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, March 30, 2006

Brain

All brains come in dual form;
{,{}}
(right,left)=(emotional,analytical)
Sometimes this mapping is flipped;
(right,left)=(analytical,emotional)
Like the split in mathematics;
(Algebra,Geometry)
Algebra for analytical, geometry for emotional.
Like the split in computers;
(Programs,Operating System)
Programs for algebra, OS for geometry.
Emotions set the overall mood by using hormones.
Thus the material base for the dual split is science;
(Physics, Chemistry)
Chemistry sets the geometry for physics (electricity) to operate in.

An M6 brain should unite around M3, like ETT.
M3=E3/H3
SE3/SH3=3!2!1!0!/2^3=12/8=3/2
Would 3/2 symmetry correspond to color perception?
Basically there is only one color to start with;
{gray}
Then it seperates into a mediated duality;
{gray} -> {black,gray,white}
Then white seperates into 3 colors;
{white} -> {yellow,blue,red}
Each color pair also get their own mediation;
{yellow,orange,red}
{yellow,green,blue}
{red,purple,blue}
Thus 3/2 colors, {yellow,red,blue}/{black,white}, define a 3D coordinate system, 3D space.
Each color gets an axis, and {black,white} gives intensity like {-,+}
Like the universe, a brain that perceives 3 colors obeys M3.

Although the color symmetry obeys 3/2, it doesn't necessarily mean light is essential for a brain to evolve.
As ETT shows, sound also obeys the same symmetry.
It seems the brain opens up a seperate 3D phase-space to solve each problem.
3D aspect of hearing is clearly visible in the conic spiral shape of cochlea in ear.
Then there must be other 3/2 modules in the brain to deal with other cognition tasks.
Sense comes in 5 types;
{see,hear,taste,touch,smell}
I think consciousness has to be added in this list.
Even then I think there is one missing, considering M6.
Taste has 3 colors;
{sweet,sour,salty}
With taste intensity we again get 3/2 symmetry.
I regarded 'bitter' as extreme sour.
Touch has 3 colors?
{elastic,stick,inelastic}
Smell comes from taste, therefore obeys the same symmetry.

Wednesday, March 29, 2006

Vertebrate body

The most obvious aspect of an animal body is that of mouth-stomach-anus or head-body-tail mediated duality.
Mouth-stomach-anus aspect would be the first to evolve for any living entity.
Life starts with birth and then eating and growth follow.
If you can't eat, you can't grow, very simple.
Eating can be as simple as in and out of a simple tube, like worms suggest.
A particular eating action evolves as a low entropy solution to many types of action that a differentiated population would use.
An eating action would have to controlled from the front since sensors would be there.
The first sense for an animal to experience would be taste.

The second most obvious aspect is left-middle-right mediated duality.
L/R seperation is necessary since most movements would have to be in a plane for a water-land margin living entity.
With head/tail and left/right setup, we are ready to navigate a plane.
A certain type of head/tail, left/right oscillation would be enough to move.
This motion would be the one that splits the first animal brain.
Then we get many appendages appear, as worm evolution shows, and these settle to head/tail, left/right and arms/legs configuration for vertebrates.

These steps map to M0-2;
SM0=1/1=1 ... mouth/anus (head/tail)
SM1=1/2 ... left/right
SM2=2/4 ... arms/legs

To continue.

M3























































Above are 3 visualisations of M3 with no perspective. It is a shape I found while playing with equal length bamboo sticks. Previously I called it Bion. I decided to call it just M3. Now I aso don't divide the outer cubes. It's outer aspect is the 3 interlaced cubes like the Escher's. Inside, it has a cube made of two reflected tetrahedrons. The number of total faces of the outer cubes is 18. The tetrahedrons have 8 triangle faces. Note the resemblance to some ancient Asian yantras, specially Shri Yantra.

I found it when I was admiring the beauty of a tetrahedron made using 6 equal length sticks. Then I rotated it pi/2 and added them to each other making a cube made of two reflected tetrahedrons. This is a cube made by it's diagonals. The diagonals of the cube make two interlaced tetrahedrons. Then I wondered what would happen if I kept this iteration of making a transformation and adding it to the parent shape. I rotated the cube pi/2 and translated it by 1/2 in each of the 6 face directions and added them all to each other. Then I completed the manifested cubes and also divided the cubes to into 8 equal pieces.

