Table of Contents

Introduction

What is a Quantum Texturology ?

Unifying Views of Events

A Brief History of our work on Quantum Texturology

What is Quantum Texturology ?

Quantum Texturology is a concept according to
- relational texture, interferences of properties and laws of texture evolution to define texturology,
- distribution, density of probability, amplitude of probability, complexe number, wave function to define the quantum aspect,
- and quaternions (reduced quinternions) to show the representation in 3D.
- arts, sciences and technologies : normaly when you are not sure of your objective work, you stay in arts to imaging freely and make imagies ; when you know that's something interesting you choice a mathematic language to think logicaly and write that you think right ; When you are sure that somethinkg important could happend you use computer to program your experience to show the objective facts ; in reality I choice every way in a cycling storm during 30 years to obtain that I feel,  that I prefeel, that I mean, that I want to obtain.
- mathematics, physics, chemistry, biology : A lot of scientists believe that mathematics revel deep things about how Nature works, they just show how we perceeve and mesure our world and exprime with a language witch model we can build. In arts the model is the nature and the work of the artist is his proper oeuvre. In sciences the model is that the scientist try to build observating and experimentating the nature.
- philosophie and politics (a fascion to live your live with the others).

Unifying several Views of Events and a thirty years of work

Trans-combination, Pretopological Texture, Texturology, Quantum Texture and Quaternions

A Brief History of our work on Quantum Texturology

A more modern writting with theorie of categories, sets, networks, wave functions, and structured quaternion

Starting from the representation of the quantum texture in the complex plan, we can see that for a great number of detectors, the information is overcharged.
For open the plan we can rotate the axis to each resulted vectors like an eventail. And with several level structured sets by inclusion we obtaine the result that is a growing spiral.
The result vector and the way of theirs components are more significative and expressive but too much superposed when detectors number increase.
figure
 

System of wave-functions

psy_{j 0}     = sqrt(int_{j i})                                                                                                                                                     |psy_{j 0}|     = sqrt(int_{j i})
psy_{j 1}     = sqrt(int_{j i})    + i sqrt(sfi_{j i})   =  sqrt(x_{j i})     * (cos theta  + i sin theta) =  sqrt(x_{j i})     * exp(-i theta)           |psy_{j 1}|     = sqrt(x_{j i})
psy_{j 2}     = sqrt(x_{j i})      + j sqrt(sfe_{j i})   =  sqrt(adh_{j i}) * (cos theta + j sin theta)  =  sqrt(adh_{j i}) * exp(-j theta)           |psy_{j 2}|     = sqrt(adh_{j i})
psy_{j 3}     = sqrt(adh_{j i})  + k sqrt(ext_{j i})  = sqrt(x_j+1)    * (cos theta + k sin theta) =  sqrt(x_j+1)   * exp(-k theta)           |psy_{j 3}|     = sqrt(x_j+1)
psy_{j+1 4} = sqrt(x_j+1)                                                                                                                                                      |psy_{j+1 4}| = sqrt(x_j+1)

For a level j of the subset and j+1 of the set.

Then we have choiced to represente it in several dimensions.
When we think of rotation with complex plans, Hamilton and Quaternions come naturaly there.
With our 5 wave-functions we obtaine a quinternion with five componentes : not one real and four complex numbers, as usual, but 1 real, 3 complex and 1 real number.
Quaternions are developed now in fractals, satelites, boats, robotics and games with the virtual or augmented and simulated reality and the apparition of 3D softwares or hardwares (Open GL of SGI).
So before studying directly a structured system of specific new quinternions it is interesting to use quaternions.
We can see that the normalisation is alwais possible (closed sets, conservation of energy, etc) but not wanted with the changing level in a quinternion.
If the classical quaternion is a  couple <scalar, vector> or the system of quaternion a couple <tensor, versor> representing the animation of a point on a sphere, the quinternion is a triplet <scalar, vector, scalar> and could be interpreted as a the animation of a point between two spheres.
If we project the quinternion into the quaternion, the animation of the point between spheres become the animation of a snake on a sphere with the start point or the end point as reference.

