Dimensional Geometry and Euclidian Geometry
The reversibility property in dimensions
The infinitesimal point in dimensions
Of ferman: Fernando Mancebo Rodriguez--- Personal page.

You can see many of my works, in the following pages:

Particular: My rural property with old house in the surrounding mountains of Malaga: for sale.

Email: ferman25@hotmail.com
Email: ferman30@yahoo.es

Property of reversibility in physical and mathematical dimensions
The infinitesimal point in dimensions

Euclidian geometry and dimensional geometry * Dimensional geometry measures and works on existent quantities of any dimension.
For example, the distance (or magnitude) defined and measured from 1 to 8 successive elements is 8 (i.e. meters), where both, the first one 1 and the final one 8 are included.

* Euclidian geometry measures and works on not existent quantities of dimensions.
For example, Euclidian spatial longitude is the distance between two not existent spatial points, because these points have zero magnitude, and so, not well-defined situation.

Properties of the dimensional geometry: divisibility and reversibility.

"The reversibility is a property that allows us to divide indefinitely to any dimension, conserving each one of the divided parts all the properties of the initial dimension; and having at the same time the property of being united and summed all these parts to build and form newly the initial dimension".

This way:

"Any physical and mathematical dimension has the reversibility property, in such a way that if we divide indefinitely any dimension in portions, all and each one of them will be converted into infinitesimals of this dimension, conserving the initial physical or mathematical properties, and not getting never the limit zero (0), in such a way that summed newly all these portions we can obtain de whole initial dimension."

This way if taking any longitude (line) we can divide it into continuous and successive portions, conserving each one of them the longitudinal property.
And later, if we turn again to their summation, all these portions can form the initial longitude.
In this case we can't reach any infinitesimal division of this line that gives us the absolute zero (0), because of in this case the portions stop to exist when being zero.
They hadn't the reversibility property either.

The same occur with the second dimension, the area.
The same occur with the volume, time, etc.
And the same with the mathematical dimension that measure to the other ones.

Dimension

Then with the anterior principles and concepts we can define the dimension as:

"Dimension is any physical or mathematical medium that contains the property and characteristic of the successive divisibility with conservations of properties in all and each one of its parts; and the property of the reversibility to return to be united and summed all these divided parts to form newly the initial dimension"
Say, property of total and indefinite division, and also total reversibility.

Although we can establish Dimensional Frames, where the divisibility and reversibility is not indefinite, but alone inside a wide frame, as for example can be the water an ocean.

The infinitesimal point of a dimension
The physical and mathematical point

In agreement to these definitions and concepts, we can establish a primary parameter on this, the point.
The point or infinitesimal would consist in an extraordinarily little quantity (for us) of any dimension, to which we can't reach never in practice, and so, we alone can consider it theoretically to be too small for us.
Although, in spite of being so little, it conserve all the properties of its dimension yet, not getting never the limit value of zero.
Say, a point or infinitesimal of a dimension is an extremely very small quantity of this dimension, but bigger than zero and with all the properties of this dimension; inclusive the reversibility property.

This way, and to distinguish it from the Euclid point, we say that a dimensional point is an infinitesimal of this dimension, bigger than zero, that conserves all the properties of this dimension.

In this case we can say that dimensionally a line is composed by longitudinal points, but not by Euclidian points.
Similarly, the interception (or vertex) of two dimensional lines must to contain a longitudinal point, but not a Euclidian point.
The same, the addition of longitudinal points give us a longitude, but not the addition of Euclidian points, which is nothing (0).

In the case of the Euclidian geometry, and concretely in his idea of point, I must to put some disagreement.
If the Euclidian point is zero (0) then this point doesn't have dimension nor situation inside space, and so, impossible of being taken as reference points to trace a straight line among two Euclidian points.

Dimensional concepts.

Mathematical concept of infinitesimal:

An infinitesimal of any dimension is a portion extremely small of the same one, theoretically gotten by successive e indefinite divisions of it, but that conserve all its properties yet.
This case, the infinitesimal is not an exact or concrete number, but a wide numeric rank that means extremely small quantity.
For example: 1/10^1000; 1/10^1000000000000; 1/10^10000000000000000, etc., can be considered as infinitesimals.

The concept (and quantity) of infinitesimal is born from the necessity of having minimum portions of any dimension or element for this way to be able obtain with these (by mean of union and addiction) bigger units and spatial and geometric figures.

Dimensions of space:

Point of space

A point is an infinitesimal of space that conserve all the dimensional properties of the space.

Longitude

It will be built by a succession and linear sum of points of space.

Area

An area is formed and built by a succession and lateral sum of longitudes.

Volume

The volume is formed by a succession and summation of superimposed areas.

Final consideration

In Dimensional Geometry the point is a common component for the other spatial dimensions, in such a way, that the addiction of spatial points can compose as well longitudes, as areas and volumes.
Contrarily, in Euclidian Geometry the points, areas and volumes are disconnected among them, in such a way that the addiction of spatial points don't give us areas, neither the addiction of areas can give volumes.