There may be more things one can do to Bion with equal length sticks as suggested by M6. There is already a large octahedron inside. I expect all the Plato's polyhedrons to appear in some iteration of Bion. I had enough of playing with sticks, too tedious.
I am thinking about a little program to experiment with Bion. Someday it may happen. It would be wonderful to make a full connection with M6.

In the middle there are two interlaced Tetrahedrons, made by rotating one of them pi/4 and adding. Each T is made of 6 sticks representing the edges. T+T(pi/4) has 2*6=12 sticks. There are 6 crosses making the faces of a cube. There are 8 corners made by the tips of Ts. Thus we got our;
12/8=3/2=E3/H3={,{},{,{}}}/{,{,{,{}}}}
T^2 is a cube made by it's diagonals.

The outer layer of Bion has 3 cubes made by sticks at the edges. There are 3*6=18=SE4/SH4 faces.

Sunday, March 26, 2006

Fundamental Theorem of Calculus

FTC states the existence of the integral;
Integral[f(x)dx,a,b] = F(b)-F(a)
An iteration of FTC gives Taylor's expansion and maps e^z to polynomials;
e^z = Sum[z^n/n!, n=0,...,Infinity]
The weighing factor 1/n! diminishes the higher polynomials and ensures absolute convergence.
It reminds the number of states of ESet;
SEn = n!(n-1)!(n-2)!...1!0!
ESet is derived from iterating a simple operation, like the e^z sum.
Then one would expect the Integral[] definition to be like the ESet operator E.
ESet accumulates its past and determined uniquely by its initial and final states, like Integral[].
ESet seems isomorphic to e^z.

Equal Tempered Tuning

E0/H0= /{}
SE0/SH0=0!/2^0=1/1=1
The root note, say C.

E1/H1={}/{,{}}
SE1/SH1=1!0!/2^1=1/2
The octave of the root, 2C.

E2/H2={,{}}/{,{,{}}}
SE2/SH2=2!1!0!/2^2=2/4=1/2
The octave of the root, 2C.

E3/H3={,{},{,{}}}/{,{,{,{}}}}
SE3/SH3 = 3!2!1!0!/2^3 = 12/8 = 3/2
3/2 is the frequency ratio of the 5th to it's root, like G/C.
12 is the number of semi-tones in an octave.
(C,#C,D,#D,E,#E,F,G,#G,A,#A,B,C)
The frequency ratio between any two adjacent notes is the 12th root of 2.
When you iterate the same ratio 12 times you reach the octave,
(Root[2,12])^12=2
This homogeneous division gives transposability.
8 is the number of notes in a diatonic scale,
(C,D,E,F,G,A,B,2C)
In fact there are only 7 different notes but only when the 8th is played it gives the ending feeling.
It can be viewed as a HFractal of depth 3;
(((C,D),(E,F)),((G,A),(B,2C)))
Like modular group.
The grouping becomes clearer when the corresponding rational numbers are considered;
(((1,9/8),(5/4,4/3)),((3/2,5/3),(15/8,2)))
G leads the second group on the way forward.
F leads the second group on the way backward.
When two scales are played, the octave 2C leads the second group;
(((C,D),(E,F)),((G,A),(B,(((2C))),2D),(2E,2F)),((2G,2A),(2B,4C)))
The two scales overlap at 2C and become linked.
There is a wonderful modularity in the scales.
This can be heard clearly when one plays the scale (specially over and over again!).

ETT compares the power taking iteration of 5th to 8th and makes them meet at some real number a/b;
(3/2)^a = 2^b
a*log(3/2) = b*log(2)
a/b = log(2)/(log(3)-log(2)) = 1.7095
12/7 = 1.7143 has a very close match.
((3/2)^12)/2^7 = 1.0136
If you take power of 3/2 12 times you get the 7th octave, almost.
If a/b is set exactly to 12/7, then all other ratios have to be adjusted.
12 is the number of semi-tones in an octave, and 7 is the number of octaves it takes for the 5th to make a complete cycle and also the number of notes in CM scale.
Thus harmony is distorted slightly to get transposability.
The miracle of ETT is that harmony and equal tempering meet with a surprisingly small distortion of harmony.
This 7 octave travel of the 5th suggests M6.

