ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Thursday, December 28, 2006

Bach's seal











Isn't it gorgeous? It has few aspects that I wanted to share with the cyber space.

Mainly it is a representation of mediated duality. Two sides reflect each other and they also merge in the middle to make the mediator. Like (-1,0,+1) and (black,grey,white). Bach is giving the second most outstanding aspect of existence, mediated duality. Unity is the first aspect.

The crown on the mediator has the numbers 4, 5 and 7 coded in a symmetry like Menorah's, the Jewish candle holder icon. There are 5 dashes and 7 knobs on the crown. The center and side knobs are specially capped, leaving 4 not capped. Is Bach giving a clue here about how to link two different symmetries? 4 is also visible in the mediator's inner square divided into 4. 5 is the number of senses, with 4 of them {taste,smell,hear,see} being mediated by {touch}. Thus Bach also gives the third aspect, mediated dual-duality. The reflected JSB on both sides also reveal mediated dualities each, S being the mediator of J and B. The universe is also built on 5 forces, {strong,weak,higgs,light,gravity}. Complex numbers have the same mediated dual-duality symmetry, (-1,+1,0,-i,+i). And as we know, complex numbers are closed.

The symmetry that Bach mastered is the symmetry of music, life and the universe. All music obey the same symmetry, but Bach knew what he was doing, not only feel. What a man. He united the western music with his 48 preludes and fugues. 48=2*24, 4!=24. When the symmetry of the universe is ever found, it may confirm Bach. Bach united music and I think also showed the way forward for science. He heard the sound of unification in his heart and mind in balance, a testament to power of good music. In doing so, he also showed the way for happiness; balance of mind and heart. It is a miracle that he ever existed.

(The seal is from Wikipedia)

...

EDIT 6/4/2013 Check this out! The expected value of Hamiltonian in Dirac formalism.


Wednesday, December 06, 2006

Sex and time

The family tree of a person is like a cone fractal infinitely expanding into past. A person in the past has both sides of the cone until the present. Special relativity defines a light cone for every locality. Sexual reproduction forms the backbone of life's space-time. Every sexual encounter that result in fertilisation forms a node in life's matrix. From this node a being emerges and goes on to make own nodes in the matrix. The distance travelled from one node to the next has a statistical spread. There would be an average speed of gene transmission into the future in this matrix, like speed of light. When looked overall, it looks like a time-cone.

Sexual dynamics may be a clue to quantum mechanics of space-time. What else for life to imitate, apart from space-time itself? Time, light and gravity are intimately linked. Light is like male, gravity like female. Light brings life to matter. Life make space by feeding and doing the chores of life. The locality defined by a life transmits itself through sex to other nodes.

Living make space, fucking make time.

Sunday, November 26, 2006

Balance

{Balance}

Balance is the middle touch and the core of being. It defines one's location in time and space. Balance in space seems to have more priority over balance in time. Evolution clearly displays the importance of territory. In physics, location (geometry) forms the backbone.

For example, hurricane takes it's vortex shape when a highly excited unstable locality brakes symmetry by collapsing and channeling energy in one angular direction. This direction is determined by the rotation of the earth. Spontaneous symmetry breaking is the mechanism used to make balance and define locality.

{OuterTouch, Balance, InnerTouch}

Balance uses inner and outer touch as its agents, when it evolves and differenciates more. Outer touch is the known touch, the screened interaction. Inner touch happens when allowed energy focus in the center. Brain performs the balancing function between these two touches. It mediates by correlating the inner to the outer.

Like reflection and refraction of light at a boundary. The boundary is closed for a being and the allowed energy makes patterns inside, creating an image of the outside. Energy comes in two modes; acoustic and electromagnetic or light. Acoustic energy gives an instant overview of the mechanics of the locality, if it is processed by an algorithm like FFT. Light energy is directional and provides 2D information. Acoustics is like gravity, globally felt. The inner brain develops ways to detect this dual energy pattern. Inner hearing and inner vision would evolve first. And then organs would evolve as the brain grows and differenciates.

{{Taste, Touch, Smell}, Balance, {Hear, InnerTouch, See}}

Then a similar splitting happens for both touch and inner touch, as the organism gets more complex and finds it's fundamental dual duality shape of the outer look. {Taste, See} and {Smell, Hear} correspondence demonstrates the level of correlation. See and Taste form the inner or local, while Smell and Hear form the outer territory. Dogs use smell and hear as the agents for defining outer territory.

