Bend coefficient of curves.
Of ferman: Fernando Mancebo Rodriguez--- Personal page.

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Bend coefficient of curves
Bend coefficient of the circumference
Circumference's fourth dimension property

From the author: * This is a new work coming from the study and resolution of my squaring Pi to which I will treat of demonstrating, by means of the explanation and application of the bend coefficient, of being the real value of Pi, I think.
Nevertheless, I will respect the process of demonstration of the squaring Pi at the moment, so later to enlarge a little in what the bend coefficient consists.
Of course, this is a first taking of contact with this coefficient (2009-Dic) that will be analyzed more deeply in the future. :

In the web of Squaring Pi I began to explain and to try of demonstrating that the algorithmic Pi is erroneous in the following way:

""Well Ferman, very interesting, very promising, very.........
We have the Squaring Pi that is function of the perimeters of the inscribed and bounded squares of the circumference; that is function of 2; that is a very promising exponential number for the development of the cosmological mathematics, etc., but:
How you would explain us the numeric difference between that Squaring Pi and the results that we obtain with the algorithms for Pi?"".

Well, I believe that this is explained and it can be understood by means of what I call "bend coefficient of the circumference", and of course of any curve.
If we notice the algorithms, let us put as example to the Liu Hui's algorithm, in this algorithm we proceed to sum in lineal form (in straight line) to all the bases of the obtained triangles, while what would proceed would be the sum in curved form applying the bend coefficient that logically has to take the circumference.
As we see in the drawing (c):

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If we unite for their sum to all the triangles' bases in that we go dividing the circumference, we see that the result of the union of these triangular bases is a straight line.
But not, what we sought to obtain was a circumference or circle with the sum of these triangles.
Then geometrically it is not the same a thing that the other one.

But what really lacked in this sum.
Of course, the adaptation to the circular form by means of the application of the bend coefficient.
In this sense, if we apply the bend coefficient of the circumference to the results of the algorithmic Pi, we would obtain the real Pi, that is to say, the Squaring Pi.

Revision of the bend coefficient of any curve.

If for example we have a series of points forming a straight line (anterior drawing, a - b) and we treat to bend this straight line to form a curve, we see that the points that compose it (b) they change their position and they join in a different way.
This alignment change of the points creates a new geometric form, which will take also accompanied a change of space distribution and of space dimension.
In this sense this author considers that this geometric change takes also accompanied a dimension and measure change, question that could take us to a conclusion or theorem on the transformation of straight lines into curved lines:

"Any straight line that is transformed into a curve suffers a dimensional variation, appraisable by means of a bend coefficient ( ."

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Now then; any straight line, to which you apply a regular coefficient of bend, acquires a degree of this curvature, with which, we can get a circumference whenever this straight line has the enough longitude.
Then:

"Any regular curve can give us as result a circumference, if its longitude is enough."

This curvature's property would take us to appreciate in the circumference tetra-dimensional characteristics as we will see.

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The circumference and the fourth dimension of space.

This author understands that the circumference contains and on it is defined the fourth dimension of space. (Cosmic model)
It is this way because we can observe that the circumference conserves the same properties, it is which is its dimension or measure, but to appreciate these qualities completely, we must locate ourselves (mentally, of course) into its dimensional level.

In this sense, if we observe a circumference of a meter of diameters, we will appreciate certain bend degree to be formed.
On the other hand if we could locate ourselves (with our short dimensions) around the circumference of a galaxy, we will observe it with inappreciable curvature due to be so big, which will appear in front us as a straight line.
Contrarily, if we observe a circumference with a millimetre of diameter, we will believe that its bend degree is enormous when seeing it as a single point.

But not, all the properties of the circumference are the same ones either at galactic level, at our level or atomic level.
For this circumstance, here we will use adapted formulas to the different appreciation levels of study of the circumference, applying some adaptive longitudes at these levels.

In this sense, we will use the different measure units as level units to study the circumferences that are created on these dimensional levels.
We will have this way, metric circumferences, decimetric, centimetric, milimetric; decametric, kilometric; of years light, of Anstrongs, etc.
All that with the intention of locating us on certain level, and there to build, to study and to understand the bends of the circumferences that we build.

