The Mathematical Dimensions.
Plane and Exponential dimensions
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The mathematical dimensions
Plane and exponential dimensions

* This study on the mathematical dimensions is born from a physical initiative to explain the fourth dimension of space.
And this is because the fourth dimension of space matches and belongs together with the mathematical characteristics.

To expose the topic firstly we should take conscience of our dimension and situation it the space, and from here, we can imagine a long trip through space to locate us at different levels of the same one, where we can see as what happens at our dimensional level also have to happen at any other level of the space, and of course, also in the mathematical field.

This way located at our level, on the floor or sand, we can draw a square of 1 meter of side.

Nevertheless, if we locate ourselves at atomic level, with the angstrom as unit, we could draw there also a square of 1 angstrom of side.
And if we make it at cosmic level we will also be able to draw a square of 1 light-year of side.

Now then, located in our level and without moving from it, we can draw a square of 1 meter of side, but also of 2 meter of side, of 3 meter of side, of 4, of 5, etc.

Therefore we notice that two levels or dimensions exist through which we can draw our square.

--- The Plane Dimension in which we are locate, and on which can make the draw directly, and the
--- Exponential Dimension to which we haven't access directly, but through our thoughts.

In the Plane Dimension or plane level, in which we are, we can draw that square of 1, 2, 3, 4, etc. meter of side.
But in anyone of the steps of Exponential Dimension, any other huge or minuscule inhabitant of these levels also could draw these squares.
For instance:
If we were of -221 meters long, we could draw our square on un electron.
This way, these two space dimensions exist, and they can be subdivided in the four dimensions of the space; the three Cartesian dimensions: longitude, surface and volume plus the exponential dimension.
Therefore the four dimensions of space will be: Longitude, surface, volume and exponential.

In Mathematics:

But off course, we are studying the problem from a mathematical point of view.
Therefore we can ask ourselves: What is the dimensional relation between space and mathematics? The mathematics, the same as space, or vice versa, also contain these two types of dimensions: The Plane dimension and the Exponential dimension.

-- The Plane dimension would be the natural numbers, 1, 2, 3, 4, etc.
-- And the Exponential dimension would be naturally the exponential numbers, although respecting the decimal metric system that we have given ourselves: 1, 0'1, 0'01, 0'001, 0'0001, etc. and in upward form 1, 10, 100, 1000, 10000, etc.

Exposed it in decimal exponents :
In descending order: 1, -11, -21, -31, -41, -51,

And in upward way: 11, 21, 31, 41, 51,

For it and summarizing, as much the mathematics as the space, they have two types of dimensions: The Plane Dimension and the exponential Dimension.
Inside these dimensions when we are formed by a certain quantity of space, we locate ourselves in the level 1 and we have capacity to move alone for the Plane dimension, that is to say, from the 1 meter toward the 2 meters, toward the 3 meters, etc., but we cannot move or enter in the exponential dimension toward the 0'1 meter, 0'01 meter, 0'001 meter, 0'0001 meter, etc.

Although this is not topic of mathematics, we must to do the observation that the exponential dimension of space also affect and is completed by the whole spatial elements, as they are matter, energy, field of forces, etc.
This way, the matter is constituted in exponential form toward the infinitely small or toward the infinitely big thing. So, stars are formed by other smaller exponential units as they are the atoms, and in turn, atoms are formed by other smaller units as they are sub-atoms, and so forth.