Then we can see that it is possible to use the two form of normalised quaternion <1,i,j,k,> if we stard at the level of the subsets of a set, or <i,j,k,1> if we start at the level of the set of subsets.
In a structure of supersets, sets and subsets, we can define a serie of quaternions with alternative both forms like an left-in or right-out imbriquation of quaternions.
May be it is a starting point of a direction of researche for why means one or the other form in structured quaternions ?

Quinternion :          psy = <psy_{j 0} , psy_{j 1} , psy_{j 2} ,psy_{j 3} , psy_{j+1 4}>         |psy|     = sqrt(int_{j i} + x_{j i} + adh_{j i} + x_j+1 + x_j+1) =
Left-Quaternion :   psy = <psy_{j 0} , psy_{j 1} , psy_{j 2} , psy_{j 3}>                         |psy|     = sqrt(int_{j i} + x_{j i} + adh_{j i} + x_j+1) =
Right-Quaternion : psy = <psy_{j 1} , psy_{j 2} ,psy_{j 3} , psy_{j+1 4}>                      |psy|     = sqrt(x_{j i} + adh_{j i} + x_j+1 + x_j+1) =

Normalization

By the set sqrt(x_j+1)
psy_{j 0}     = sqrt(int_{j i} / x_j+1)                                                         |psy_{j 0}|     = sqrt(int_{j i})
psy_{j 1}     = sqrt(int_{j i} / x_j+1)    + i * sqrt(sfi_{j i} / x_j+1)                   |psy_{j 1}|     = sqrt(x_{j i})
psy_{j 2}     = sqrt(x_{j i} / x_j+1)      + j * sqrt(sfe_{j i} / x_j+1)                  |psy_{j 2}|     = sqrt(adh_{j i})
psy_{j 3}     = sqrt(adh_{j i} / x_j+1)  + k * sqrt(ext_{j i} / x_j+1)                 |psy_{j 3}|     = sqrt(x_j+1)
psy_{j+1 4} = sqrt(x_j+1 / x_j+1) = 1                                                   |psy_{j+1 4}| = 1

Quinternion : <psy'_{j 0} , psy'_{j 1} , psy'_{j 2} , psy'_{j 3} , 1>
Left-Quaternion : <psy'_{j 0} , psy'_{j 1} , psy'_{j 2} , psy'_{j 3}>
Right-Quaternion : <psy'_{j 1} , psy'_{j 2} ,psy'_{j 3} , 1>

By the set sqrt(x_{j i})
psy_{j 0}     = sqrt(int_{j i} /x_{j i})                                                      |psy_{j 0}|     = sqrt(int_{j i}/ x_{j i}) = Rint_{j i} <=1
psy_{j 1}     = sqrt(int_{j i} / x_{j i})    + i * sqrt(sfi_{j i} / x_{j i})                |psy_{j 1}|     = sqrt(x_{j i}/ x_{j i}) = 1
psy_{j 2}     = sqrt(x_{j i} / x_{j i})      + j * sqrt(sfe_{j i} / x_{j i})                psy_{j 2}|     = sqrt(adh_{j i}/ x_{j i}) = Radh_{j i} >= 1
psy_{j 3}     = sqrt(adh_{j i} / x_{j i})  + k * sqrt(ext_{j i} / x_{j i})               |psy_{j 3}|     = sqrt(x_j+1/ x_{j i})  = Rens_{j i} >= 1
psy_{j+1 4} = sqrt(x_j+1 / x_{j i})                                                      |psy_{j+1 4}| = sqrt(x_j+1 / x_{j i}) = Rens_{j i} >= 1