I referred to Michael Rubinstein's page for the algebra;
http://www.math.uwaterloo.ca/~mrubinst/index.html

M6 suggests further division of the octave, like some Arabic scales which divide a tone into 9 commas.
12/7 equal tempering seems to correspond to the outer shape of M6, since it is centered around E3/H3.

Mirror Set and the elementary particles

Now revised;
http://a-theory-of-nothingness.blogspot.com/2006/11/mirror-set-rev-1.html

Mirror Set of order n is the (n+1)*(n+1) matrix quotient set;
Mn=Mnkl=Ek/Hl, (k,l)=0,...,n

M6 seems interesting with the following main diagonal state quotients and proposed mappings to elementary particles;
SE0/SH0=1 ... Higgs?
SE1/SH1=1/2 ... Light
SE2/SH2=1/2 ... Weak
SE3/SH3=3/2 ... Quarks
SE4/SH4=18 ... Leptons
SE5/SH5=1080 ... Space-Time-like?
SE6/SH6=388800 ... Mass-Gravity-like?
The digits of 4,5,6 all add to multiples of 9.
388800/1080=360
1080/18=60
18/3/2=12
3/2/1/2=3
1/2/1/2=1
1/2/1=1/2

3/2 is the symmetry of quarks and equal tempered tuning.
Quarks come in 3 generations and 2 types;
{{u,d},{c,s},{t,b}}
I put 3/2 in the middle and proposed M6 since energy seems to be balanced around quarks. This balancing is achieved by the large mass ratio Mp/Me=1836.
When the state quotient is expressed more explicitely it becomes clearer;
E3/H3={,{},{,{}}}/{,{,{,{}}}}
SE3/SH3=3!2!1!0!/2^3=12/8=3*2*2/2*2*2=3/2
There is 2*2*3 ness (3 generations, 2*2 types) and 2*2*2 ness (3 color/anti-color pairs) at the same time.

18=2*3*3 is the number for leptons.
Leptons also come in 3 generations and 2 types;
{{e,ve},{m,vm},{t,vt}}
18 is pretty obvious in the periodic table, displaying the 18-slot lepton wrapping of the nucleus.
6 is is kinda explained but how about 3*6=18, why does it get multiplied by 3?
Maybe leptons are satisfying the color multiplicity of quarks.

1/2 is degenerate curiously.
The first 1/2 has to be light, electric and magnetic together make one unit.
The second 1/2 then stands for weak, with W+,Z0,W-.
The degeneracy may stand for the realtively easy light-weak unification.

1 must be Higgs which is assumed to be scalar and not detected yet.

I made a guess about what the other two are, 1080 and 388800, space-time-like and mass-gravity-like.
Gravity seems to be a deeper field as black hole theories suggest.
Space-Time stop at the event horizon but Mass-Gravity don't.
Putting quarks in the middle suggests the couplings;
{Higgs,Mass-Gravity}
{Light,Space-Time}
{Weak,Leptons}
Weak-Lepton coupling is observed in the decay of neutron;
n->p+e+ve
Light-Space-Time coupling is pretty obvious.
We define Space-Time with light.
Higgs-Mass-Gravity coupling remains to be seen.

Saturday, March 25, 2006

Exponential Set

This set should be called Factorial Set of order 2, in line with the updated definition of Mirror Set;
http://a-theory-of-nothingness.blogspot.com/2006/11/mirror-set-rev-1.html

E0=,
E1=EE0=E={,}={}
E2=EE1={,{}}={{},}
E3=EE2={,{},{,{}}}
E4=EE3={,{},{,{}},{,{},{,{}}}}
...
En=EE(n-1)

Where E is a reflection operator. It gathers the past under one roof. Although it has multiple content, it also has an overall duality like HSet. ESet looks like e^z since the self similarity in breadth. HSet is self similar in depth, for example H3={,{,{,{}}}}. Self similarity of exponential function is evident from differentiation, (d/dz)e^z=e^z. Taylor's expansion of e^z is an iterated process, like ESet.

