination and can see the connections with everthing that we have been saying does not mean that anybody/everybody/you can see the connection. We need more than imagination.
We cannot go making "outrageous" claims without some kind of support.

Look at what I suggested
QUOTE
Packing and kissing numbers make symmetry. Symmetry arized from kissing numbers and packing.  I see this as the underlying and assumed principle. I have not found anything linking the two subjects,
I ask for your help in proving (right or wrong relevant or irrelevant ). Maybe your seach will be better than mine.
The web is good but it does not have everything.
The mathematicians have probably got tons of papers on it.. (that my friend is a presumption) 

Do we go on and on presenting one idea/theory then another....then...? Where will we be at the end of the day?
OR... we can pick a possible crossroad/funnel/pinch point/etc and analyse it to see if it could be a common ground for us to proceed.
You got my suggestion already.
Symmetry 101.
I know... I have overstate/generalized/dramatized/exagerated to make the point. smile.gif
StevenA is right.
WE ALL have tonnes of tantalizing ideas. smile.gif
jal
Excal
Well, one of the reasons RC divided up the project into various threads is so we can focus on the different aspects of the theory in dedicated threads. So, let's talk symmetry 101 and mathematics in this thread. Starting with two, reciprocal, rates of change is a beginning that not only is very fundamental, but its logical development, being purely mathematical, is highly inflexible and constrained, which is what we want in a good theory. As Weinberg explained in his book The Dream of a Final Theory, as reported by one reviewer:

QUOTE
Like a Chopin waltz, whose fragile beauty would be destroyed if a single note were changed, we cannot tinker with a beautiful scientific theory without destroying it. Like a Rembrandt portrait, a fundamental theory is in essence its simplicity - built on a few simple ideas, free of superfluous adornments. Another aspect of a theory's beauty concerns its practical utility. Just as part of the beauty of the racehorse Shergar depended on its ability to win races, so the beauty of a theory depends on its ability to give an immaculate account of how nature works.


So, from the point of view of the mathematics aspect, can we beat the beauty of the exponential expansion of reciprocal numbers?

Excal
jal
Okay....
I've done some checking on your links and searched your topics
1. ohmy.gif DEWEY B. LARSON
Reciprocal System of Theory capable of explaining all physical phenomena from subatomic particles to galactic clusters.

2. ohmy.gif New book!!
DIVINE PROPORTIONS : Rational Trigonometry to Universal Geometry
by N J Wildberger
Home Page of N J Wildberger
3. ohmy.gif Eric's Random Writings
The Utility of Mathematics
An essay, originally written for the Extropians list, on why mathematical formal systems are so mysteriously applicable to the real world.
4. ohmy.gif This website describes some new theories and ideas backed by experimental tests on physics topics which are by far incomplete in conventional science.
Gravity, Wave structure of matter, Physics related sites
did I make my point? smile.gif
Therefore,
QUOTE
Do we go on and on presenting one idea/theory then another....then...? Where will we be at the end of the day?
OR... we can pick a possible crossroad/funnel/pinch point/etc and analyse it to see if it could be a common ground for us to proceed.
You got my suggestion already.
Symmetry 101.

have you done/found anything on my question...
QUOTE (->
QUOTE
Do we go on and on presenting one idea/theory then another....then...? Where will we be at the end of the day?
OR... we can pick a possible crossroad/funnel/pinch point/etc and analyse it to see if it could be a common ground for us to proceed.
You got my suggestion already.
Symmetry 101.

have you done/found anything on my question...
Packing and kissing numbers make symmetry. Symmetry arized from kissing numbers and packing.   I see this as the underlying and assumed principle. I have not found anything linking the two subjects,
I ask for your help in proving (right or wrong relevant or irrelevant ). Maybe your seach will be better than mine.
The web is good but it does not have everything.
The mathematicians have probably got tons of papers on it.. (that my friend is a presumption) 
smile.gif
jal


StevenA
For symmetry of a ToE, here's a link I saw that might be interesting:
http://www.virtualchaos.org/science/mark-r.html

(I lost a long post here but I'll repost it outside this thread. It gives a possible macroscopic scaling of, rate of time ~= mass^.5)
jal
QUOTE
For symmetry of a ToE, here's a link I saw that might be interesting:

This thread is about math. not other theories
123 - Big Bang Blueprints
That post belongs somewhere else smile.gif
jal
Excal
Jal,

I've read a little on it, but I don't know what to do with it. To me, the idea seems unrelated to the task at hand, at least at this stage. Can you be a little more specific in asking your question.

Excal
jal
Excal...
Read my post over again. (LET'S ALL STOP FOR A MINUTE! )
QUOTE
Posted: Feb 22 2006, 03:24 PM

If we want to discuss other theories it should be in an other place.
Here, one of the things that we should be trying is finding an easier way to communicate with the math that is "accepted".(symmetry)
Re-hashing what is already out there will get us nowhere. We must make the link/connection with symmetry Otherwise all you'll hear is,
QUOTE (->
QUOTE
Posted: Feb 22 2006, 03:24 PM

If we want to discuss other theories it should be in an other place.
Here, one of the things that we should be trying is finding an easier way to communicate with the math that is "accepted".(symmetry)
Re-hashing what is already out there will get us nowhere. We must make the link/connection with symmetry Otherwise all you'll hear is,
...To me, the idea seems unrelated to the task at hand,...

The only way that the "doctorates" will listen or even consider your idea will be because YOU have shown how it is related in their "language". (symmetry)
Why "packing and kissing numbers "? It is a much easier concept to understand.
It must have been shown, by the mathematicians, that this is the underlying principle of symmetry.
Therefore, if your idea "violate" packing and kissing it would mean that your idea would "violate" symmetry.
Re-read my post again. smile.gif

jal
jjac
Hi Everyone,

[QUOTE]The kind of symmetry I am introducing is “resonant symmetry.” How it ties to geometry is important too.

T.Roc,

I agree. Your "resonant symmetry" concepts refer to properties of the natural system (Everything), and the encoding of these concepts in a formal system, such as a geometric system, is important too. Your approach, identifying properties of a natural system. is the first step in the approach (Complexity Theory) developed by Robert Rosen called Model Relation (MR). Encoding "resonant symmetry" in a formal system would be the next step in Rosen's MR approach. The hoped for outcome would be that when your formal system is decoded it demonstrates correspondence (commutes) with your "resonant symmetry" concepts of the natural system. The degree to which it commutes is the degree to which it represents the formal system of Everything.

How to encode "resonant symmetry" in a formal system is the question if one is following Rosen's MR. My suggestion would be to seek a formal systemization of "resonance". Being an artist/sculptor, my approach would be qualitative as opposed to quantitative. What are the qualitative properties of "resonance"?

A vibrating string exhibits resonance. What are its qualities? I would propose its qualities to be straightness with rotation, speculating that the rotational quality corresponds with the 3-dimensional form of the vibrating string. Assuming the element required for the formal systemization of "resonance" must exhibit the qualities of straightness with rotation, and incorporate all possible notes and scales, the question is how to model this concept.

A structural element that models straightness with rotation is a flat, twistable stick. The longitudinal line down the center of the flat stick represents the straightness, and the edges, a pair of helical lines moving around the staight line, represent degrees of rotation. As you know, this is the structural element of my Spiral Geometric System, which decodes to demonstrate that it commutes with the qualitative properties of a system of molecule-thin liquid membranes, and those of molecular structure.

I believe that the quantitative properties of the SGS will decode using a novel mathematical term of measure (distance/rotational degrees) and commute with your "resonant symmetry" concept of Everything.

jjac



jjac
I have a one page diagram of Robert Rosen's Model Relation, but haven't yet learned how to download images??? Heeelllp!

jjac
jal
jjac...
You have posted in the the wrong thread.
This is another theory....
Robert Rosen

QUOTE
The purpose of this site is to provide information and links regarding the work of the theoretical biologist Robert Rosen (1934-1998), who showed that biological systems exhibit complexities and physics beyond the strictures of Newtonian mechanics

Go post it in COMPARISONS between OUR TOE and OTHER THEORIES, Discuss TOE in light of CURRENT 'MODELS'
Have the discusion there. Not here.
jal smile.gif
TRoc
jal, jjac, ...



jal, your argument applies to you, me, & Excal as well: we are all talking about other theories; if that were not the case, we wouldn't be having the "communication" problems that exist from the different terminology.

I invited James to discuss with us, please don't send him away. He is specifically connecting what he posted with what I have said.

Excal & myself have "agreed" to continue here, after wondering if we should "start another thread". I, for one, find it more difficult to have 3 or 4 conversations at once, covering the same topic, on different threads.

Essentially, what we are discussing, in our different formats, is Mathematic & Number theories, as they apply to a build-up of a "TOE". At this point, more specifically, we are talking about the "beginning" of any theory that is or can be described with quantity. Binomial expansion, 1 quantity vibrating into 2, either of which can then be measured with "the third" quantity that represents either "both" or "neither". This is the simple and logical start. If you are not using one of these quantities at this point in our conversation, you are probably "too early".

That's my take on it; if I am wrong, then I can continue my conversation on the other thread.

smile.gif

T.Roc
jal
TRoc...
I'm gone... smile.gif
my post are there for evaluation.
jal
jjac
OK, smile.gif so anyway, assuming we're on the same page, my last post ended with:

[QUOTE]I believe that the quantitative properties of the SGS will decode using a novel mathematical term of measure (distance/rotational degrees) and commute with your "resonant symmetry" concept of Everything.

