*Dimensional Geometry and Euclidian Geometry *

*The reversibility property in dimensions *

The infinitesimal point in dimensions

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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.

"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.

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. 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.

Differences between Euclidean geometry and dimensional geometry

Simplifying we could say that Euclidean geometry is eminently abstract, while dimensional geometry is rather practical and applied.

Euclidean geometry.-

To begin with, Euclidean geometry contemplates the point as an element without dimension.

Then consider the line as an infinite set (inf.) of points.

And consider the volume as an infinite set (inf.) of lines.

These assumptions have certain contradictions, which will begin when we use the term infinite in these compositions.

The infinite concept in operations almost always leads to indeterminacy and the difficulty of achieving true simple measurements with points, lines and volumes.

For example, if we say that a line consists of infinite points of 0 dimension each, then any composition of points to create a line is not possible. Nor is the creation of a volume with line composition.

Line = 0 x inf. = undeterminedNow, if we want to start building a geometric figure on paper, we already need the space dimension to build it.

If we start by putting a point on the paper, this point has to be anchored to a point on the paper that does have dimension.

If we mark a line between two points of the paper, this line is anchored to the paper-space and when we measure the line what we measure is the marked paper.

Therefore, we see that the abstract of geometry produces certain inadequacies in geometric constructions.

Dimensional Geometry.-

In order to understand it a little better, here we will start by practicing.

If we measure a rod with a meter which measures 30 cm, what we measure is the length of the rod component material, from centimeter 1 to centimeter 30, both included.

We do not measure from an assumed imaginary point (and without dimension) that is at the beginning and end of the rod, that is, the inner material element of the rod is the one who expresses its length.

The question here could be: And why the application of this dimensional geometry?

Well, so that there can be a continuous and staggered composition of more complex dimensions by adding more simple ones.

Points make up lines; lines make up surfaces and surfaces make up volumes.

Graphic or visual point

Although this theory considers the point as infinitesimal, for practical use in drawings we can use approximate graphical points of 0.1 millimeter in length.

For surfaces, the graphic "surface point" could be 0.1 mm^2

And for volumes we would use the graphic "cubic point" of 0.1 mm^3.