Part 1         

                                                                  Fourth  Dimension

At present the term " fourth dimension" is used mostly by science fiction writers. Some scientists only  assume hypothetic possibility of the fourth and even the fifth and so on dimensions existing. But there is no authoritative and proved grounds to this assumption. It should be noted that speaking about the fourth and so on  dimensions their geometric-spatial  version is supposed. The notion of three dimensions: length, width, height (three dimensional system of co-ordinates) is an axiom. There is no subject to be discussed, but! Let's try to make some short analysis. 
              Point of departure
a)  beginning of co-ordinates
It is infinitesimal point (nought)  arranged in any point of space. It means that we can always  dispose it in such a way that any body, process  can be described in positive. (Mathematics in physics "doesn't want" to work with negative numbers: mathematics: 2 - 5 = -3, it's O.Key. But in physics  2 bodies - 5 bodies = -3 bodies, it's absurdity).
b) first dimension
It is a straight line, rather a ray, originating from the point of  the beginning of co-ordinates.(fig.1)                                       We take arbitrary length of a piece as a unit of measurement, for example: a meter, an inch, a mile etc. This measurement determines the length of pieces. Here mathematics is arithmetic.

c) second dimension
This ray originates from the point of the beginning of co-ordinates angle-wise 90degrees  to the ray of the first
dimension. (fig.2). We take the same unit of measurement like in the first dimension,  but it's not necessary. We
may take FOOT for example, it will be m x foot. We may also take a unit of area not in the form of
a  square,  but in the form of a parallelogram, a triangle, a hexagon and so on. It's important to understand that  the unit obtained is independent, non-derivative.
We may know nothing about the first dimension. But if we take a certain area (a sheet of veneer, for example); this sheet  is an independent example of the unit. A square meter  isn't   m x m  in the least.  It can be in the form of a  rectangle with its sides 0,5m and 2m, in the form of a triangle, in this case its
Fig.2                      area must  be equal to the above mentioned example. In general,  multiplication of a meter  by a meter   is similar to multiplication (or division) of a cow by a cow. Such units as an acre,a hectare and so on should be used . Therefore with the help of the second dimension  we can describe  plane geometric figures and lines and their mutual disposition  in the plane. Mathematics here is  planimetry and algebra.
    d) third dimension
            Look at the figure.                                
   You have already understood that cubic measure  here can be in the form of a parallelepiped, a pyramid and so on
    and can be named : a litre,a gallon, etc. The obtained unit is independent.                              With the help of the third dimension we can describe solid figures, spatial  lines and their         mutual disposition.  Here  mathematics is stereometry, algebra, differential calculation.    

                  Here comes the analysis itself.
It consists in successive comparison of each  dimension with the previous one and the searching  for possible  regularities on this basis. Don't forget that  absolutely everything  equals nought  in zero dimension.

1. The first dimension (linear) differs from  zero dimension so that  when we add it we get  the first independent  (non-derivative!) unit - the unit of length, scope of dimension from  0 to infinity. This dimension  includes  the previous 0 dimension. 
2.The second dimension(plane) differs from the first one by appearance of a new independent unit  - the unit of area, range of measuring from 0 to infinity and this dimension includes all the previous dimensions. 
3. In third dimension(space) a  new independent unit appears again.  The  unit of  cubic content, range of measuring from 0 to infinity. This dimension includes all the previous ones.
Here we have such  phenomenon as NEST-DOLLS.
Note: it's very important to understand every current  unit is independent.  For example: your parents are John and Mary, you may be called "the son of John and Mary" but you are a new independent personality with your own characteristic features. You are Steve.  And your son should be called "the grandson of John and Mary plus another grandma, another granddad", and his son should be called… It's not only because it's inconvenient , but dealing with you  I  deal concretely with Steve, not with somebody's son. I appreciate exactly your characteristics, but not the product of ýour parents'characteristics. The same is with characteristics of the unit of area : its characteristics are not  characteristics of the product of a meter by a meter .
      Conclusions:   during the analysis a certain regularity of consistency in dimensions' construction  was revealed:
    1. Units of measuring are positive numbers in a range from a zero to infinity.
    2. Each following  dimension includes all the previous  ones.
    3. Every dimension  creates a new, independent unit of measuring..
                The obtained regularity determines algorithm of consistency in dimensions' costruction.
4.Fourth dimension.
Three dimensions are depicted in fig.4. In order to construct fourth dimension one more axis should be drawn.  Drawing it  in the domain of  these three dimensions doesn't have any sense. And drawing it in the domain of negative numbers contradicts to algorithm. It should be noted  that Decart  system of co-ordinates ( axes at angle 90˚ ) isn't a dogma. For example, axes can have 60˚ as it is shown in fig.5. To draw here the fourth axis seems quite possible . Please try if you've got time. One needs to understand  that all the dimensions  are abstraction, created by ourselves for describing something. There is objective reality: space, material bodies, processes occurring in substance. We                                                                          have described space in three dimensions, it's time to describe substance.
  Let's put everything in order:
a) First dimension
    Here the unit is  - an agreed piece.