We find back the known ratios Rint_{j i}, Radh_{j i} and Rens_{j i} that exprime the aggregation, the dispersion and the proportion for each subset in a set.
Quinternion : <psy'_{j 0} , psy'_{j 1} , psy'_{j 2} , psy'_{j 3} , psy_{j+1 4}>
Left-Quaternion : <psy'_{j 0} , psy'_{j 1} , psy'_{j 2} , psy'_{j 3}>
Right-Quaternion : <psy'_{j 1} , psy'_{j 2} ,psy'_{j 3} , psy_{j+1 4}>
 

We know that the 5 wave-functions are independant and should be normalized each one.
By the sets sqrt(int_{j i}), sqrt(x_{j i}), sqrt(adh_{j i}), and sqrt(x_j+1)
psy_{j 0}     = sqrt(int_{j i} / int_{j i})   = 1                                                                                                             |psy_{j 0}|    = 1
psy_{j 1}     = sqrt(int_{j i} / x_{j i})      + i sqrt(sfi_{j i} / x_{j i})     =  sqrt(int_{j i} / x_{j i})      + i * sqrt(1 -int_{j i} / x_{j i})        |psy_{j 1}|    = 1
psy_{j 2}     = sqrt(x_{j i} / adh_{j i})    + j sqrt(sfe_{j i} / adh_{j i}) =  sqrt(x_{j i} / adh_{j i})    + j  * sqrt(1- x_{j i} / adh_{j i})      |psy_{j 2}|    = 1
psy_{j 3}     = sqrt(adh_{j i} / x_j+1)  + k sqrt(ext_{j i} / x_j+1)   =  sqrt(adh_{j i} / x_j+1)  + k * sqrt(1-adh_{j i} / x_j+1)     |psy_{j 3}|    = 1
psy_{j+1 4} = sqrt(x_j+1 / x_j+1) = 1                                                                                                                |psy_{j+1 4}| = 1

Quinternion : <1 , psy_{j 1} , psy_{j 2} ,psy_{j 3} , 1>
Left-Quaternion : <1, psy_{j 1} , psy_{j 2} ,psy_{j 3}>
Right-Quaternion : <psy_{j 1} , psy_{j 2} ,psy_{j 3} , 1>

Texturology, quaternions and fractals

We have obtaine a model to represente the mesurement of the quantum texture and texturology with 5 wave-functions and then define a process of information analysis.
If we want to simulate the process we can inverte it and in this terms we can say that the system have to be controled by the parameters following the 5 wave-functions.
Then the control equation is a  complex polynome with five degrees where each wave-function is a root.
For the normalized quinternion we have :
Z_n+1= (Z_n - 1) * (Z_n - psy_{j 1}) * (Z_n - psy_{j 2})* (Z_n - psy_{j 3}) * (Z_n - 1)
Z_n+1= (Z_n - 1)² * (Z_n³ - (psy_{j 1} + psy_{j 2} +psy_{j 3}) * Z_n² + ( psy_{j 1} * psy_{j 2} + psy_{j 2} * psy_{j 3} +psy_{j 3} * psy_{j 1} ) * Z_n - psy_{j 1} * psy_{j 2} * psy_{j 3})
 
 
 

// 3 wave functions
// psy1= sqrt(inti/Xi) + i sqrt(sfii/Xi)
// psy2= sqrt(xi/adhi) + j sqrt(sfei/adhi)
// psy3= sqrt(adhi/X) + k sqrt(exti/X)
#declare X=10;       #declare Xi=3;          #declare Adhi=7;
#declare psyr1=1*Xi; #declare psyi1=0.75*Xi; #declare psyj1=0.25*Xi; #declare psyk1=0.33*Xi;
#declare psyr2=1*Adhi; #declare psyi2=0.25*Adhi; #declare psyj2=0.2*Adhi; #declare psyk2=0.5*Adhi;
#declare psyr3=1*X;    #declare psyi3=0.4*X;     #declare psyj3=0.1*X;    #declare psyk3=0.9*X;
// constante C = -psy1*psy2*psy3
#declare psycr=psyr1*psyr2*psyr3; #declare psyci=psyi1*psyi2*psyi3;
#declare psycj=psyj1*psyj2*psyj3; #declare psyck=psyk1*psyk2*psyk3;
// somme psy
#declare  srpsy=psyr1+psyr2+psyr3; #declare  sipsy=psyi1+psyi2+psyi3;
#declare  sjpsy=psyj1+psyj2+psyj3; #declare  skpsy=psyk1+psyk2+psyk3;
// somme produit psy
#declare  sprpsy=psyr1*psyr2+psyr2*psyr3+psyr3*psyr1; #declare  spipsy=psyi1*psyi2+psyi2*psyi3+psyi3*psyi1;
#declare  spjpsy=psyj1*psyj2+psyj2*psyj3+psyj3*psyj1; #declare  spkpsy=psyk1*psyk2+psyk2*psyk3+psyk3*psyk1;
// produit psy
#declare  prpsy=psyr1*psyr2*psyr3; #declare  pipsy=psyi1*psyi2*psyi3;
#declare  pjpsy=psyj1*psyj2*psyj3; #declare  pkpsy=psyk1*psyk2*psyk3;

