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written and published by Renzo Diomedi // UNDER CONSTRUCTION

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3 space-dimensions cannot explane the dark matter and the dark energy. 
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Conventionally, we use two series of cartesian complex conjugate planes intersecting each other and on each of them. 
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Then we fix a point in every plane. 
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We name J-planes the first set of planes and 
we name K-planes the set of intersecting planes.
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the set of J-planes is the physical three-dimensional space made by 
particles of energy imprisoned in the mass. 
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The set of K-planes is the electromagnetic space made by photons, particles of massless energy.
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So, the K-planes intersect J-planes and vice-versa. 
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The distance beetween the planes is quasi null, so, the hypercube consists of two Strings intersected among themselves 
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Every String is composed by points (one for each plane) , the string join all the points of all the 
J and K-planes vibrating in a two-dimensional surface called <a href="polyakov.htm"><i>worldsheet</i></a>, 
not a 3-dimensional space 
as it would be immediate to think. 
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Every set of planes creates a worldsheet. So the hyper cube created is composed by 4 space 
dimension created by 2 worldsheets intersected. 
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But the fourth spatial dimension can not be perceived by our senses 
created to perceive 3 dimensions only. 
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Considering 2 points on a sheet of paper, their distance along the surface will remain unchanged also bending the sheet, 
as it will always remain the same also curving the whole book, id est the geometric variety, of which it is part. 
We note that the whole universe is dotted with curved surfaces, and even when we draw a straight line on a sheet we have 
to remind us that the Earth's surface is bent and that the euclidean plane considered, is valid 
only by approximation locally.
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If the measuring system is a Cartesian metrics, in a two-dimensional space, the <b> axes x, y </b> are perpendicular 
to each other, but in a non-Cartesian metrics, <b> µ, v axes </b> are generally oblique


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Length of the Strings: 
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<img src="69.png"> <img src="110.png">
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<i> n = number of all the planes in first string , m = number of all the planes in second string , 
p = planesheet, x = Real axis , iy = Im axis </i>

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The massless electromagnetic particles and particles 
with mass constitute the universe. If the distance among every point on 
J-plane and every point on K-plane is zero, the massless particle intersects and coincides with particle with mass . 
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using the metric tensor we can calculate the sum of the vectors of coincident points of J and K :
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<img src="70.png"> 
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we can calcualte the distance among every point of every plane J and every point of every K:   
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<img src="124.png"> 
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<i> X is a generic component of the reference axes, j= series of planes , k= series of intersected planes, I = intersection point in the plane , x = Real axis , iy = Im axis </i>
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we could also consider the two sets of crossed planes as cross-cut and intersected lattices

considering the Strings belonging a worldsheet

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we need to use tensors that in a geometrically bendable structure
keep equal distance beetween the lattice points
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<img src="112.png">
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<img src="113.png">
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then, also using the Einstein notation
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<img src="111.png">

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then <img src="114.png">
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then <img src="115.png">
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then <img src="116.png">
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then <img src="117.png">

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The Metric Tensor <b> g </b> expresses the property of a structure
geometrically curvable with the points of its lattice at a distance
always equal in relation to the structural components themselves

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then <img src="118.png"> contravariant metric tensor
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then <img src="119.png"> covariant metric tensor

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to be continued...........................






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<img src="120.png"> <img src="121.png">
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to be continued............
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the series of the points of the j-planes creates a curve function that we can use as a Line Integral = <img src="108.png">
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the series of the points of the k-planes creates a curve function that we can use as a Line Integral = <img src="109.png">

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TO BE CONTINUED....................
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the series of the planes J is a cube, also the series K

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<img src="74.png">

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each cube composes a string that is the result of the action of so many vector fields.
Each cube, therefore, can be seen as a tensor. 2 intersecting cubes (hypercube) are a super-tensor
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considering <img src="95.png"> , then we have an Identity tensor: 


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<img src="75.png">
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<img src="76.png">
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<img src="88.png">
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<img src="89.png">


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then
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<img src="90.png">
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<img src="91.png">
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<img src="92.png">
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<img src="93.png">



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to be continued....