What I know about the SGS is that its quantitative properties correspond with the quantitative properties of the the system of plane right-angled triangles. Both express the natural trig functions. The SGS expresses the trig functions in 3-dimensional space, making the plane angle of rotation into a helical angle of rotation along the x and y helical coordinates. The cosine of the helical angle of rotation of x and y is the length of x and y. So all points along x and y are identified by two numbers, one corresponding to distance and the other to degrees rotation. The grid or matrix defined when rotational degrees is included corresponds with the natural forms of a system of molecule-thin liquid membranes. When the rotational degrees is excluded then the grid or matrix corresponds with the straight-lined form of the hyperbolic paraboloid. This would be the argument for using both quantities, distance and rotation, if one chooses to quantitatively decode the SGS, and maintain a correspondence with a natural system.

I wonder what a 3-dimensional Pythagorean Theorem would look like?

Also, there would seem to be an equivalence between the distance-rotation inverse relationship of the SGS and that of the wavelength-frequency inverse relationship.
Distance : rotational degrees :: wavelength : frequency?

Does it seem to anyone else that a quantitative analysis along these lines is a good idea?

jjac



StevenA
I wonder if dimensions can be defined in terms of what form of non-linearity they use?

For example in the Reciprocal System http://www.reciprocalsystem.com/dbl/index.htm, you basically have a combination of linear and reciprocal operations on spacetime (reciprocal would seem to equate to multiplication and division but not easily provide exponential/fractal/recursive structures)

It seems like a lot of heterodyning/AM modulation would be occuring as many wave actions would interact in a non-linear manner and alias into different frequencies.

phi=7/5/pi/e (I know you mentioned a 5/7 symmetry ... maybe there's a clue in that)

You can generate binomial expansion in the exponents:

For example if waves interacted multiplicatively, you could generate the sum and difference of frequencies. This would appear as a convolution of the exponents if frequencies were exponentially spaced and had intervals of phi somewhere along the line. You wouldn't have to pick phi but phi^1/n so there might be a way to add some other symettry in there by picking n.

But just to keep it simple, if you had frequency spacings of phi, then to operate in reciprocal space, you'd simply negate the exponents:

Multiplying the signals cos(t*phi^0)*cos(t*phi^1) would give 1/2*(cos*(t*phi^-1)+cos(t*phi^2))

Alternately, doing this as a complex rotation could be:
e^(i*t*phi^0)*e^(i*t*phi^1) = e^(i*t*(phi^0+phi^1)) = e^(i*t*phi^2)

These could be represented as vectors being colvolved and starting from a frequency of phi^0 up to phi^2):

100 (x) 010 = 001

Binomial expansion would be a recurring convolution with 11, so

1 (x) 11 = 11
11 (x) 11 = 121
121 (x) 11 = 1331 etc.

Alternately using a non-linearity of x^phi to perform this operation, I believe would operate like this (I'm not going to bother with scalings but am just throwing out the general form I think it could take. Someone else can do the exact math if they want. I hope there's not an octave component ... it would be nice if some phi^j/k was a power of 2 wink.gif)

(cos(t*phi^A)+cos(t*phi^cool.gif))^phi ~= cos(t*phi^(A+B+1?)) + cost(t*phi^(A-B+1?))

(Using summations and single input non-linearities would be a nice option to have, if possible but not necessary ...)

Anyway, converting these vectors between time or space domain would be simply reversing them:

So a time domain of:
05112 would be a spacial domain of 21150 (assuming the central vector was the unity time/space scaling)

Phase information might not be critical if we're simply analyzing steady state characteristics, though you could probably treat a phase vector similarly.

I don't know how you could combine linear spaced frequency operations with this though but again, you could pick phi^1/n instead. If you picked phi^1/8, that's ~1.062 which after 12 steps gives 2.058 ....

Here's an interesting one ....

phi ^ 1/(5^2) (yes, I know that's large) ~= 1.9996^1/(6^2)

(25 and 36 are likely useful ... I used to design ICs for a music company ... I wish I would have known of this close relationship back then and could have used this)

I don't know if this helps but 1/x = 1/y + 1/xy. (I'm just trying to see if there's some way to simplify this 36/25 ratio). Or 25 * 1.2 = 30 and 30 * 1.2 = 36. So there might be some geometric symmetry around 30?

phi^(30/25 or 36/30) = 1.781
2 ^ (25/30 or 30/36) = 1.782 biggrin.gif

There have been many claims that the exponent for gravity doesn't remain the same over all scales either so nature might be using a tradeoff between a linear true tempered/octave scaling and an exponential phi scaling. Gravity might be the reflection of this asymmetry to EMF forces and redshift could be a slight shift in frequencies over large scale differences. If this value existed, it could likely be approximated from observations though an apriori value could be estimated.

If you see some relationship you'd like, let me know and I'll crank through some numbers.

Also, check this out:

User posted image

I did some math and found that each symmetric double rotation averages rather close to a ratio of phi (~1.7) per layer when using atomic numbers.

For example, taking the double rotation from Ga to Y and comparing it to the similar double sweep from Tl to Ac goes from a ratio of (81/31) = 1.616^2 to (89/39) = 1.510^2.

Going from Ti to Zn and comparing these to Hf to Hg, 2 rotations away, gives ratios from (72/22) = 1.809^2 to (80/30) =1.633^2

Al to Ca compared to I to Ba goes from (49/13) = 1.941^2 (56/20) = 1.673^2.

To get a larger scale estimate of this ratio, let's try the 4 rotations in active metals from Li to Rn (skip hydrogen because it's not scaled right) 87/11 = 1.677^4

Picking steps of phi^1/n that allow for some correlation to atomic orbital structures might be good.

User posted image

Ok, I don't want to drift off topic too much but I wanted to post those in case they give some clues to possible structures.

Again, I wonder if there might not be some spaces that correlate with these:

1) Linear (spacial)
2) Reciprocal (time)
3) Exponential (fractal level/binding forces?)

Here's a link to a site with a lot of ideas along these lines (half of the things tend toward metaphysical correlations but you can skip then. There's at least one geometric 5/7 symmetry listed but it's beyond me):

http://www.soulinvitation.com/physicsofphi/PhysicsofPHI.html

Here's a site with a ton of phi observations (including a few ways of spacial tiling with phi relationships also), there are links on the upper left:

http://goldennumber.net/mandalas.htm

No, I'm not a phi fanatic! But at a minimum, doing frequency heterodyning as simply a convolution of ratios of phi sounds like a good path to look at - for the reciprocal system it ends up being a reversal of the vectors to translate between time and space, and for scaling, it's simply a shift operation. If the universe appears similar on all scales, this would make sense too. Wouldn't it be wierd if the universe simply repeated a convolution/shift operation, like a pseudo random number generator but interpreted these shifts as an expanding spacetime? I'd almost bet there's some universal computational equivalent between the two models.

To compare this to the reciprocal system (BTW, thanks for posting that link), I think the tap density would correlate with the quantization scaling he uses.
StevenA
After sleeping on it, I've got to offer a couple critiques of the last post, mostly because of it not being a linearly scaled system.

1) Though, it could possibly be rescaled rather easily to describe phenomenon on various scales like galactic or solar evolution as well as atomic and quantum observations. Not being a linear scaling would make it less suited to simultaneous simulation on different scales.

2) Not having true integral relationships, steady state conditions would be difficult to work with. Without rescale the exponent used, the complexity of the desciption would only grow by the log^3 of the scale used, for 3-D spacetime. (I've got a bias in that I think the complexity grows approximately by the square of the scale and though it might even be smaller but it's hard to envision it being a log relationship, (though it may not be impossible it easily be an unrealistic limitation). Basically, you'd be viewing a rather layered representation of some space giving finer and finer detail but would find it necessary to rescale to analyze anything specific in that space, which would simply be an approximation. (This could be a great CAD/CAE tool but possibly too restrictive for a ToE? ... then again that might be missing the forest for the subatomic particles in the way)
Excal
QUOTE ("StevenA"+)
For example in the Reciprocal System http://www.reciprocalsystem.com/dbl/index.htm, you basically have a combination of linear and reciprocal operations on spacetime (reciprocal would seem to equate to multiplication and division but not easily provide exponential/fractal/recursive structures)


Actually, there's a lot more to those combinations than one might expect. For instance, the only number that exists initially is 1/1 = 1. But then the physical concept of this number, as the unit proportions of two rates, makes it possible to get two more numbers, 1/2 and 2/1, from the original 1/1. Combining these three numbers results in two more numbers, 2/3 and 3/2. From this point, we have five numbers and one combination of numbers that can be combined, but, again, how the combinations are formed has to do with the phyisical concepts. The mathematical possibilities are many, but we want to be constrained by the physical concepts.

This means that, while mathematically, 2/1 is twice 1/2, and 1/2 is a 3D vibration, and thus 2/1 is an octave above 1/2, frequency wise, we have to consider that 1/2 is a speed less than c, while 2/1 is a speed greater than c. Therefore, s/t = 2/1 is really t/s = 1/2, because time has no direction in space, and space has no direction in time. So, we can't really regard a 3D time vibration as just a high-speed 3D space vibration. They are different motions. One is space motion and the other is time motion.

QUOTE ("StevenA"+)
It seems like a lot of heterodyning/AM modulation would be occuring as many wave actions would interact in a non-linear manner and alias into different frequencies.


I think so too.