b) Second dimension
  It can be represented  in the form of one axis, because this dimension has already included the    previous one. Here the unit of  measure is an agreed area.   
c) Third dimension
  It can also be represented in the form of one axis. Here the unit of measure is an agreed cubic  content.
                                  / The same order is valid  for the following dimensions / 
d) Fourth dimension  
In order to describe substance which is characterised by density, let's draw a corresponding axis. As a result we obtain a new independent unit - mass. It is possible to accept: a kg., a pound, etc. By means of four dimensions we can describe any body, its form, its disposition in space and  the disposition as to other objects. Algorithm is observed.
  It is actually the fourth dimension. Properties of substance: state of aggregation of matter ( for                                           example: gas, liquid), atomic structure,etc. is presented in Chemistry.
     Let's proceed and create the fifth dimension.
e) fifth dimension          
According to algorithm  it must be the product of mass… into what? As substance can move then  obviously into the unit of motion. We know that generally accepted unit in this case is velocity. But it doesn't correspond to our algorithm because velocity is derivative quantity V= s/t. That's why let's  temporarily take the existence of an abstract unit of motion as a fact. As motion is always directed then this unit has a vector. Let's denote it  .  We know that maximum velocity of substance motion is velocity of light in vacuum. It means that our unit  ()   must have limit , but it doesn't correspond to our algorithm. Have we come into the deadlock?  Not at all.  If we take it not as a dimensional unit but as a coefficient then everything will be put in order. Let's name this unit - a coefficient of  processes intensity.We can take maximal value of a coefficient = 1 or for convenience = 300thousand (meaning the velocity of light is 300.000km/sec),  it 's not important, the matter is this coefficient can describe any motion without ambiguity.
For example,  (Kmax = Klight): a car moves at the speed of 100km/hour  - the coefficient here = 0.027, and the speed of the rocket = 2000km/hour, here the coefficient = 0.55 and so on. You aren't accustomed to it yet?  But it's correct.
You'll make sure of it in the further account. Product of mass into a vector coefficient gives a new independent unit of dimension. Let' denote  it   . The dimension range is positive numbers from 0 to . How shall we call it? The correct name can be a pulse, but this term exists already. Our unit has quite different meaning,  that's why let's name it  vector mass  ().
In fifth dimension  rectilinear motion of a body or several bodies , their possible interaction ( everything is going on  in one line) is described.  We are on the boundary between classic and quantum physics and have entered  the area of vector algebra. Algorithm of dimensions' construction has been observed, we can start with a new dimension.
(By the way, those who would like to travel in 'other' dimensions, the fifth one suits here well: make some steps  - and you are there).

 f) sixth dimension
Substance can move not only along a strait line but also in the plane.You'll get convinced in it when you pour some water on any surface. We'll create the required dimension for describing such  processes. I think it's not necessary to explain what axis should be added.  As a result we get a new  independent unit  which equals the product  ×.  As you can see this is the product of two    vectors. Sorry, but let me remind you:  in vector algebra there are two  multiplications:
                                                       The first   one is scalar ?×b×cosφ, as φ=90?, in this case the product equals -0. We aren't going to  examine  this version here, bit it should be pointed out that this formula is the basis for the unified field theory. We can discuss it if you 'd like to.
                        The second  is the vector product. This is a new vector with scalar quantity  which equals a×b×sinφ, we have k×i×sin90? i.e. i1 =  k×i. But its direction is perpendicular to the plane of co-ordinates . Moreover, this vector is non-commutative. Simply speaking, it can be directed to both sides. (fig.a). What does it mean? I am going to explain it but at first let's transform  this figure. Let's take one of two possible vectors - the lower one at first. According to the rules of vector algebra we can carry  vector (preserving it being parallel). See  figure b.
 We have already mentioned a jet of water falling down on the plane. Here we describe similar processes. The jet, when falling down, spread out  at the angle of  φ = 360°, but if we pour it in the corner of the room it will  spread out at the angle of  90°. Similar processes  take place  if we take the second vector. Let's direct a jet of water up from a hose (ignoring Earth's gravitation) .It concerns non-commutativity.
Please pay attention to the fact  how clear the regularity of dimension's presenting in the form of one axis  is exhibited here.
We have created the next dimension, algorithm has been observed,but we can't see a new unit of dimension. In fact   multiplying quantity  by a coefficient we will not get a new unit . But multiplying this quantity by a vector  coefficient we get a new unit. This unit has different direction of a vector. So algorithm has been observed.  Here  mathematics is  vector algebra and everything  that we used before.
( Travelling in this dimension is possible, but  very undesirable).
g) seventh dimension
Substance can move not only along the straight line or in the plane but also in the volume: sound, light,
explosion and so on.
It's not necessary to describe the construction of this dimension , it is similar to the preceding dimension.
Let's talk of  NEST-DOLLS. The last three dimensions exhibit this phenomenon explicitly: substance when moving  "finds" necessary dimension itself depending on conditions. For example, a flying stone as it struck the wall scatters - goes from fifth to sixth dimension. Water running down by a hole from the surface -  goes from sixth to fifth dimension,
a gun shot - goes from seventh to fifth  dimension, and a mere explosion  - goes from  fourth to seventh dimension and so on.
Now let me explain what sinφ is. Shortly speaking this sinφ  forms wave gist in any motion.You can ask me a question: what about  rectilinear motion?  Let's use vector mathematics here. You know that two and more vectors in the plane or in space  can be converged  into one vector by means of adding together. But we can present one vector in the form of  two or three ones. Thus  rectilinear motion is a particular case of  motion.
                             So the attempt to explain "the fourth dimension"  has brought us to the creation of a certain system.                                  This system permits us to answer many questions.
                                                                       Some additional information and examples
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