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Nonagon – only with compass

NONAGON – INSIDE (TRUNCATED METHOD)
ONLY WITH COMPASS

EXAMPLE OF TRISECTING

Although there are very many possibilities to acquire trisecting codes, as well as possibilities to acquire the nonagon and logically, the codes of all other polygons can be acquired. However, it being high time to manifest solutions of the remaining enigmas (doubling the cube, squaring the circle, constructing the heptagon and other polygons, prime numbers) we still hold that some more time must be spent on the initial nonagon solution – only with compass and unmarked straightedge because of one more element that is basic in the sacred geometry of ancient times. In addition to all the fundamentals that we have so far shown and learned, and these are known as geometrical combinatorics some of which we have “learned” by means of the ancient artifact called “the flower of life”, using only one of its elements (the described circle) whose radius divided into six parts divides its initial circle into eighteen parts in tandem with its six poles. We used this combinatory element as a scheme for trisecting. Besides, we have to comprehend and assume that the ancient geometrical artifacts are a part of “a message”. In the language of geometry these artifacts reached us as an aid not only in expanding our geometrical views but much more. One of the possible messages is to teach us about ourselves and the micro and macro world around us to which we also belong. Sure enough, the many “impatient ones” among us will promptly search for a numerical explanation for all these geometrical presentations. They certainly do exist but everything must have an ordered agenda of occurrences, a time and a place in this huge geometrical opus, and that’s why we are going step by step. It will then be easier once we comprehend the fundamentals and thereafter proceed further on to the “tones, rhythms and notes”, as musicians would call them. That is why we will continue with trisecting and the nonagon precisely because of combinatorics. In this chapter only with the compass and in the next chapter we will connect with one more essential element of the ancient artifact – the Star of David – or the hexagonal star-shaped polygon (combined curved drawing with compass and straight line drawing with unmarked straightedge) and use it as another method of creating the “flower of life” artifact, which lightly touches (the basic – the window) of the other realm of sacred geometry. Of course, in this chapter we once again have to rely on an altogether truncated form of applying sacred geometry at the price of losing an array of information. This construction of the nonagon is considered one of the simplest ways of manifesting nonagon trisecting.

* * *

A circle of arbitrary radius divided with the radius – flower-like pattern – hexagon.

* * *

Connectives – pole of hexagon – every other one; produces star-shaped hexagonal polygon. It creates six new intersections inside the basic circle.

* * *

We enter the magnitude of the radius into the compass – the newly emergent intersection (inside the circle)– the second pole on the basic circle (this time we designate it as r2 in spite of the fact that this should not be done in sacred geometry). Then with this radius we connect the poles of the basic circle with all six intersections of the star-shaped polygon. We acquire two kinds of points where lines intersect. We take the first ones – closer to the basic circle and circumscribe their circle – their poles from the basic center.

* * *

That circle has the magnitude of its radius and divides the basic circle into nine equal parts (starting the division from the peak pole of the basic circle).

* * *

On the other hand the intersecting points of the nonagon closer to the basic center have a circle the length of its radius.

* * *

That radius once again divides the basic circle into 18 equal parts and “confirm” the nonagon (every second pole of the 18 poles overlaps each 9 poles on the basic circle. Thereby we have in the simplest way “unsealed” two “impossibilities” of constructing a nonagon and the  Archimedean assertion that it is not possible to construct a 20° angle only with a compass and unmarked straightedge – or solely a compass – using combinatorics.

* * *

And this is what a full circular presentation without an 18-sided polygon presentation on the basic circle would look like. No matter how complex it may seem, the person who wants to construe it step-by-step will not only comprehend some, but all of the information and products of ancient geometrical presentations of Sacred geometry.

* * * *

In light of the fact that mathematics call for permanent exercise (repetition), we too will constantly repeat the fundamentals by making newer and newer additions (repetitions), hence incessantly reiterating the fundamentals so that sooner or later every child at the age when it starts learning geometry will be able to understand it.

So, let’s recapitulate:

  • every randomly given (basic or initial) circle has a radius that divides it into six equal parts;
  • every second radius divides the basic circle into some other number of parts;
  • every division commences from one point in the circle and ends in that same point;
  • the basic circle is a mirror of all “occurrences” inside and outside of it;
  • „occurrences“ are the intersection points arising from combinatorics and then they are radii of other magnitudes that are smaller or greater than the basic radius that divides the circle into 6 parts;
  • logically, these are other magnitudes of radius, some other number of segmentations of the basic circle;
  • divergent are the magnitudes of peak angles whose peak is the midpoint of the basic circle or basic center – the initial point of any presentation in an unlimited space;
  • combinatorics are the means by which we acquire other radii to divide the basic circle;
  • the sizes of these radii are not arbitrary but are determined by the intersection points (at least two same ones) and they can serves as an example;
  • the midpoint of the basic circle are the intersection points arising from the connecting of poles, hence those poles become the poles of the basic circle, etc.

Now we shall apply the emergent nonagon for another (single) trisecting of an arbitrary angle; an angle greater than 90° and smaller than 120°. For angles in excess of 120°, as we have learned, we will apply the duplicating mode.

* * *

Randomly given angle with its arbitrary arc, chord and bisector.

* * *

Chord size of arbitrary angle of the equilateral triangle.

* * *

The bisectors of the inscribed circle’s equilateral triangle form the center that circumscribes the triangle.

* * *

We divide the circle with its radius starting from the peak pole which is also the endpoint of the length of the chord of the arbitrary angle.

* * *

We connect the poles of the emergent hexagonal division of every other pole. We acquire a hexagonal star-shaped polygon.

* * *

From the intersecting points of the star-shaped polygon – we connect every second pole of the inscribed circle’s poles.

* * *

We circumscribe the newly emergent intersections that are closer to the arc of the basic circle (the described equilateral triangle).

* * *

With this radius we divide the basic circle which came forth as the described equilateral triangle with the magnitude of the chord of the arbitrary angle. By division from the peak pole we acquire the nonagon.

* * *

Thus we get trisection of a point on the basic circle. With half-lines through them towards the peak of the arbitrary angle we have divided the arc of the arbitrary angle into three equal parts.

* * *

Thereby we have trisected a randomly given angle into three equal parts with the aid of the nonagon scheme emergent by means of combinatorics, in the shortest way and solely with a compass.

* * * * * *

3 Responses to “Nonagon – only with compass”

  1. Ramin says:

    There is a much easier way to divide a line into three. I will not tell you my solution, since I wish to publish it on my own, but I advise you to think about it.

  2. Anubis says:

    Thanks for a wonderful demonstration, I cant help but wonder what is the logic behind this progression? How did you (or whoever did) reach this method, by what reasoning?

    Thanks
    Anubis

    • admin says:

      It’s simple:
      This is what The Lord says, hi who made the Earth, The Lord who formed it and established it: “My name is Jahve. Call to me and I will answer you and tell you great and unsearchable things you do not know.” (Jerremiah 33; 2-3)

      My prayer (in the year 2007.)
      Show me, please, Jahve, simple things unknown between us people, to make them real to serve for Goodness of Earth descendants, I pray you in the name of Jesus Christ. Amen.

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