QUOTE ("StevenA"+)
phi=7/5/pi/e (I know you mentioned a 5/7 symmetry ... maybe there's a clue in that)

You can generate binomial expansion in the exponents:

For example if waves interacted multiplicatively, you could generate the sum and difference of frequencies. This would appear as a convolution of the exponents if frequencies were exponentially spaced and had intervals of phi somewhere along the line. You wouldn't have to pick phi but phi^1/n so there might be a way to add some other symettry in there by picking n.


However, a significant point to consider is that pi and e have not yet emerged from the system at this stage. Indeed, even the 7/5 (5/7) numerical ratio does not yet exist. We can combine the 1/2 ratios with the 3/2 ratios and that gives us a series of integers:

1) 2/3 (3/2) + 1/2 (2/1) = 3/5 (5/3)

2) 3/5 (5/3) + 1/2 (2/1) = 4/7 (7/4)

3) 4/7 (7/4) + 1/2 (2/1) = 5/9 (9/5)

...

which gives us integer speed-displacements,

1) -1 0 1

2) -2 0 2

3) -3 0 3

...

which one can prove is an infinite series. So, the heterodyning/modulation possibilities are endless.

QUOTE ("StevenA"+)
You can generate binomial expansion in the exponents:

For example if waves interacted multiplicatively, you could generate the sum and difference of frequencies. This would appear as a convolution of the exponents if frequencies were exponentially spaced and had intervals of phi somewhere along the line. You wouldn't have to pick phi but phi^1/n so there might be a way to add some other symettry in there by picking n.


Great observation, but pursuing its implications beyond this seems a little premature at this point. I think of it as something to put on the shelf for later consideration, so as not to get side tracked in light of the possibility that we might discover the fundamental properties of radiation, matter, and energy in these number combinations, if we don't get side tracked. After all, such a break-through would be staggering, and there are tantalyzing hints that we are on the right track.

For instance, the RNs, as a new, reciprocal, interpretation of the mathematics that ultimately leads to representations of Lie groups, the foundation of current theory based on guage symmetry, is compelling. So, we ought to find out how to combine these initial RNs and see where that leads us. What I've found intrigues me, but also perplexes me too, because there are no established rules to follow as far as I know.

Again, when we think of combining physical entities, there has to be a physical basis for the combination, not just a mathematical basis. So, what is the physical basis for combining RNs? If we consider the initial unit dispalcements, 1/2 and 2/1, we see that one is stationary spacewise, while the other is stationary timewise, relative to the unit space/time ratio. We can illustrate this on a world line chart:

user posted image

Then combining these two ratios creates an entity that moves at unit speed relative to either of its constituents, a 1D RN, as shown on the chart below:

user posted image

The question is, how are subsequent Ss and Ts combined to these 1D RNs? If we add them,

1/2 + 1/2 = 2/4 = 1/2,

seems mathematically inconsistent, but only from the orthodox quantitative point of view. From a reciprocal (operational) point of view, it makes perfect sense: The ratio of the expression

1/2 + 1/2 = 2/4

equals 1/2, but the speed-displacement is twice that of 1/2. In other words, s/t = 1/2 is a speed-displacement from c-speed (s/t = 1/1) of one unit in the negative "direction" (recall the 1D RN (1/2 + 1/1 +2/1) is equivalent to (-1 + 0 + 1)), but 2/4 is a displacement of two units in the negative "direction," even though the space/time progression ratio in the two values is the same (1:2).

But what if we multiply them instead of add them:

1/2 * 1/2 = 1/4.

Again, quantitatively, the mathematical result is misleading, but taking the reciprocal view, the product of these two displacements is 1^2; that is, it's one unit in each dimension, but the displacement of this new, 2D, number is 3! recalling that the displacement value is always 1/(n-1), so that 1/(4-1) = 3 negative units. There is much more to say about this, but one thing that's fascinating right from the get-go is that we see from this that by addition we get a 2 unit and a 3 unit (previously) value that are 1D versions of the number, but we also get a 3 unit value that is a 2D version of the same value!

Quantitatively speaking, this is no big deal, because every scalar has an n-dimensional equivalent, but reciprocally speaking, this is astounding, because we are saying that (1/2)^2 = 3 displacement units, equivalent to a negative integer 3, not 1/4, or half of a half of a whole! We are also saying that 1/2 + 1/2 = 2/4 = 2 displacement units, which is, again, a negative integer, not (1/2 + 1/2) = (.5 + .5) = 1, a whole.

Looking at our 2D RN, i.e. (1D RN)^2, as a 3x3 matrix, gives us four 1D quantities (2 pluses and 2 negatives) and four 2D quantities. The four 1Ds are:

|***|2/1|***|
|1/2|1/1|2/1|
|***|1/2|***|

but the four 2D quantities are (sorry, trying to do this with ascii characters since tables don't render here):

|2/2|***|4/1|
|***|1/1|***|
|1/4|***|2/2|

which, given the polarities, means a +3 and a -3 in one diagonal and +-1 and -+1 in the other, 2D wise, but formed from two 1D sets of (-1+ 0+1), or a 1D RN squared. Extending the same thing to a 1D RN^3, yields a +++ and a --- 7 unit quantities, but I haven't had time to figure out what the mixed polarities are yet.

What this gives us is a 0D 0 unit quantity, a 1D 1 unit quantity, a 2D 3 unit quantity, and a 3D 7 unit quantity, all with two polarities, plus much more that I can't take the time to get into yet.

The point is, I guess, the physical concepts constrain the mathematical, as well as vice versa. These polarized quantities may end up leading to theoretical entities with the properties of observed photons, electrical charges, magnetic moments, and gravitational masses, with the spins, energy, frequencies, etc, as appropriate in each case.

Thus, there may be, for the first time, a new, much simpler, means for developing a physical and mathematical model of radiation, matter, and energy, compared to the current nuclear and standard models.
StevenA
I know just enough to find these ideas tantalizing, Excal, but sadly not enough to be able to 'talk shop' in reciprocal spaces ... yet. I'm tempted to buy a book or two ... any recommendations, or a good starting spot on-line? I read a lot of chapters on-line from that Reciprocal System site but I saw no diagrams. I appreciate the images you posted as they were truly the first I saw (I was imagining a polar diagram with a unit circle being 1:1 space/time motion and distance being interpreted as either space or 1/time), so the interior and exterior of the circle would be reciprocally related.

That exponential growth you showed could correlate with binding forces.

Pardon the possible drift but I just had an insight into how this could possibly be transformed along the lines of my small/resonant universe idea:

You might be able to transform this into a sphere (or cube for quantization like you're using). Imagine a circle of diameter 2, with the center being considered 0, the unit circle being 1 and a distance of 2 correlating to infinity (a d/(2-d) transformation). Then you could pull in the edges at distance 2 (infinity) and connect them together as the top of a sphere, or lets switch to a cube to simplify it. If observations occured from a stationary reference in here, then that would correlate well with ratios of velocities between space and time as being close to integer fractions. For example, like a 1:1.001 ratio, would only be detected as a rare event as it quickly aliased across the "surface of real space", and only be observed once every 1000 cycles (truly most likely once every 1000^3 cycles for a 3-D volume) as it "bounced" around inside this small universe). So science, as a collective pursuit, would only be able to objectively witness those recurring integral relationships, whereas these would be more of a stochastic average of the most, in a sense, "self-reinforced"/coherent phenomenon. Though maybe with enough understanding or sensitivity in detection, it might be possible to predict the periods of these rare events and alter their trajectory to make them "real"? It might be viewed like diffraction or reflection, except you dynamically adjust the diffraction/reflection to select which path you want to oscillate stably in real space.

In reciprocal space, he calculates a unit step in time and space, that I think's on the order of 45 nM or something. Look at the peak wavelength of blackbody radiation ~1000 nM. There might be a way to correlate the diameter of the universe in space and time using these unit steps also (though I don't think it's quite as simple as a division). The entire multiverse could possibly be connected within some relatively finite multiple of this unit time step. You could determine the stability or rarity of these relationships by how well they resonate with the width of our universe as only certain step sizes would remain stable (you can get spiral arm galaxies by plotting primes in a polar coordinate fashion) and such aliasing across space could explain entanglement. Matter might be related to how well 4-D "dark matter" type energy in a multiverse can form stable resonances inside the 3 dimensions of our universe.

For example if energy had a velocity of x,y,z, in 3 dimensions and the universe were a cube of dimensions j,k,l, if an observation of this energy occured at 0,0,0 and it's velocity was 1,1,1 with dimensions of the universe as 20,20,20, then its path would fill the diagonal from 0,0,0 to 19,19,19. If one dimension was time, then it would appear to be at an x,y location of 0,0 at t=0 and 1,1 at t=1 etc. Though the velocity could be 11,1,1 in which case the path could alias as either 0,0 at t=0 and 11,1 at t=1 and then back to 2,2 at t=2 (for a wrap around universe, or 18,2 for a reflective one), so you could have energy appearing to alternate between two phases over time while "simultaneously" interacting with distant objects. The width of this "universe" could be determined by looking at what modes of oscillation are stable. The non-linear mechanism is the toughest part.

Ok, I'm going to mess around with some primes (maybe pythagorean triplets/quadruplets for unit velocities) and atomic orbital shells and see what correlations this small resonant universe idea has with Reciprocal Spaces. It appears there's a transformation between the two models.