The identity (E0=,) is a bit strange but my hand was forced by 0!=1. But it also makes some sense since the seperator is defined first. (E0=) is even stranger.

The number of states;

SE0=0!=1
SE1=1!0!=1
SE2=2!1!0!=2
SE3=3!2!1!0!=12
SE4=4!3!2!1!0!=288
SE5=5!4!3!2!1!0!=34560
SE6=6!5!4!3!2!1!0!=24883200
...
SEn=n!(n-1)!(n-2)!...2!1!0!

Self similarity also shows up in the state calculation. The factorial is calculated to depth 2.

Note the curious 'nineness' of 4-6;

2+8+8=18=2*9
3+4+5+6+0=18=2*9
2+4+8+8+3+2+0+0=27=3*9

H-Set

H-Set is like H-Fractal or the binary tree.

H0={}
H1=HH0={,{}}
H2=HH1={,{,{}}}
...
Hn=HH(n-1)

Where H is an inclusive-exclusive reflection operator.
It reflects the previous.

The number of states;

SH0=2^0=1
SH1=2^1=2
SH2=2^2=4
...
SHn=2^n

The Mandelbrot iteration;
z(n+1)=z(n)^2+c
is expanded in a H-Fractal;
z0=z0
z1=z0^2+c
z2=(z0^2+c)^2+c

Nothingness ain't nothing













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

Thing mediates nothing and everything, and becomes Thingness;
Thingness = {nothing, thing, everything}
Or rather more simply;
T={,Infinity}
Where the humble comma "," stands for thing. It seperates everything from nothing. Thus the seperator is the origin of everything. Language and being are inseperable.

Nothing and everything do not make sense by themselves. They exist as trancendental entities mediated by thing, however small but never nothing. Thingness comes from nothingness.

It is 0 (or rather epsilon) for mathematics, water for taste, air for smell, space-time for action, dx for calculus. This is an unpolarised state. Like hermaphrodites before sex evolved. They are still around. Nature does not like sameness. Like in quantum mechanics, two particles in the same quantum state are not allowed. Therefore there would always have been slight differenciation in the reproductive techniques of a hermaphrodite population. Difference make a population as much as similarity. 0 in math indicates that idealisation of very small quantities by a trancendental concept is useful and it works. We treat space as nothing in most cases, but we know that it is an entity. This state corresponds to Grey color.

Then comes another point of differentiation. This time we get a boundary by polarisation.
{,}
A boundary has at least 2 elements which reflect each other. Like {-1,0,+1}, {+Charge,Light,-Charge}, {Male,Hermaphrodite,Female}, {Black,Grey,White}, {No,Maybe,Yes}, {Nothing,Something,Everything}. Then comes a rule;
{,}={}
That is, sameness is not allowed in a set. Or rather, the seperator is negligible. Polarisation wins out. And we get our empty container to fill with content.

It can also grow;
{,{}}
It can be collapsed to expose it's ordered elements;
{,{}} == <,<>> + <<>,>
Reminding Quantum Mechanics. It seems it can be used for classifying things. Classification is the game. {} is the player. An empty container in some context seems to be a fundamental concept, like space. An iteration process would represent time. ESet and HSet are generated by simple iterations. Both breadth and depth are generated by one operation. We get breadth-depth (space-time) unification in one simple process from the start.

This triplet nature is apparent in many things;
{no,maybe,yes}
{black,gray,white}
{nothing,thing,everything}
{0,1,infinity}
{left,middle,right}
{blue,green,yellow}
{cold,mild,hot}
{-,0,+}
{geometry,probability,algebra}
{space,energy,time}
{solid,liquid,gas}
{heads,side,tails}
{inside,border,outside}
and so on. All dualities come with a median, whether obvious or not. It is not surprising that our number system is based on a triplet, {0,1,I}. The whole universe seems to obey it. It displays itself more explicitely in the first iteration of {};
{,{}}
There are two things inside seperated with a comma. A triplet within a triplet. Sameness is not allowed in a set;
{,}={}
Thus the container defines it's own seperator. Reflecting the hidden nature of median, comma ",".

Maybe Set Theory went too complicated and ignored the potential of {}.

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