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Nonagon & 18-sided polygon

FLOWER OF LIFE (a component)

NONAGON & 18-SIDED POLYGON

–         TRISECTION EXAMPLE
–         120° ANGLE
–         240° ANGLE

FLOWER OF LIFE
In chapter 9 we made reference to the Flower of Life without any comments, even though it is one of the oldest artifacts in the history of mankind that one can encounter in its “hidden” form throughout the world; saved and veiled in secrets to this very day. However, it is not up to me to speak about it and its powers, but only about what I know from a geometrical perspective, the theme and aim of which is at present the nonagon and trisection; so now I am adhering to this. It is partially for these reasons that I demonstrate the Flower of Life from some other aspect in a manner that has thus far never before been presented, as can be ascertained from existing documentation. By using the cognitions of sacred geometry that I have so far learned, I’ve noticed that the descriptive circle of the Flower of Life, its hexagonal division, its “floral” pattern, separates the basic circle into eighteen equal parts, which corresponds to logic (3 full basic circles x 6 = 18 circles). This is another “clue” that the manner of dividing the basic circle is the division with diameters larger than those of the basic circle’s. That the Flower of Life in Abidos (Egypt) was not diagrammed by the Pythagoreans is also ascertained by the fact that otherwise a much greater geometrical knowledge would have been extended to our times. It is, however, likewise a fact that the flower of life is a symbol and an artifact that is prevalent worldwide, in various cultures and is profoundly prestigious. Concretely, for me, the flower of life is a geometrical acknowledgement of perfect evolutionary concordance, a point in space – a bordering of space – a radial division, an elementary beginning, unique and alone in space and time yet its products proceed into infinity.

But, lest we turn this into just some other story, let’s make it clear and recapitulate: the Flower of Life, its structure, and then let’s apply one of its products – the descriptive circle (shortened) as the trisection basis of the scheme or code.

* * *


The basic hexangular circle with its congruent circles – floral pattern.

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Floral patterns begin to form from the exterior intersections of the congruent circles of same radius.

* * *

From these new intersections of the same circular radius, floral patterns of six congruent basic circles form and create the pattern of the “Flower of Life”.

* * *

Now with semicircles of same radius we form floral patterns of all the circles.

* * *

Floral patterns of all the circles of the Flower of Life are constituted.

* * *

We circumscribe the Flower of Life with a descriptive circle.

* * *

We divide the radius described circle starting from its vertex pole (shortened) into six parts. Its floral pattern together with the polar points of the basic circle divides the basic circle into 18 equal parts only with the use of the compass.

* * *

If we take every second divisional point, we then have 18 ÷ 2 = 9. In other words, we have a nonagon that is our scheme for solving the trisection of angles up to 120°. We display the example of angles up to 120° and greater than 90° and of turning angles greater than 180° and less than 240° by an entirely shortened method according to the Flower of Life scheme.

* * * *

Example of angles up to 120°:

The arbitrary angle is greater than a right angle and less than an angle of 120° with its random arc, chord and bisector.

* * *

Its equilateral triangle has sides that are the length of its chord.

* * *

The bisectors of the equilateral triangle’s sides constitute the center of the triangle.

* * *

From the center we circumscribe a circle.

* * *

On its peak centerline we circumscribe circles of same diameter so as to get a total radius of the Flower of Life. We disregarded plotting the full Flower of Life since we now know its magnitude – three radii of the basic circle.

* * *

We circumscribe a descriptive circle the magnitude of which is three radii of the basic circle.

* * *

From the peak centerline we again divide the circumscribed Flower of Life with its radius into six parts.

* * *


We know that with that pattern we separate the basic circle into 18 partitions (every second one into 9) – the arc above one side – the chord into three equal parts. From these intersections towards the vertex of the arbitrary angle we divide the arc of the arbitrary angle with rays. Divided into three equal parts – we have three isosceles triangles with vertex angles of same angle magnitude. Which is? We don’t know, just like we don’t know the magnitude of the randomly given angle, but we have its trisection with the help of the “tool” of the descriptive circle of the Flower of Life. (If we want to know its magnitude we can apply chapter: “Determining the magnitude of an arbitrary angle”).

* * *

With this plotting we now explain the full shape of the Flower of Life, hence the division of its basic circle into 18 partitions, or 9 respectively, or the division of its arc into three equal parts, respectively.

* * *

From these points drawing half-lines towards the peak of the arbitrary angle we divide the arc of the arbitrary angle into three equal parts, only with compass and unmarked straightedge.

* * * *

Second example:

(for the first time in written history) a randomly given  angle greater than 180° and lesser than  240° with its arc and bisector.

* * *

This time we use its chords (not only one but both) of same length and separate them with centerlines.

* * *

Each chord is a side of each equilateral triangle that overlap (partially).

* * *

To find the center of each triangle, we plot the centerlines of its sides in a shortened mode so as not to “get lost”.

* * *

From the center of each triangle we circumscribe its descriptive circle.

* * *

To each circle we add one more circle. Their intersection is on the first bisector of the arbitrary angle and this is a confirmation of the correctness of our plotting.

* * *

First of all we divide one circle from its vertex.

* * *

Thereafter we do the same on the other circle from its centerline peak, to avoid “getting lost”.

* * *

In that way we divided each circle with the Flower of Life scheme. In this case we doubled two circles of the Flower of Life, meaning each fourth divisional point. Then, with rays leading towards the peak of the arbitrary angle we divided its arc into three equal parts.

* * *

 

And so we get three isosceles triangles or trisection of an angle greater than 180° with one element of a Flower of Life, its division into six parts of its circumscribed circle (twice shortened), i.e. simplified in order to make it comprehensible.

In doing so we eliminated the so-called “impossibility” of constructing a nonagon only with compass and unmarked straightedge (that mathematicians have been asserting during the past millennium), and in a similar manner the constructing of the eighteen-sided polygon (Archimedes). Since the Flower of Life is an exceptionally expansive area of geometry, it calls for a separate book. However, we will continue with the nonagon and its congruent angles in several other different ways.

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2 Responses to “Nonagon & 18-sided polygon”

  1. n13L5 says:

    Its a thing of beauty!

    But tough to follow in spite of your simplifications 🙂

    Was looking around the internet to make a round window with a nonagon flower in it.

    But now you got me thinking of your flower of two diverging circles..

    I know that wasn’t the destination of your work 😉

    Either way, I’m glad I found your site, I plan to return to it – the next time, I’ll come just for understanding, rather than a practical purpose of making something…

    Your work is much appreciated. Thank You.

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