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NS, composed by a conservative continuityequation <img src="106.png"> and a non-conservative 
Momentum equation <img src="87.png"> not exactly measurable as a scalar field, but divisible by 
3 scalar equations laid and projected along the directions <i> x, y, z </i> which returned 
values not coinciding. 
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<img src="104.png"> ; <img src="81.png"> = Density , <img src="80.png"> = Viscosity
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But the fourth space dimension could be considered as the synthesis of other 3 so in substitution of
<i> x, y, z, </i> so <img src="72.png"> , <img src="98.png"> , then <img src="99.png"> 
then we get a single equation in substitution of three, to get a single output <img src="105.png">
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Demonstration:
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Considering the 4 independent variables <i>x,y,x,t</i> and the 4 dependent variables <i>u,v,w, </i> 
and <i> p </i> (pressure) and the 6 given by the Stress Tensor considering <i> u, v, w </i> components 
of the shift vectors <img src="78.png"> along axes <i> x, y, z, </i>, we have:







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<img src="79.png"> 
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<img src="77.png"> 
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then, if the components of the viscous stress state are linearly linked 
to the components of the deformation velocity through Stokes' relations,  

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<img src="82.png">
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<img src="84.png">
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whereas <img src="83.png">
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then 
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<img src="100.png">
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so, using <img src="94.png"> we have <img src="101.png">
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then
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<img src="102.png">
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so, <img src="103.png">


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momentum scalar equations:
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<img src="104.png">
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using <img src="94.png"> we have <img src="105.png">


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TO BE CONTINUE...

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Let the Times of the series of the planes , not coinciding 


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TO BE CONTINUE...



































<!--


<a href="polyakov.htm"><i>String theory</i></a>



If we assume that the <b><i>xiy </i></b> plane Time and the <b><i>aib </i></b> plane Time may be different. 
Is it possible 
that this temporal distortion nullifies the fourth dimensional and its HyperDensity that it creates?
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to be continued................
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But if the points of the strings coincide , the dimensions are reduced to three.
but How could they coincide?
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The <i>xiy plane</i> time and the <i>aib plane</i> time may be different.
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Assuming that we can manage the time of each single plane autonomously, we can verify if there is a moment 
with <i>x,iy </i> and <i>a,ib</i> points coincide where the planes intersect
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So , in this case, the differential is no longer the difference in the position of the points ,
but it is now given by the time difference of the two planes
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We get 3 dimensions through a temporal distortion that nullifies the fourth dimensional 
and its HyperDensity that it creates

The fourth dimension is the synthesis of the other 3, and we can use in substitution of x y z,
So the resolution of difficult equations becomes simplified.
Look at NS: we can reduce the three equations to one

10 variaBILI dipendenti diventano 7

le variabili indipendenti diventano 2

We also need an integral function that describes the tensorial variations

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to be continued................

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<b><i> y, b, </i></b> can be <img src="57.png"> zero (to be continue...) 

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to be continued....

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Every of the 2 manifold worldsheet, can have its different time that gives the position of the point 
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to be continued...............

<img src="tetra1.png">
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<img src="tetra2.png">
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<img src="tetra3.png">
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<img src="tetra4.png">

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the Delta of LaPlace becomes the Tetra (or delta-nabla) <img src="rhombus.png"> that describes the tensorial variations

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how calculate the volume of this object?
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<img src="64.png">

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<b>f(x,y)</b> is the set of all points of all planes <b>x, iy</b> 
considering them as a single one plane.

note that if <b>y</b> or <b>x</b> has value = zero, the four dimensions compact theirself 
and become two only , but we can consider the Im axes with only positive values and so the cube 
has not border in its two faces.
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<b><i> co-vector , covariant plane </i></b>
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the axes joining the coordinates to the point, remain perpendicular at the main axes, 
(but not parallel) 
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conventionally, indicated with lower index 
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<b><i> contravariant plane </i></b>
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the axes joining the coordinates to the point, remain parallel at the 
main axes (but not perpendicular) 
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conventionally, indicated with upper index
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