Thanks to all you guys. I must admit this stuff is fun to mess around with. I don't want to sidetrack anyone though but simply hope to compliment or possibly stimulate new ideas if there seems to be a roadblock though truly it doesn't seem like there's any showstoppers except getting something we can plug numbers into.
Jimmy Roberts
There is an article that compares Rational Trigonometry to Classical Trigonometry.
See:
http://www.geocities.com/xyz.abc_123/ratio...comparison.html
Excal
QUOTE ("StevenA"+)
I know just enough to find these ideas tantalizing, Excal, but sadly not enough to be able to 'talk shop' in reciprocal spaces ... yet. I'm tempted to buy a book or two ... any recommendations, or a good starting spot on-line? I read a lot of chapters on-line from that Reciprocal System site but I saw no diagrams. I appreciate the images you posted as they were truly the first I saw (I was imagining a polar diagram with a unit circle being 1:1 space/time motion and distance being interpreted as either space or 1/time), so the interior and exterior of the circle would be reciprocally related.


Larson's books are available online from Amazon. However, as you say, there is not much in the way of diagrams in them. Those that I have, I have developed myself. Also, the reciprocal concepts of the numbers were also developed by me, using Larson's ideas, so there are no other diagrams that I know of other than what I've done.

The major diagram that I use is a simple one I developed as the output of an algorithm called a "progression algorithm," or PA. This shows the reciprocal aspects of a 1:1 scalar progression:

user posted image

It's based on Wolfram's cellular automata rule 254:

user posted image

Here, Wolfram's rule 254 produces units of space that grow uniformly from 1 to 3 to 5 to 7, etc, over time. So, the number of units of space is represented on the horizontal line, or columns, and the number of units of time is represented by the number of rows.

I modified this by dividing the columns into two, opposing, sets of columns, the left and right columns with respect to the center column, and designated the left set as "space" units, and the right set as "time" units. The rows then become units of "motion" and the diagram represents the progression of these units. As it grows from 1 to infinity, the number of space units and the number of time units are always equal, because the rate of the space to time progression in this algorithm is 1:1.

However, this space/time ratio changes to 1/2 or 2/1 if the space or time units in a given unit of motion changes "direction;" that is, if it decreases/increases rather than continually increasing, while the reciprocal aspect continues to uniformly increase in the normal manner:

user posted image
"Direction" reversals in space aspect.
user posted image
"Direction" reversals in time aspect.

(Ignore the extraneous line, at the bottom, underneath the space/time arrows. It has no meaning.)

As you can see, taken together, these three diagrams represent the 1D RN:

(1/2 + 1/1 + 2/1) = user posted image + user posted image + user posted image

QUOTE ("StevenA"+)
That exponential growth you showed could correlate with binding forces.


Exactly, but here's the wierd part: the square of the 1D RN, the 2D RN, has two components, the postive (++) and positive/negative (+-) component, and the negative (--) and negative/positive (-+) component. While this corresponds to the positive and negative charges with spin, such charges separate from one another.

When a gamma ray photon has enough energy, it too can separate into two charges with spin, one negative and one positive, the so-called pair production of an electron and the positron from a gamma ray. Here's how Wikipedia describes pair production:

QUOTE
By interaction in the vicinity of the coulomb force of the nucleus, the energy of the incident photon is spontaneously converted into the mass of an electron-positron pair. A positron is the anti-matter equivalent of an electron; it has the same mass as an electron, but it has a positive charge equal in strength to the negative charge of an electron. Energy in excess of the equivalent rest mass of the two particles (1.02 MeV) appears as the kinetic energy of the pair and the recoil nucleus. The electron of the pair, frequently referred to as the secondary electron, is densely ionizing. The positron has a very short lifetime. It combines within 10-8 seconds with a free electron. The entire mass of these two particles is then converted into two gamma photons of 0.51 MeV energy each.


Disregarding the energy aspect for a moment, the obvious idea that occurs to one first is that the magnitudes of the 3x3 matrix split, or are redistributed. We can understand this only when we analylize the magnitudes of the 2D RN in terms of their combined values:

|***|2/1|***|
|1/2|1/1|2/1|
|***|1/2|***| ------->

|2/2|***|4/1|
|***|1/1|***|
|1/4|***|2/2|

but because the center value, 1/1, is actually, (1/1)^2, this can be written as,

|2/2|***|4/1|
|***|1/1|***|
|***|1/1|***|
|1/4|***|2/2|

which is to say that a 2D RN, with sufficient energy, is equivalent to

(1/4 + 1/1 + 1/1) + (1/1 + 1/1 + 4/1), ratio wise.

Mathematically, this is the same as the 2 1D RNs; that is, the constituent terms of the two 1D RNs,

[(1/2) + (1/2)] + [(1/1) + (1/1)] + [(2/1) + (2/1)] =

[(-1) + (-1) + 0] + [(0 + 0] + [(1) + (1) + 0] =

[-2] + [2] = 0,

but they are physically manifest as,

[(1/2) * (1/2)] + [(2/1) * (1/2)] + [(1/1) * (1/1)] + [(2/1) * (2/1)] + [(1/2) * (2/1)] =

{[1/4] + [1/1]} + [(1/1)^2] + {[4/1] + [1/1]},

in the 2D RN.

This might seem like a stretch to get an electron and a positron out of a gamma photon, but it's just a thought at this point. It does seem to be consistent, though.

Excal

StevenA
Thank you for the post, Jimmy. I noticed from that article that by selecting either rational/classical approach for some calculations you can avoid irrationals altogether. That might give a clue as to how nature calculates them.

Excal,

I wanted to give you an update, and see if you can possibly offer some advice as well. I'm basically trying to use pythagorean quadruplets to keep all velocities and energies integers.

For the photon/mass collision example you gave, I've been trying to model particles as:

(velocity in time, velocity in space, spin velocity) with rational values for all of these including energy, like this

e^2 = vt^2 + vs^2 + sv ^2.

So I'm assuming, for example, a mass of 13 velocity/169 energy could be the triplet (13,0,0) and a photon of identical energy could be (0,13,0). With interactions being rotations like producing a pair (12,5,0) and (5,12,0) but I want to split the secondary particle into complimentary pairs. I believe I might be missing some spin information or my assumption that the mass can be stationary is wrong, but I've been searching for triplets that follow this relationship:

(m,0,0) + (0,p,0) = (m,v,0) + (a,b,c) + (a,b,-c)

and trying to assure energy is conserved with no irrational values used. I might need a scaling value of sqrt(2) for spin .....?

Anyway, I've found integer values for all these that seem to conserve everything but the resulting energies in the production particles aren't integers.

The main question I had was whether or not the zeroes I have for spins of the mass and photon that enter the collision is correct. I did searches up to energies around 70 and found none that were perfect pythagorean quadruplets, so I think I need an extra spin or velocity to play with, though also, I'm not too familiar with subatomic physics to know what else I should consider. I welcome any suggestion you might have.

Basically, I'm thinking once I can find particles like these that they can be scaled to reciprocal space equivalents and see if they might correlate with real particle interactions. If all values can be integers, then they can easily fit in a small resonant manifold and then I'm hoping to figure out how the fractal unfolding works.

(BTW, I love the Wolfram site ... all except for one moderator on the forum biggrin.gif)
Excal
Steven,

The main thing to keep in mind is that the integer values of speed-displacement are a measure of a motion relative to the unit progression value, 1:1. Therefore, a s/t = 1/2 displacement is a c vibration at rest relative to a fixed, spatial, reference system, while a s/t = 2/1 displacement is 2c vibration at rest relative to a fixed, temporal, reference system. However, the 2c value is not on the velocity side of unity, but on the energy side, so it is not a velocity at all; that is to say, velocity is limited to c, and energy is limited to c as well, but from the other "direction."

Therefore, the numerical 2c value is a measure of energy, not velocity. Now, all of our conventional units of energy and mass are based on arbitrary units, so going from the integer velocity and the integer energy units of the new system to the conventional units of the Newtonian system is not as straightforward as it might seem at first. The main thing we have to keep in mind is that the energy associated with the motion of mass, is not the same as energy defined as time motion, though they are related. The best way to distinguish them for now is by their dimensions: the energy and momentum associated with the motion of mass is a function of one-dimensional motion, f(x), whereas the energy of time motion is a function of three-dimensional motion f(t/s)^3. Thus, the conversion between the two is 2D (c^2).

However, at this point, all we have are numbers. How we can identify physical magnitudes such as charge, mass, spin, or energy with these numbers is not yet clear. What is clear is that we can identify magnitudes of space, i.e. lines, planes, and volumes with these numbers, but motion in a reciprocal system is defined as a ratio, as a change of space over a change of time, or vice-versa, so we have a long way to go.

Combining the 1-unit magnitude of velocity associated with the negative polarity of the 1D RN (s/t = 1/2), with the 1-unit magnitude of energy associated with the positive polarity of this number (t/s = 1/2) requires an assumption of what it means to combine velocity and energy. Since, by definition, one is the inverse of the other, a mathematical combo means 1/2 * 2/1 = 2/2 = 1/1 = 1, but what does this mean physically?

I believe that the physical interpretation has to consider that time has no direction in space (i.e. energy is scalar), but velocity does have direction in space. Since the "direction" reversals in the universal space/time progression are necessarily scalar, the direction of the scalar space vibration with magnitude c is all directions simultaneously, or an oscillating volume. However, it is still a velocity, and thus has vectorial 1D, 2D, and 3D components that cannot be combined with the scalar energy magnitude that has no direction at all, but only magnitude. So, just as a complex number is a combination of a real and an imaginary number, and the geometric product of Geometric Algebra is a combination of a scalar and a vector, so too the 1D RN is a "complex" combination of a unit of velocity in all directions and a unit of energy in no direction.

