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Final Angle Trisection (second part) “Angelic” Codes


CHAPTER 30.

FINAL TRISECTION (2nd part)
„Angelic“ Codes

I should have written all this in the form of an enigma in a manner that the ancient mathematicians have passed on to us (so they say). I express the supposition “they say”, for who knows whether they knew or were imparted to know; because surely they were not equipped with the “instruments” (read: a compass for precise drafting and a base, i.e. a sheet of paper of adequate thickness so that the leg of the compass needle can be pressed into the same point for a number of times without causing a shift). Hence, even if they did have the knowledge, I rather suspect how and in what way they could have arrived at a completion. Even our present-day contemporaries are not able to “relatively” draft this since I think that we hae not yet reached the degree of development of the mind and technology to be able to pass on our knowledge to future generations with an “exactness” that had once upon a time existed somewhere. When and where, for the time being there is no answer. Namely, what is it all about? Various geometrical forms of the past have reached our time, artifacts wrapped in “secrets”, some of them simply because they were “replicated” from generation to generation or because of a “beauty” that was pleasant to the eye, emanating a harmony, and so they became ornaments on various temples and with the passage of time they turned into styles. And today, nobody asks what are they? Pictures, we might say… but nevertheless from another aspect I’ve coming to the possibility that from a point a triangle can be constructed, and the center of its circumscribed circle can be determined, the center of its height and everything else that can be derived from it! The prerequisite: only with a compass (requirement of “angelic” geometry or read: of spherical geometry). It may at first glance seem that we failed to interpret it, for example due to our incorrect formulation of the trisection enigma and division of the given arbitrary angle into three parts. Three equilateral triangles that overlap with complete identity – requirement: only a compass and unmarked straightedge – which is “incorrect” or needless. Triangles: triangle are a result of the division of an arc of some angle, and not the base of sectioning as if we are starting to build a house from its roof down to its ground-floor. And the ground-floor is the spherical geometry of every enigma. The perception of our mind is the fundament and that is why everything starts from there. Whatever instigates the mind, we still call “the spirit”. About it, we know even less. As regards the “spirit” it is simply inevitable to know that it knows! And once it “pushes us” onto its own path, it is then probably the right time for us to know as well. That is why in this chapter we will announce how to correctly construct a triangle, how to determine the center of its circumscribed circle, and by the way how to trisect a given arbitrary angle and go onward from there. And that will be the first time in the new era of our civilization – only with a compass. When it is here, than is is certainly the right time.

* * *

But to avoid getting lost in this “new world” we will in the beginning use some of our world’s geometrical features for triangle vertices (letters A,B,C), until our mind  “gets used” to the “spherical”. Therefore, here is point (A) in space. 

* * *

Set any arbitrary magnitude in the aperture of the compass. From point (A) circumscribe a full arc – a circle. 

* * *

On the arc from point (A) determine another point (B) and from it circumscribe a circle of same radius as from point (A). 

* * *

With their intersection, circles from A and B produce a third point (C). From point C we circumscribe a circle of same radius as from points A and B. Thus, points A, B and C are the vertices of triangles produced only with a compass.

* * *

Now our aim is to find their center, again only with a compass. It won’t be as simple as it is with rectilinear drawing, since it involves geometrical authenticities of its own. So, let’s follow them. Starting with the external intersections of the circles of the same radius (dotted line), all three circles are divided into their six segments with semicircles.  

* * *

Three circles divided into six segments. Triangle (A, B and C) is in their center. Partitioning the perimeters of the circles we get the poles of this division. 

* * *

Now from point A, we take into the compass aperture the magnitude of the subtended intersections of the circles. This is immediately the radius that we will use from all points, intersections and vertices in sequential order. 

* * *

At first, from the vertices of the spherical triangle, i.e. from A, B and C (by means of partial semicircles that do not exceed the perimeters). 

* * *

Now, take the same radius of the intersections from the three circles. 

* * *

And then, the same radius of poles resulting from the division of circles with their radii into six segments (division goes the whole way to the bordering perimeters of the three circles (as dotted). 

* * *

From the poles of triangle A, B and C, thereafter from the intersections of the three circles and ultimately from all the poles that resulted from the division of circles with their radiuses, we have acquired three star-shaped spherical hexangular polygons but we did not stop at points A, B and C but we go on to the perimeters of the three circles. On these perimeters we acquire 6 points. 

* * *

In the compass aperture we immediately take from one of them the magnitude of its adjacent points.  