Taste --> {Mouth, Stomach, Anus}
Smell --> {Nose, Lungs ? }
Hear --> {LeftEar, InnerEar, RightEar}
See --> {LeftEye, InnerEye, RightEye}

Then each sense splits to create it's organs. When organs split, they leave the boson of the relationship in the center as the mediator. InnerEar combines two sides to make a stereo image like model of the sound-scape outside. Duality in hearing gives depth, like also in seeing. I am not so sure about the nose-lung connection, in comparision to mouth-stomach. Maybe nose took dual role of mouth and anus at the same time, since it is only air.

{{Strong, Space, Weak}, Higgs, {Gravity, Time, Light}}

The energy expressed by gravity balances that of light. This is known but not integrated into physics properly yet. Potential/kinetic duality of energy comes from this. I suspect a similar thing is happening with {Strong, Weak}. {Light, Weak} and {Gravity, Strong} correlate strongly. The 3*2 symmetry of space is closely related to 3/2 symmetry of Strong. Higgs forms the background field which supports everything. Space is inner touch, time is outer touch.

Light --> {+ECharge, Light, -ECharge}
Strong --> {3 Color Charges, 3 Color Gluons}
Weak --> {W+, Z0, W-}

Then each field splits similarly to form an expanding tree, forming own charges. Strong seems to be like Taste. It branches deeper. Taste has mouth, stomach, anus, liver?, intestines...

Sunday, November 12, 2006

Being, music and universe

ATON-M6 classification thus meets a serious challenge from a modular group like symmetry, {{a,b},{c,d}}, which is also in line with the new multifactorials and Zeta function formulation.











================================











==================================

Time/Space takes the center stage
as Not-Doing/Not-Being.









=================================

A typical cycle of action;
{Feed,Feel,Think,Judge}
displays the 4/4 measure of being
as also done by musical conductors.
Christians do the same sign.





===================================

Do/Be expanded to reveal
a model of nothingness or slumber.
My kind of model this one.

Mirror Set, rev-2

After discovering the multifactorials, or Barnes G-functions, a revision is necessary since G-functions are more general. Mirror Set maps pairs of naturals to rationals;





Where m,n are natural, G(m,n)=m!n is multifactorial of m to depth n. Revising and broadening Mirror Set previously defined as M(m)=M(m,2). This form is more general and combines G-functions and Zeta function. It has a ying-yang look and variables do not commute, G(m,n)=/G(n,m). Here are some small values;

M(1,1-4) = (1, 1/2, 1/3, 1/4)
M(2,1-4) = (2, 1/2, 2/9, 1/8)
M(3,1-4) = (6, 3/2, 8/9, 3/4)
M(4,1-4) = (24, 18, 4^4/3, 6^4)

Previously I took M(3,2) as the center. Now M(2,2) =2/4=1/2 takes the center, standing for time/space. {2,2/9}={light,gravity}, why 2/9 for gravity? (3/2,1/2}={strong,weak} is reasonable.

Applying continuity by setting m-->z and summing gives a harmonically weighed G-function. For G=1 it is the Zeta function;






Assuming continuity holds, G may be generalised to a function of 2 complex variables;






Where z!w stands for the multifactorial to depth w, G(z,w). The form of M seems to imply the extension of G but it remains to be seen if it can be made continuous. I think G works like a local correlation function.

Thus naturals take the center stage of ATON, instead of sets. Maybe not surprising. Don't they (mathematicians) consider number theory as the purest?

Saturday, November 11, 2006

Multifactorials

Lately I found, to my great surprise, that ATON Exponential Set is in fact already defined in the number domain as the second multifactorial, also called Barnes G-function;

G2(z+1)=Gamma(z)*G2(z)
G2(1)=1
Gamma(z+1)=z*Gamma(z)

Higher multifactorials follow. To see how it works, expand 3 to depth 6;

G0(3) = 3 = 3*2^0
G1(3)=3.2.1.1 = 6 = 3*2^1
G2(3)=(3.2.1.1).(2.1.1).(1.1).(1) = 12 = 3*2^2
G3(3)=[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)] = 24 = 3*2^3

G4(3)={[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]} = 48 = 3*2^4

G5(3)={{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.