The mathematical sum of the velocity and energy units then is

(1/2 + 1/1 + 2/1) = (-1 + 0 + 1) = ((-1) + (1)) = 0,

but this is misleading since the +1 is physically different than the -1. Taking this difference into account, the -1 is 1/2, because the space "direction" reversals are effective in all directions, but the +1 is not 2/1, but 2/2, because time cannot reverse in space (it has no direction in space). Thus, the physically correct sum of these ratios would be

(1/2 + 2/2) = 3/4

on each side of unity, due to the fact that time cannot reverse in space and space cannot reverse in time. So, what is the velocity|energy space/time ratio of the (1D RN)^2? It has to be

(1/2 + 1/1 + 2/1)^2 = (1/2 + 2/2)^2 = (3/4)^2,

on both sides of unity; that is, if the velocity|energy combo space/time ratio is 3/4, then the square of this value should equal the 3x3 matrix of values explained previously in the posts above. Thus,

(1/2 + 1/1 + 2/1)^2 = (1/2 + 2/2)^2 = (3/4)^2 = 9/16 =

|***|2/1|***|
|1/2|1/1|2/1|
|***|1/2|***|

Now, if we consider the absolute values of the four magnitudes formed by the four poles of these two dipoles, we get

|*4*|*|*4*|
|***|1|***|
|*4*|*|*4*|

which is a 3x3 = 9, 2D, matrix of 4^2 = 16, and if we go to the next higher dimension, to the 3x3x3 = 27, 3D, matrix,

(1/2 + 1/1 + 2/1)^3 = (1/2 + 2/2)^3 = (3/4)^3 = 27/64

we see that each of those four 4s turn into two 8s, for a total of eight 8s altogether in the 3D matrix. Thus, we see a trinomial/binomial mathematical progression:

I) When one, 3^1, unipolar|bipolar (3-pole) unit of 2-velocity|2-energy, (s/t = (3^1)/(2^2) = 3/4), is squared,

II) we get one, 3^2, unipolar|bipolar|quadrapolar (9-pole) unit of 4-velocity|4-energy, (s/t = (3^2)/(4^2) = 9/16), and when it is cubed,

II) we get one, 3^3, unipolar|bipolar|quadrapolar|octopolar (27-pole) unit of 8-velocity|8-energy, (s/t = (3^3)/(8^2) = 27/64).

However, this time the binomial expansion is not the simple geometric expansion by dimension, but the velocity|energy polar expansion by dimension; that is, because it takes two units of motion to produce a differential of one unit of motion, the normal geometrical unit progression 2^1, 2^2, 2^3, becomes the tupule unit progression of motion, (2^1)^2, (2^2)^2, (2^3)^2.

Of course, we have to deal with the polarities eventually, but this analysis shows us that the absolute magnitudes, so-to-speak, of the n-dimensional velocity|energy poles, are a numerical progression equivalent to the n-dimensional geometric magnitudes, but in terms of the 2-unit tupule values of velocity|energy combined.

The important aspect of this analysis is the assertion that, while we normally cannot combine scalars and vectors mathematically, we can do it physically. In other words, the 3/4, 9/16, and 27/64 RNs, are the space/time ratios of the velocity and energy combo in one, two, and three dimensions respectively, on the velocity side of unity, and the RNs 4/3, 16/9, and 64/27, are the inverse time/space ratios on the energy side of unity.

If this is a valid analysis, then the velocity side magnitudes may be the fundamental magnitudes of the radiation|submatter|matter entities, and the inverse magnitudes on the energy side may be the anti-radiation|anti-submatter|anti-matter entities.

Notice that the anti-entity of the radiation entity, or 1D RN is not itself, just as the inverse, or mirror image of a 1D rod is not itself, but the inverses of these n-dimensional RNs are different; that is, the mirror images of these entities are distinguishable in terms of their polarities/inverse polarities.

If we can consistently express these n-dimensional tuples in terms of space/time progression ratios, so that the properties of given entities of radiation and matter, charge, spin and mass, etc, can be calculated from the value of these numbers and their possible combinations, imagine what that means.
TRoc
jjac,


QUOTE
"Encoding "resonant symmetry" in a formal system would be the next step ..."
"How to encode "resonant symmetry" in a formal system is the question..."
"What are the qualitative properties of "resonance"? "
"Assuming the element required for the formal systemization of "resonance" must exhibit the qualities of straightness with rotation, and incorporate all possible notes and scales, the question is how to model this concept."
"I believe that the quantitative properties of the SGS will decode using a novel mathematical term of measure (distance/rotational degrees) and commute with your "resonant symmetry" concept of Everything."
"Also, there would seem to be an equivalence between the distance-rotation inverse relationship of the SGS and that of the wavelength-frequency inverse relationship.
Distance : rotational degrees :: wavelength : frequency?"


First, I think that the term "resonance", as used in Science today, needs to be further defined, and added to. Everything will be resonant to a degree, or dissonant to a degree; really they are an inverse pair the same as mass and energy. A "pure" resonance, measured in terms of cycles (or 360 deg rotations), will have no wasted "partial" cycles; zero beat frequency allows for 100% energy transfer. This is duplicated geometrically in the right angled triangle. It (zero BF) can only happen with 2 identical values, or a value and double that quantity, as in the "traditional" definition of resonance, and with 3 (or more) values with the correct ratios to each other. It also has a distance or area constraint: the velocities, and vectors must be such that the values have enough time to interact to produce zero BF. The obvious limitation is that when the values no longer share the same space, they can no longer interact. The other side of that coin is that very special vectors (given = velocity as in EM waves) will allow for the "triangulation" resonance to occur, and alter said vectors into circular form, where they will remain. With a small enough radius, these values could not be separated in time or space (electron), yet with a larger radius, they could be momentarily separated into the 3 symmetrical values (quarks from proton/neutron). In either case, the angular momentum, or opposing vectors of the orbit cycles and direction of travel would combine to create some inertia to the initial, or preferred velocity of ©.

I think that modeling this with SGS, and using 1 rotation per hertz might be a little difficult? With that in mind, perhaps using 1/2 rotation in the SGS model per Hz x10^e3 would scale it down a bit, and still conform to resonant quantities.

The problem is that there is also a "drift" associated with these quantities in order for mirror symmetry to manifest. A good example is going from "visible" end quantity of 3.86^e-7 (just above violet): it drifts 5 columns over while increasing by an order of n x 10^e21 to end at 3.83^e14 (just below red). I'm not sure you can follow that without having my resonant matrix in front of you.


Also to Excal and StevenA: both of your last posts are very interesting, but I'm not sure where to go with them? But let's continue...


T.Roc


Excal
T.Roc:

The idea of discussing mathematical concepts in this thread is to off-load that part of the TOE construction project. However, the project is totally inactive. Realitycheck, in spite of many promises to the contrary, appears to have abandoned it entirely, while he engages in vitriolic exchanges over steel beams on a daily basis.

We have to decide if we are up to this challenge or not. If the author of it isn't, then I don't think there's much sense in continuing.

Excal
TRoc
Hi Excal,


I agree that the purpose of this thread is to off-load the sub-topic from the main TOE thread. However, my personal quest began long before that thread started, and will continue with or without it. I welcome bright, open minds, such as yours, into conversations because it helps me along the way.

I would really like to understand your POV better, and feel that I must generate a few "ah-has" in other people's minds to know that I have found the correct terminologies for my theory. I am still working on that.

We could just continue this conversation, along with Jal, StevenA, JJAC, and others, and if the main TOE thread never goes any further, would we be any worse off?

I don't really have the time to adopt the project, or even co-moderate. BTW, what happened to RC's co-moderator? Keep that in mind too; he may have felt secure in tackling this project with help, but now?? Apparently, RC felt the need to "defend Science" against would-be attackers, but I just go my way. Have you seen the next HOTTEST thread on this forum? The "plane on a conveyor" debate is not for me. The same philosophy for TV, being mean, drinking too much, etc. -- you just go your own way. Change the channel, smile, save the next glass of wine for tomorrow.

Also, my last thread was in hopes to get James (jjac) back into this; I was away for a week and unable to continue our conversation.


T.Roc

Excal
Hi TRoc,

It looks like RC is back now and determined to carry on. Here's my POV in a nutshell:

1) Understanding the true nature of space and time is the crucial task we face.

2) There are two choices: space/time is a background or it is not a background.

3) I choose not a background.

4) Without a background, 1D vector motion (a function of position on a background) cannot be defined, initially.

5) Therefore, we define a universal motion as the starting point and space and time are two, reciprocal, aspects of it; that is, we begin with a universal increase of space and time: two, reciprocal, increasing numbers.

6) The initial ratio of the two rates of increase is 1:1

7) A theory of change in this ratio, in specific instances, is what we investigate.

Fortunately, the nature of this investigation is mathematical, because, as it turns out, a little known feature of numbers has traditionally been overlooked that seems to make a revolutionary difference. This is the feature of mathematics that there are two interpretations of number possible. The first is the interpretation of number as a quantity, how much of something. It is called the "quantitative" interpretation. This is the traditional interpretation of mathematicians.

The second interpretation of number is the "operational" interpretation. It is the value of the relation of one number to another. Hence, the value of n/n, for instance is zero, because there is zero imbalance between the two numbers, but a non-zero imbalance, n/m, or m/n, can exist on either side of zero, in two "directions" we might say.

The first possible non-zero value of n/m (m/n) is 1, as in 1/2, or 2/1, but n and m can be any numbers with a difference of 1: 11/12, 10001/10000, etc. The value of the number is the difference between the two numbers that comprise it.