#declare wk=0.022;   // 4th dimension
#declare iter=10;       // Number of iterations. With Quaternions this can be a small number
#declare bailout_value=2000; // voisinage belonging to the QuaternionSet or Fatou set

// calcul de z^2
         #local temp=xs+xs;
            #local xs2=xs*xs-ys*ys-zs*zs-w*w;
             #local ys2=temp*ys;
            #local zs2=temp*zs;
             #local w2=temp*w;
// calcul de z^3
           #local temp=xs2+xs;
            #local xs3=xs2*xs-ys2*ys-zs2*zs-w2*w;
             #local ys3=temp*ys;
            #local zs3=temp*zs;
             #local w3=temp*w;
// calcul de Z^3 -(psy1+psy2+ps3) * z^2 + (psy1*psy2+psy2*psy3+psy3*psy1) * z - psy1*psy2*psy3
     #local znr=xs3 - srpsy * xs2 + sprpsy * xs - prpsy;
             #local zni=ys3 - sipsy * ys2 + spipsy * ys - pipsy;
             #local znj=zs3 - sjpsy * zs2 + spjpsy * zs - pjpsy;
             #local znk=w3  - skpsy * w2  + spkpsy * w  - pkpsy;
             #local m=m+1;
             #local lengde=znr*znr+zni*zni+znj*znj+znk*znk;
 
 

Texturology and System of Quaternions

A quaternion is composed of Tensor and Versor : Q= <T,V>
The composition of Versor is equivalent to the sum of vector-arc on unit sphere. Not commutative.
Incrementation of Vector'angle is equivalent to exponentiation : expt(i n theta) = cos (n theta) + i sin(i n theta) = (cons theta + i sin theta)ⁿ  Moivre's formula
Substration of vector-arc is equivalent to a quotient of Versors.
 

Case of two subsets complementary in a set.
 

Case of three sets
 

Case of several sets
 

Case of increasing inclusions of 3 sets
 

Case of decreasing inclusions of 3 sets
 
 

Quantum Texturologic Computer

|psy> = <psy_{j 1} , psy_{j 2} ,psy_{j 3}>
with :
psy_{j 1}     = sqrt(int_{j i} / x_{j i})      + i sqrt(sfi_{j i} / x_{j i})       =  sqrt(sfi_{j i} / x_{j i})       | 0 >  + sqrt(1 - sfi_{j i} / x_{j i})        | 1 >
psy_{j 2}     = sqrt(x_{j i} / adh_{j i})    + j sqrt(sfe_{j i} / adh_{j i})   =  sqrt(sfe_{j i} / adh_{j i})    | 0 >  + sqrt(1 - sfe_{j i} / adh_{j i})   | 1 >
psy_{j 3}     = sqrt(adh_{j i} / x_j+1)  + k sqrt(ext_{j i} / x_j+1)   =  sqrt(ext_{j i} / x_j+1)    | 0 >  + sqrt(1 - ext_{j i} / x_j+1)    | 1 >