.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}} = 96 = 3*2^5

G6(3)={{{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.
.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1)]}}} = 192 = 3*2^6

Therefore Gn(3)=3*2^n.

Instead of multiplying, one could add the expanded elements (with no apparent benefits);

GS0(3)=3
GS1(3)=7
GS2(3)=14
GS3(3)=25
GS4(3)=41
GS5(3)=63
GS6(3)=92

Thursday, October 26, 2006

Future >= Present/Past










The uncertainty principle (by Heisenberg) is expressed by a dual duality in physics,
D={{P,M},{T,E}} = {{Position,Momentum},{Time,Energy}}
It states that the uncertainty in the measurement of "coupled" entities is always larger than a cetain limit, Planck's constant divided by 4pi. Momentum is also a function of position, and energy is also a function of time.

It is a statement of prediction. It predicts that nothing can be measured beyond a certain limit, with the available means of D. It applies to every measurement process, therefore to everything. For example, measurement of pitch in sound is optimised by balancing accuracy (the length of the buffer (window)) with timeliness. Speech recognition and synthesis is an event that has to happen in real time. Therefore parameters like pitch are measured in flight. Longer windows give more accurate measurement of pitch, but the location is lost. Shorter window locates the pitch better but misses length for accuracy of intensity. Therefore an optimum buffer length is used.

Note the black hole form in the action divided onto a spherical surface. 4pi is the amount of solid angle around a point in 3D space.

Since it is a statement of prediction, it suits quantum mechanical language well. QM's language is probability. It has an involved derivation but I will go through it, since it is so important...

Tuesday, October 24, 2006

Complex Normal Distribution

I called it CND but I haven't seen it defined. I just put complex variables in ND;

z=(1/c1*sqrt(2*pi))*e^((-1/2)*((z-c0)/c1)^2)





Where c1 and c0 are standard variation and mean. In the following c1 is the free parameter. A surprise here is that Mandel and Julia are similar in large scale. Although ND has that property of looking similar after a frequency transformation.

As before, restricting expansion also limits domain. I think I made a mistake with the expansion of CND. Fractal Explorer's Exp(z) does not give Taylor's expansion to a number. The expansion is not so trivial to calculate. Too many terms. I will keep the "expanded to 6" ones until I correct them. Currently I am trying to learn some Java.

Mandel
e^x expanded to 6
Sitting Buddha, hmm
















Julia
e^x expanded to 6
c1=(1.01,-1.11)
Doesn't it look like infinity sign?
Unfortunately the above Mandel
is not at the same c1, Mandel is below









Mandel
e^x to 6












Mandel
bailout=49













Mandel
bailout=324











Julia
bailout= 25












Mandel
bailout=25












Julia
bailout= 361












Mandel
bailout=361












Mandel
c0 free












Mandel
c0=(0,-1)
e^x expanded to 6
Mandel as mouth

Performance and nothingness

Performance of a difficult piece needs higher concentration and relaxation at the same time. Regardless of what piece. Musical piece, sexual piece academic piece. A paradox. How can you relax and concentrate at the same time? How can you create on demand? Even a purely mechanical performance needs a lot of juice. The key it seems is passion with wisdom. Passion knows the destination, wisdom knows the way. The destination is beauty, the way is nothingness. Like drops making an ocean.

Playing guitar needs a lot of concentration and relaxation at the same time. To learn a difficult piece, the fastest way is to play as slow and light as possible. A paradox. As Abel Carlevaro states clearly in his guitar method, you gotta give your full attention to balance from the start. Balance is achieved fastest by building it from little. So to learn you decrease tempo and power as much as possible but keep good balance and control. The {position, amplitude} duality of Delta Function is here as {tempo, power}. After all, learning is locating. Slowing down is as difficult as playing faster. You have to hold notes longer with your left hand, but the real difficulty is to remember. It is easier to remember when you play in a relaxed tempo which happens to be the natural tempo of the piece. The brain likes different speeds for different things. Intensity or power is also difficult to reduce. You have to play your right hand really close to the strings, which is really hard. So it is as hard to play slow and light as fast and powerful. But to learn, or to teach the piece to your brain, you have to slow down. Like learning a language.