Thus, the system of signed integers, -n..., -1, 0, 1...n, is equivalent to n/m...0...m/n. Using this operational interpretation of number constitutes a huge change in traditional mathematics, because it means that we don't have to invent an imaginary number, i, to get a number that squares to a negative number. In traditional mathematics, -n * -n = n^2, but this is due to the quantitative interpretation of number. In the operational interpretation, n/m * n/m = (n/m)^2, which is a negative number. Hence, no imaginary numbers are needed, not the imaginary number 'i' of complex numbers, or the imaginary numbers 'i', 'j' and 'k' of quaternions.

In fact, instead of having the traditional system of numbers, the reals, complexes, quaternions and octonions (two sets of quaternions), based on the quantitative interpretation of number, we have a new system of numbers consisting of unipoles, dipoles, quadrupoles and octopoles (two sets of quadrupoles), based on the operational interpretation of number. In this new system of mathematics, called the reciprocal system of mathematics, RSM, the unipole is a single ratio of two natural numbers interpreted as a quantity, and it can take the value of any number in the system of signed integers, either a negative (n/m), or a positive (m/n) value.

The dipole is the combination of two unipoles of opposite "direction;" that is, it is the combination of a negative and positive unipole,

[(n/m) + (m/n)].

A quadrupole is the product of two dipoles,

(a + cool.gif * (c + d) = (ac + bc + ad + bd),

where a, b, c, and d are unipoles. An octopole is a combination of two quadrapoles that are the product of a quadrupole and a dipole,

(ac + bc + ad + bd) * (e + f) = (ace + bce + ade + bde) + (acf + bcf + adf + bdf).

Significantly, since all these numbers are composed of reciprocal real numbers, operationally interpretated, all algebraic properties hold for them and thus their algebra is completely commutative and associative, unlike the traditional system in which quaternions are non-commutative and the octonions are non-commutative and non-associative.

It may not be immediately clear what all this means, but, believe me, it's revolutionary, because this simple change in perspective changes, in turn, the foundation of modern science and technology. Where would science and engineering be without complex numbers? All of electronics is based on the complex plane. The vector of vector algebra would not exist without it, and we cannot even describe quantum mechanics without complex numbers.

The implication is that physicists cannot afford to leave mathematics of physics to mathematicians. The mathematics of physics must be guided by physical principles. Unless it is, theory can become unconstrained mathematics, just as we see it has in string theory today. The illustrative example that we have before us now demonstrates this conclusively. The principle of reciprocity and symmetry, first found to be so powerful in physics, is now found to have the same power in mathematics.

Our work is cut out for us.

Excal
Montec
Hi Excal

Could "RSM" be equivalent to vector math that is bound to one dimension?

You have both magnitude and direction but the direction is limited to "plus" or "minus", "up" or "down", etc.

smile.gif

Excal
Hi Montec,

I don't know anything about "vector math" as you describe it. Of course, a vector is always bound to one-dimension in a complex plane. The resultant of any number of vectors is always a one-dimensional vector. Limiting the vectors to orthogonal directions might be a special case of this.

However, the numbers of the RSM are n-dimensional numbers in the mathematical sense, just as polynomials are n-dimensional; that is, each term is a dimension. However, they are also n-dimensional in the geometric sense; that is, a unipole is a zero-dimensional number of one term, a dipole is one-dimensional number of two terms, a quadrupole is two-dimensional number of four terms, and an octopole is a three-dimensional number of eight terms.

This is the pattern of Pascal's triangle, the binomial expansion:

2^0 = 1 = 1
2^1 = 11 = 2
2^2 = 121 = 4
2^3 = 1331 = 8

Let me add that the numbers of these first four layers of Pascal's triangle were regarded by the ancients as sacred. The question is, "why the number 2?" I think its because a quantity, a scalar, can only increase or decrease; that is, there are no negative quantities. We can't have negative things, like cabbages. We can only have more or less of something, but which is more and which is less depends on the point of view one takes. Thus, one number can be both more and less than another.

These two "directions" are actually a property of numbers and that's why it is so important. If we take the operational interpretation of number, then the number 2 raised to a power would actually become (2/2)^n, which can have several different values, not just 2^n. On this basis, we could rewrite the expansion as follows:

(2/2)^0 = 1/1
(2/2)^1 = 2/2
(2/2)^2 = 4/4
(2/2)^3 = 8/8,

and now we can interpret it as

(2/2)^0 = (1/1) = 0
(2/2)^1 = (2/2) = 1
(2/2)^2 = (2/2) (2/2) = 2
(2/2)^3 = (2/2) (2/2) (2/2) = 3

from this we can see that 1/1 has no degrees of freedom, because it's not 2/2. Therefore, it has no direction. However, (2/2), on the other hand, has one degree of freedom. The two can be decreased to one in either the denominator or the numerator, creating the possibility of an increase/decrease, or one "direction." Clearly, the 1/2 is a value that is opposite in "direction" from the 2/1 value.

The next number, 4/4, has this two of these bidirectional entities,

(4/4) = (2/2) (2/2),

and the final number, 8/8, has three of them

(8/8) = (2/2) (2/2) (2/2).

In each case, however, the entity (2/2) has a total of three possibilites: the two "directions" or no "direction" (1/2, 2/2, 2/1). Therefore, we can also write the expansion as

(2/2)^0 = (1/1) = 0 = 1
(2/2)^1 = (2/2) = 1 = 3
(2/2)^2 = (2/2) (2/2) = 2 = 9
(2/2)^3 = (2/2) (2/2) (2/2) = 3 = 27

Which is to say that, with no degrees of freedom, only one value is possible, but, with one degree of freedom (DOF), three distinct values are possible, corresponding to (-1, 0, +1), and, with two DOF, nine distinct values are possible, corresponding to

__|+1|
-1| 0 |+1
__|-1|

With three DOF, twenty-seven distinct values are possible, but I won't try to write them in ascii. The important thing is that these "directions" and values are natural properties of the operationally interpreted numbers, not arbitrarily assigned properties. Given the number 2/2, these properties must follow, as the dimensions are increased from zero to three.

This is just huge.

Excal




TRoc
Excal, Montec


I agree with your 7 points from the prior to last post.

The one question I have is what would happen if you "quantisized" your ratio? Especially with an irrational?

1:1.05946

giving 13 equal steps in order to get to 1:2


This would seem to "marry" our conceptual approaches.


T.Roc

Excal
TRoc,

Of course, mathematically you could do it, but physically I don't know how it could be done. Kronecker said, "God made the integers, all else is the work of man." I first heard about him from a talk Atiyah gave at Santa Barbara last year wherein he quoted him, but Atiyah said that he didn't go quite that far himself, which intrigued me, because a statement like that indicates a controversy. It took me a while to get around to looking into it, but recently I noticed that the title of Hawking's new book is taken from this observation of Kronecker's. I searched the book in vain, however, looking for an explanation.

So I decided to search the Internet. I found out that Kronecker hated irrational and transcendental numbers, following Pythagoras who called them "unutterable," and I guess there's a real drama to the story of these numbers that we could make into a TV show, called "Real or unreal" (see: http://everything2.com/index.pl?node_id=1548022).

Apparently, Kronecker's position was considered extreme and unrealistic for a long time, but now mathematicians and physicists are relunctantly coming to the same conclusion: Ultimately, irrational numbers are physically unreal. Hence, if we assume that space and time are a quantized progression, the fundamental progression ratio must be an integer ratio, if it is to be regarded as something that actually exists.

Excal
TRoc
Excal,


I'm confused on the "..physically I don't know how it could be done" part. Are not mathematics just a systematic form of communication to convey quantity? If we go to the other side of the coin, is the quantitative difference between an integer and a same bodied irrational qualitatively real?

Would we notice if I had 1 orange, and you had <1.05946 ? (same body example)

Would an electron encountering 2 eV do anything really different than one being hit with <2.11892 eV ?

Really, I think that I take the opposite stance from the one you mentioned from Kronecker.

If everything in our Universe is interconnected, through the workings of the electrons' interaction with space, time, or other particles, then is it possible to describe an interaction with the quantity of One?

This is similar to my thoughts on Zero: there is not a situation that can be contrived where, from some frame of reference, Nothing (or zero) exists. It could always be described as an equilibrium between two entities. (-1, 0,+1)

If you follow/believe that, then where does that leave One? In the same boat, from my perspective. You need the second point, second electron, second person, to describe/communicate anything. Our consciousness throws a wrench into this perspective, because we think before we speak, and hear our own voice, as it is sent out carrying the information. Is talking to yourself real communication?

Of course, this is philosophy; I don't want the inconvenience of ONLY having irrationals to compute quantity. That would be, ... well ... irrational. tongue.gif

However, in searching for alternate views in which to re-evaluate the world that we live in, and Science has yet to properly, and completely describe/define that world, it is worth the effort. As you said, " ..this simple change in perspective changes, in turn, the foundation of modern science and technology." Your ratio system is quite useless with just One thing as well.

In the end, we view time as linear, and real, because that is the way we live and die. With a linear system of quantity, how can we expect anything different? 1 through 9, then add a digit, and keep going. Easy? Yes. Reflective of the Universe? Producing natural resonant quantities? No.

My take on One is: it is a harmonic of Two. Sounds simple enough, maybe even harmless. It will form Pascals triangle going up. It will form Pythagoras ratios going down. Now for the catch: you can't produce the quantity of Three without irrationals (in this system). Three is just FAR too important in the scheme of things. No triangles, no chords, no circle (3-point-irrational), no third party observer to describe an interaction between 2 things, even the fruit of reproduction can not exist without Three.