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In the previous formula, we have different parameters to build circumferences starting from a certain straight line.
In the formulas we see the following terms:

--L that will be the dimension and measure of the straight line with which we will build the circumferences.
This term at the same time locates us on the level in that we want to make the study.
For instance, if L goes in centimetres, then we will use this centimetric level to build the circumferences; if L goes in light-years, we will locate ourselves at galactic level to build them; etc.

--L' will be the measure of the curved line that will form the circumference, semi-circumference, circumference of multiple whorl, etc.

--Cf will be the bend coefficient that we apply to L to get the curve or circumference.
I should clarify here that Cf is a relativist value; relative to us and our observation level, or in any case, relative at the metric level that we are using.
If we give different bend coefficients, it doesn't means that we are changing the tetra-dimensional qualities of the circumference neither of its coefficient ), but rather we are giving different relative bends to us, with which we will obtain the idea that different types (more or less curved) of circumferences exist, question that is not real but apparent, since the values and qualities of the circumference and its curvature coefficient ) are always the same, and relative to: ).2Pi.R.

--Cn will be the portion or number of circumferences that we build with L and with the coefficient Cf.

--NÂș will be the degrees that the portion or number of created circumferences contain.

-- ) is the Bend Coefficient of the circumference that represents the loss coefficient of longitude to build a circumference of L' = 2Pi and at the chosen level.

-- R will be the radius of the obtained circumference (circumference of multiple whorl; circumference arc, etc.) that we have formed with the previous parameters.

Referring us again to the tetra-dimensional properties of the circumference and its influence in the cosmic structuring, we see that infinite circular forms exist through the whole Cosmos and at multiple and serial levels.
This way at our level, we would have planets as circular elements, built by means of the properties and parameters of the circumference.
At superior level we would have stars, later on galaxies, and so forth.
At inferior level we would have atoms, later on particles, and so forth.
I say "so forth" due to if the mathematics don't have numeric limit toward the very small numbers neither toward the very big ones, and the circumference neither has, because in the same way space, time, energy, matter, etc., neither they should has.
This way, we see that the circumference is the method of construction of all the basic elements of the Cosmos and that the same quadrature of the circumference (squaring Pi) is the one that would take us to build different cosmic levels, with different types of basic spherical elements as we have seen: Particles, atoms, planets, stars, galaxies, etc.
These considerations are also coincident with my cosmic theory, of which I put the following drawing, Tetra-coor, in which the different levels of the Cosmos are represented in hexahedron form: Each hexahedron represents a level.

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*** As we have pointed when we spoke of the applied coefficient Cf, the uses of different coefficients (bigger or smaller) to the circumference it is more mathematical and geometric theory than cosmic reality, since as we have seen, the specific and tetra-dimensional properties of the circumference make it to be built with the same bend coefficient in any place and levels of the Cosmos.
Another different question is that for its study and good comprehension we can observe the curvature processes from different positions and with different considerations.

In this sense, we can consider to the applied coefficient Cf as a relativist and three-dimensional parameter by means of which, and from our level in the Cosmos, we can express our perspective of observation of the circumference in its different levels.
This way at deep inferior levels (i.e. at micrometric levels) and from our perspective the bend coefficient would be enormous; while at galactic levels, and from our perspective and situation, the bend coefficient would be almost inexistent.
On the other hand, the circumference that has tetra-dimensional characteristic, it can be located in any level containing and using the same bend coefficient.

But at the same time it is a work method that allows us to build circumferences at different levels, of course, starting from the metric level of work L that we have chosen previously.

The following drawing helps us to understand a little better the invariability of the circumference to any level and dimension.
In it we can see as the angle of curvature to build circumferences of different dimensions is always the same ones; and so, the curvature coeffcient is the same one also.

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With all that exposed previously, I would express a personal conclusion, maybe little accepted or respected:
"When we stay analyzing deeply multiple circumferences of very different radios, what we are really observing is some qualities of the fourth dimension of space."

Ferman: 2009-Dic