Performing for musicality is similar. If you don't relax you can't bring the beauty out. Improvisation must be the real test for any musician. Improvisation, or creating on the go, is the way to go. The way is nothingness.

In the end everything is mediation, or compromise, or balance. It is difficult to play powerful and fast or light and slow at the same time. So you find a middle way that suits you best. But you keep pushing the difficult limits to keep your domain nice and roomy. Build strength around your comfort zone, and extend your comfort zone to meet your difficulties.

e^((z^2+c1)/c2), expanded to 6

Expanded the exponential to power 6, like M6. Bailout does not expand the domain in this case. The set looks similar from either of c1 and c2, which is interesting. Limiting the expansion of e^x to 6 seems to restrict the domain. A bit like the top of the pyramid flying off in the Egyptian mythology. Hunab Ku also seems to imply that ying-yang part is seperate.

Expansion to 6 is faulty. I found that expansion of ND is not easy. It has too many terms and difficult to calculate. I am trying to shift to Java at the moment.

Mandel
c2=(1.36,-0.66)












Julia
(0.5,0.5)
c2=(1.36,-0.66)











What do you reckon?













Mandel
c1=(0.26,-0.35)












Julia
(0.5,0.05)
c1=(0.26,-0.35)












I always wondered what those points were inside.

Monday, October 23, 2006

Smiling Buddhabrot

e^((z^2+c1)/c2)
c1=(-3.17,0.02)
bailout=25
















How about that?











Hahaha


e^((z^2+c1)/c2)

c1=(2.05,-2.01), max-iter=194, bailout=25,100,1024,10000

I like this form because it looks like the normal distibution. It also has a similar form to Mirror Set, as exponential for given H-fractal of z^2+c form. Totally unexpected looks though. Look at the smiling Buddha Mandel!

I made this with Fractal Explorer which is free. The diameter increase in both Mandel and Julia (which are contained in circles) corresponds to the bailout. Thus the domain is half the complex plane which is weird.

Check out the amazing similarity to Mayan icons.

Mandel
bailout=25












Julia
bailout=25













Etznab









Hunab Ku


















Mandel
bailout=100













Julia
bailout=100













Mandel
bailout=1024













Julia
bailout=1024













Mandel
bailout=10000













Julia
bailout=10000













Another Julia
bailout=25












Julia up close













Bronze age
Gold

Thursday, October 19, 2006

M0







In the beginning there was nothing and everything at the same time. (anon)

M0=,/,
SM0=0!/0^0={1/1,1/0}
0^0 is double valued, {0,1}. A special case of x^y. From x it looks like 0, from y like 1. We are forced to consider two variables {x,y} since exponential is hierarchical. exp(y*ln(x)) is the dual of x^y, reflecting the duality 0^0={0,1}. M0 has either 1 state or infinite states at the same time. If you want fireworks it seems you gotta measure zero.

The duality of measurement (or counting) is exposed here. Combinatorial for given exponential. Measurement is relative to a base which is the normal case. The norm is exponential in the universe. Everything is expressed as compared to a base which is always quantised. 0^0 expression is attempting to take 0 as base and this creates problem. This will be resolved by transforming to M0+ and taking a boolean base, 2^0. The nominator is factorial since it measures complexity of the meta structure on top of base.

A state is needed to express just a point. But this state is only needed in retrospect to M0+=E0/H0 to explain H0={}. M0 reflects the identity element. In fact (,/,) reflects the concept of nothingness better. Something really insignificant compared to itself. Then Higgs is M0 since (,/,) looks more like a self interacting scalar or dust field. No dimensions and infinite dimensions live together here. A bit like the impulse function, or rather Dirac Delta Function;
























Like we had to consider x^y form when looking at 0^0, here we have a duality, {position, intensity}. In fact Delta Function is the algebraic definition of a point;





It can be defined in any space, the simplest being one dimensional.

Although it looks thorny, it is possible to deal with it analytically and an essential part of dynamical system analysis. In electrical process control, a system is defined by it's impulse response. It seems ATON has landed on it's analytical bottom. This is the place to dwell for a while and maybe look at some quantum mechanics.

There is also the discrete equivalent of DF, Kronecker Delta Function with similar properties to DF;














KDF behaves like DF in definition of position;





I always thought Tarot cards looked strange. Look at the card-1 of the "major arcana" of the magician. Can you see the Delta function?


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