Three is a higher harmonic of 1.5 ; that quantity, of course, resides between One and Two. The question is, how does the quantity "grow" from One to Two? When I suggest "quantization", everyone will think that I am borrowing that from modern Physics, but this is NOT true. The quantization of waves occurred BEFORE QM; in fact very close to Newton's time. It was "hidden" in the "ad hoc" postulates of musical theory, and I have lifted that veil. One can make a good argument that Pythagoras also "quantized" these resonances, but did so in great fear of the "irrational" numbers. The name conveys the emotion attached.

In the end, I am using integers as the qualitative, represented by steps of discreet quantities. This is necessary to produce the spectra of sound and of light, as well as the product of © using symmetric values from the quantitative background. Non-symmetric values result in inertia to the natural product.

These are very large steps towards our goal; I hope that I can interest you in this idea.


T.Roc

TRoc
Hi Excal,


I found a chart that saves me from doing a large amount of math to make my point on the convenience of the irrational quantum over ratios of integers.

Again, integers are used here in steps, giving the ease of use that our minds prefer. But nothing else lines up with the frequency pattern like the 12th root of 2.


Note__Ratio___Cents__ et-Cents__ Interval

Eb__256:243__90.22___100___minor second
Bb__128:81__792.18___800___minor sixth
F___32:27___294.13___300___minor third
C____16:9___996.09__1000__minor seventh
G____4:3____498.04___500___perfect fourth
D____1:1______0_______0______ unison
A____3:2____701.96___700___perfect fifth
E____9:8____203.91___200___major second
B___27:16___905.87___900___major sixth
F#__81:64___407.82___400___major third
C#_243:128_1109.78__1100___major seventh
G#_729:512__611.73___600___augmented fourth
[D#][2187:2048][113.69][100] [augmented unison]

This can be found, along with further discussion, at Wikipedia
http://en.wikipedia.org/wiki/Pythagorean_tuning


The system I am advocating uses 12 equal parts, and 13 equal steps to the octave.


T.Roc

Excal
QUOTE ("TRoc"+)
'm confused on the "..physically I don't know how it could be done" part. Are not mathematics just a systematic form of communication to convey quantity? If we go to the other side of the coin, is the quantitative difference between an integer and a same bodied irrational qualitatively real?

Would we notice if I had 1 orange, and you had <1.05946 ? (same body example)

Would an electron encountering 2 eV do anything really different than one being hit with <2.11892 eV ?

Really, I think that I take the opposite stance from the one you mentioned from Kronecker.


The position that mathematics is more than "a systematic form of communication to convey quantity" is at the heart of much philosophical debate. I believe that mathematics is a form of truth, and the "unreasonable effectiveness of mathematics in physics" is not unreasonable at all when we understand this.

Irrational numbers are certainly part of this truth, but because they emerge from the rational numbers, not because they are fundamental. This is probably closer to Kronecker's own position. If we quantize motion, then we have to start with integers, because whatever irrational number might otherwise be selected as the fundamental unit of space or time, becomes an integer by definition; that is, if one or the other unit, say time, is 1.05946, then whatever the other is, they relate as units.

For instance, say that the unit of time is assumed to be 1.520655 x 10^-16 seconds and the unit of space is 4.558816 x 10^6 centimeters, two irrational numbers, yet the 1/1, or 1/2, or 2/1 relation of these units in a progression is a rational number, expressing units of speed. So it's the units of speed that are rational, while the corresponding units of time and space are actually irrational.

This means that we may be able to have our cake and eat it too. The important thing is that we have a reciprocal number system that corresponds to our reciprocal physical system. What we have to do is see how the number system evolves as the physical principles constrain it. Right now, the big physical constraint is that time is always unidirectional in space and vice-versa.

If we take the first four numbers as number subsystems in the traditional number system

1
11
121
1331

and replace them with four reciprocal number subsystems in the reciprocal number system, we get

1/1
1/1 2/2 1/1
1/1 3/3 3/3 1/1

but we know that these can be expressed in terms of "poles," monopoles

(m/n, or n/m, or n/n),

dipoles

(m/n + n/m),

formed from two monopoles,

quadrupoles

((m/n + n/m) + (m/n + n/m)),

formed from two dipoles,

and octopoles

(((m/n + n/m) + (m/n + n/m)) + ((m/n + n/m) + (m/n + n/m))),

formed from two quadrupoles. Thus,

1/1 = 1mp
1/1 1/1 = 1mp + 1dp ~ 3mp
1/1 2/2 1/1 = 1mp + 2dp + 1qp ~ 9mp ~ 4dp
1/1 3/3 3/3 1/1 = 1mp + 3dp + 3qp + 1op ~ 27mp ~ 8dp ~ 2qp

where mp, dp, qp, and op are monopole, dipole, quadrupole, and octopole respectively. Hence, we have all the reciprocal numbers necessary to make any conceivable n-dimensional reciprocal number by combining these basic n-dimensional reciprocal numbers.

Interestingly enough, if we break the periodic table of elements into groups of quantitites, we see that the first group contains four quantities in four subqroups (4x1^2 = 4)

a = 1
b = 1
c = 1
d = 1

while the next group contains sixteen quantities in four subgroups (4x2^2 = 16)

e = 4
f = 4
g = 4
h = 4

The next group contains thirty-six quantities in four subgroups (4x3^2 = 36)

i = 9
j = 9
k = 9
l = 9

and the last group contains sixty-four quantities in four subgroups (4x4^2 = 64)

m = 16
n = 16
o = 16
p = 16

Since the square of a reciprocal number is a quadrupole, this would indicate that these quantities might correspond to four times 1qp, four times 2qp, four times 3qp, and four times 4qp, or in terms of qp

a = 1qp
b = 1qp
c = 1qp
d = 1qp

e = 2qp
f = 2qp
g = 2qp
h = 2qp

i = 3qp
j = 3qp
k = 3qp
l = 3qp

m = 4qp
n = 4qp
o = 4qp
p = 4qp

that is, each element contitutes a magnitude that is part of a sequence of quadrupole numbers. However, since a quadrupole is a combination of two dipoles, and a dipole is a combination of two monopoles and octopoles consist of two quadrapoles, each of the magnitudes corresponding to a given element should also be equivalent to some number of these various poles, as a single, n-dimensional, reciprocal number.

If this is true, maybe we only need to find the right sequence to fit it all together.
Excal
QUOTE ("TRoc"+)
Again, integers are used here in steps, giving the ease of use that our minds prefer. But nothing else lines up with the frequency pattern like the 12th root of 2.


I see what you're saying. Interesting. Let me get back to you on it.

Excal
TRoc
Excal,


Just to be sure, I am not saying "either/or" here. I think I mentioned before that we are talking about 2 sides of the same coin; "same coin" being the key words.

The label of the irrational is an integer; so we can communicate "movement" through a system with these easy values.

What I am also curious about is, what are you describing, in Physics terms, by your reciprocal number system?

For clarity, I am using my matrix to describe resonant interactions between values in "frequency"; which can then be translated into mass, energy, etc.

So, mainly this is a "scalar" system, because direction coordinates don't come into play, except for indirectly determining interaction times of waves or wave-centers (mass) that are NOT traveling on the same heading. In that spherical idea, you only need 1/4 of the circle, and then rotational & mirror symmetry takes care of the rest. Essentially, there is a reaction equivalence time for 22.5, 45, 90, and their opposites. You end up with 4 groups of "triads"; this is why I am convinced of the correctness of the 12th root of 2: not only does it produce the resonant ratios by a discreet quantum irrational, but it translates into spherical coordinates as well. It does both with the MINIMUM complexity. There are other roots that will produce the resonant ratios, but at the expense of either simplicity, or thoroughness. AE's "..as simple as possible, but no more.." Historically, the octave (8 steps/7 parts) worked well for single instruments, but the thoroughness of the harpsichord & piano demanded more. Expanding on this, Newtons 7 part rainbow/spectrum had the same limitation. It leaves a few unanswered questions as to the production of magenta, and of its' opposite, green: both have unique properties in the spectrum.


Regards,

T.Roc

Guest_Excal
TRoc,

We assume that the sole component of the universe is a universal motion. Then, space and time are simply its two, reciprocal aspects. Of course, being motion, we know its equation:

motion = delta space/delta time, or

m = ds/dt, which is usually written with the deltas understood,

m = s/t. So, the unit ratio of this motion is an increase of one unit of space for every unit of time, or

s/t = 1/1. (think of clock space together with clock time as two reciprocal aspects of clock motion, forming a unit progression)

The only thing that can change this unit ratio of the progression is a reversal in the "direction" of one or the other of the two aspects; that is, instead of an unending increase, one of the two aspects begins to alternate from increase to decrease to increase, etc. In the case of the progression of the space aspect alternating "directions" like this, the unit progression ratio is changed from 1/1 to 1/2. If it's the time aspect reversing "directions," ratio is changed to 2/1.

These values can then be combined, forming dipoles, quadrupoles, and octoples as explained earlier. The thing is, they are 3D vibrations, or resonances. Thus, a dipole is a chord of two tones. A quadrupole is a chord of four tones, and an octopole is a combination of two chords of four tones each. However, because the fact that time cannot reverse in space, the chords are altered.

For instance, the four tones in the fundamental quadrupole (quadrutone?) are

1/4, 2/4, 2/4, 4/4,

which is altogether equivalent to 9/16. The next one is formed by adding a 1/2 monopole to the velocity side of the first dipole ((1/2 + 1/2)+ 2/2) = (2/4 + 2/2), which, when squared, changes the quadrupole to

4/16, 4/8, 4/8, 4/4 = 16/36.

