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, 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?


Wednesday, October 18, 2006

M3 (take-2)



















M3=E3/H3={,{},{,{}}}/{,{,{,{}}}} is the form that shapes all the rest since it has the overall symmetry of SM3=12/8=3/2. 3/2 is the symmetry that makes quarks/gluons and equal tempered tuning in music. I used call it Bion but given up on that. It is made by translating M2 by half and rotating by pi/4 in each of the 6 faces. And then completed the 3 manifested cubes giving M3 an outer shape like one of Escher's works.

It has 3 distinct aspects;






































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