Now, the question is, "how do we interpret this?" The ratios are actually unchanged,

s/t = 4/16 = 1/4 and
s/t = 4/8 = 1/2,

but the time-displacement, that is the difference between 1 and 16 as well as 4 and 8, are very different from the time-displacement of 1/4 = 3 and 2/4 = 2; that is, the intensity has changed, because

s/t = 4/16 = 4/4 * 1/4 and
s/t = 4/8 = 2/2 * 2/4.

Thus, the intensity of the first tone quadrupuled and the intensity of the second and third tones doubled. Now, what happens if we add a second monopole to the energy side of the equation? We get

(2/4 + (2/2 + 2/2)) = (2/4 + 4/4),

balancing it out again (remember, time doesn't reverse on velocity side, so 2/1 becomes 2/2, even though it is actually 2/1 on the energy side that is added to the combo). So, squaring this now, we get

4/16, 8/16, 8/16, 16/16 = 36/64,

which, again, only changes the intensities, not the frequencies. However, the intensity of the first term is left unchanged, while the intensities of the second and third resonances are doubled again. But when we look at the frequencies of the three quadrupoles, there is a slight dissonance between the first and second and the second and third, while the first and the third are identical tones, though the intensity of the third is four times that of the first:

9/16 = .5625
16/36 = .4444
36/64 = (4/4 * 9/16) = .5625

Notice, that the second is slightly off the double the first: 2/2(9/16) = 18/32, as compared to 16/36. So, although we are far from being able to identify the magnitudes of the n-dimensional RNs with physical properties at this point, we can see that both the intensity and the frequency of their constituent harmonics vary according to a straightforward addition of integers. Specifically, adding monopoles to dipoles changes both the freqency and intensity of these sums in a predictable manner, but we could ever guess that the outcomes would be so irrational.

Analyzing these and similar, but more complex, results, in terms of octaves and scales that might compare to audio and optical frequencies is yet to be done. The major task now is to examine these numbers carefully, to ensure that I'm not just a crazy old man. Does this stuff really make sense, or have I made a foolish error somewhere along the line that makes it all nonsense?

Warm regards,

Excal



TRoc
Excal,


QUOTE
We assume that the sole component of the universe is a universal motion. Then, space and time are simply its two, reciprocal aspects. Of course, being motion, we know its equation:

motion = delta space/delta time, or

m = ds/dt, which is usually written with the deltas understood,

m = s/t. So, the unit ratio of this motion is an increase of one unit of space for every unit of time, or

s/t = 1/1.



That is the logical root of the ratio perspective.

The next thing that comes to mind, and being fundamental to Physics, is the product of the terms.

Now we are talking VELOCITY. This is a constant in the system of EM. All products (wavelength x frequency) MUST = 299,792,458 . There are a few important points that relate to our discussion.

1. Rarely, if ever, will the F or W be an integer. This is behind my thoughts on putting irrationals on equal footing with the integers.

2. Normal mathematics seem to me to play a secondary role to symmetry here. If you organize a system of quantities, you will have a very hard time coming up with something that ALWAYS has the product of ©. A series of integers WILL NOT do it; the "plus one" mathematics of our lineal 1-9 set does not work here.

3. There is a second version of determining these quantities, that pre-dates Physics. It is purely biological: one set is a spectrum of frequencies that would produce the rainbow, or fundamental colors that our eyes perceive. The other is a set of frequencies that would produce the octave, or scale of independent sounds (A to G notes)


Although I frequently use musical terms, it is only because there has been more (even though "ad hoc") work done in that area.

My main standing points are RESONANCE, AUTOPOIESIS / QUANTUM, and VELOCITY.

Asking the question: "What 'discreet quantum', or specific quantitative step, infinitely repeated, would produce resonance, the biological perceptions of discreet color and sound, and a constant velocity of 299,792,458?"

As far as I can tell, there is only one value that can do this, and I have determined it.

There is not a lot more analysis to be done. Any change in symmetry results in a change in velocity, and that in return, will create a "drag" on the preferred velocity that we label "MASS".


Regards,

T.Roc

TRoc
Excal,


Just for a refresher, go back to page 2 of this thread for instructions on how to build my resonance matrix of values. matrix

We can then discuss integer steps on this background matrix of resonant values.

I can not put this on this forum because it is too large, and the symmetry would not show if done partially. I can put a condensed version up, but the accuracy is lost without the length (digits) of each number.


T.Roc



Why Not?
Hey TRoc,

As I stated in the Puzzling questions forum, I have created your matrix - based on the instruction provided on page 2 of this thread. (For anyone else reading this, I recommend you do the same).

I was wondering how you derived the CENTER of you matrix? I understand the frequency/wavelength quadrants and the role of i. I understand, but I cannot figure out how you found the center by anything other than trail and error while looking for c as the product of the (x,y) and (-x,-y) and/or (x,-y) and (-x,y). The vertical axis is intuitive, it's the horizontal that is giving me problems.

TRoc
Greetings why not,


If you have an intuitive picture of the vertical axis, but not the horizontal, I am going to assume that you did not make the matrix to the MINIMUM size that I recommended. Of course, that assumption could be wrong, but I will explain nevertheless.


If you take the values at the 2 rows that begin with 2.88_e14, and 5.77_e14, you will "see" the visible spectrum values in frequency. The first thing that I did was to color the appropriate values with art pencils; this helps the instinct work faster.

Now, take the values at the 2 rows starting with 2.44_e-7, and 4.89_e-7, and do the same thing.

Immediately, you will notice a few things:

1. The inverse relationship of frequency & wavelength shows up here as right-to-left vs. left-to-right.
2. When you make a line from any color, from f to w, it crosses in the same area. This shows the rotational, or mirror symmetry.
3. There are 70 rows between the visible bands as measured in f & w.
4. The area that the symmetries cross over is, therefore, 35 rows from either band.

That is how the horizontal center was derived. In fact, I already had a line drawn there because I use the matrix for a non-linear vibrational spectrum chart, similar to the one found in Physics texts, showing the EM spectrum linearly. That line marks the end of effective radio frequencies for Earth bound transmissions. (between VLF & ULF)

Very interestingly, this line is the same line that caps the range for human hearing. This is another reason why I prefer to include the biological evolution of our sensory range into the spectrum, as well as the evolution of wave resonance. After all, it is capped by the “visible light” measurements, and sandwiched in the middle, is our hearing bands, and neurological levels of consciousness. This last part I don’t think I’ve mentioned here yet, because it is hard enough to get people to see what is being conveyed here, without bringing the subject of consciousness up.


I did not add the “position coordinates” until later (x,y), so you are right about that I would have had to do some serious trial & error to get to the center of such a body of quantities. I was fortunate that Mother Nature had already done most of that for me.


T.Roc

Why Not?
Hey TRoc,

ohmy.gif I am kicking myself in the head. I built your matrix as specified. But keep in mind, you gave us the CENTER and you have stressed the importance of the mirror and rotational symmetries "built into" your matrix. In fact, the first products I derived were the diagonal corners at the CENTER and then the diagonal corners at the corners, just to make sure I had not screwed up any equations. They equaled ~299,792,458 (to fourteen significant figures, I am at 299,794,457.995931 w/ R=1.02670929945999). My task was to figure out how (mathematically) your matrix worked and then deduce your rationale in it's creation. I never gave a thought that you graphed it! Gotta love experimentation!

By the way, I had deduced the vertical access before I created your matrix. Based on the explanations you've posted, it is easy to visualize the printed matrix, rolled into cylinder, with columns 1 and 13 overlapping with a 1 row displacement. You create a split between columns 1 and 2 and rotate 180 degrees. Using the same logic, I guessed that you assumed 13 steps down (count "1" as you step on row 2) and 180 degrees rotation. Or thirteen steps from the creation of the base, which began with ~1. How's that!?!

To move on... I have a question about your aversion to zero (as a quantity). I understand your rationale, but I speculate that the more up rows you create (with larger and larger significant figures) the closer you get to zero. And, of course, you approach infinity in the down direction. I my opinion, both zero and infinity have their place (or else all that calculus is for naught!) and I see them at the vertical poles of your helix. Thought?

Why Not?
Hey again TRoc, et al,

Have you, by chance, read this article?

Variable Constants

If true, it would have very interesting consequences for you model....

As well as any TOE mathematic/number theory.
TRoc
Why Not,


I am not against the use of zero in general, or mathematically. Without going into deep philosophy, it is just in the terms of vibration: there is no zero in a cyclical model.


I did read that article. I don't think that it is enough to cause any problems though.


T.Roc

David Argall
Sorry to possibly intrude here, but ...

Some years ago, I ran into [& possibly misremember] a reference to some theories that all of math is statistical, that 1+1 does not in fact =2, but it merely statistically averages out to that, and since huge numbers of dice are being "rolled", the real world operates as if it were an absolute law.

Much information on this or other theories questioning 1+1=2 likely falls under the law of turtles ["This book tells me more about turtles than I want to know."], but I do need some information on the point, and an idea of where to go to get more.

Thanks for your assistance.

David Argall
PhysOrg scientific forums are totally dedicated to science, physics, and technology. Besides topical forums such as nanotechnology, quantum physics, silicon and III-V technology, applied physics, materials, space and others, you can also join our news and publications discussions. We also provide an off-topic forum category. If you need specific help on a scientific problem or have a question related to physics or technology, visit the PhysOrg Forums. Here you’ll find experts from various fields online every day.
To quit out of "lo-fi" mode and return to the regular forums, please click here.