* * *

And along with the aperture from its external points we acquire a “floral pattern” that has its own center.

* * *

This center forms the center of triangles A, B and C hence just with a compass we have found the center of (spherical) triangles A, B and C. We circumscribe them with a described circle. We could “pause” here for a moment just to see where this is leading us. 

* * *

We have already noticed on occasion of drawing the hexangular star-shaped polygons of the three circles, that within the A, B and C triangles a pyramidal shape arises. Thus (from point B dotted) we take the radius to the pyramidal peak into the compass aperture.  

* * *

Therefore, the radius from A, B and C is described the whole way to the perimeter of the circumscribed circle of the triangle.

* * *

Thereafter, from these points we proceed with the same radius. This radius (the described magnitude of the triangle center is larger than described triangles) divides the described circle of the triangle into 9 segments. Therefore, into one of the trisection codes for (simplex) angle magnitudes up to120°.

* * *

Allow me a rectilinear explanation, as well as a trisection example at the end of this chapter, but before that let’s see what else we have here. 

* * *

Within the A, B and C triangles some other intersections resulted during the construction of the star-shaped hexangular polygons of the three circles. Describe them with a circle. A smaller radius than the radius of the described A, B and C of the spherical triangle comes forth. 

* * *

That radius divides the described circle of the A, B and C triangle into 10 segments (only from peak pole „C“). So, from three poles that would be 30 segments.  

* * *

Curiosity leads us further on. The “floral pattern” solution of the center intersects the described circle of the A, B and C triangle. We take into the compass aperture from the vertex of the third C intersection and with that radius we divide the described circle of the A, B, C triangle.  

* * *

We divided it into 5 segments (only from vertex C), and if it were yet from vertices A and B that would be 15 segments. 

* * *

And the division into 15 segments can be carried out from the radius of any vertex (A, B or C) of the second passage of the floral pattern that composes the center. 

* * *

Pentadecagon. We can already perceive the sequence of number 5. 

* * *

So let’s go back to sequence 5 aperture of vertex C and the fourth passage of the floral pattern, the creator of the triangle center. 

* * *

Decagon. Here we have more and more sequences of products but we are still to establish just how far the concept of the described circle of triangle A, B and C goes.  

* * *

And when the perimeter of the described circle of the A, B and C triangle was divided into their 6 star-shaped segments, other points were constituted. Let’s take into the compass aperture the second point C of peak C. 

* * *

Here we exit from sequence 5 because that radius divides the described circle into 31 segments only from peak C. 

* * *

Certainly the concept of the described circle of the ABC triangle (or any other equilateral triangle – spherical or rectilinear) creates the concept of three, the “flower of life” respectively. 

* * * *

But before we go on to the promised example of angle trisection, I can declare that the pores of beauty of the “flower of life” are the continuation of everything that brings forth one of the overall most beautiful geometrical presentations that when viewed holds within itself the thing we could call the peace of the soul, and its attractiveness and richness of expression (products – a merger, a pyramidal concept and sequences of others) induced me to call it “Solomon’s Lily”. On the other hand, when observed geometrically, many other things will be found here, from the secrets of the pyramid concept to the unifications (fusions) of sky and earth, to procreation. May my readers not bear a grudge, because I’m so high spirited. But, I ask you, have you ever known what has been presented in this chapter? To conceptualize a triangle from a point only with a compass, and determine its center – again only with a compass. The former, maybe (only the triangle), but the latter I doubt it. We would find ourselves in a problem (or as folks would say “in a mess”). But, I can explain one of Christ’s comparisons concerning endless human adaptability and angels in reference to geometry. In what way? When the former part, the triangle, is spherically conceptualized, the center is conceptualized rectilinearly. And in this manner we then determine the aperture of the compass from center to any point, intersection or passage that “complies with” the  conception of the spherical (angelical) center. This system of our adaptability can indeed be applied in many situations (my comment: in a positive and a negative sense). But let us proceed with the example of angle trisection by applying the hitherto presented concept. 

* * *

We draw a given arbitrary angle of indefinite magnitude with its angular arc and to the very end we leave it in a state of “inactivity”. We are solely interested in the points of its arc (endpoints) A and B. 

* * *

We conceptualize the radius of triangle AB and thereby acquire point C of a spherical equilateral triangle with its three circles.

* * *

With the same radius we divide it into 6 segments (we are looking for the triangle’s center)… 

* * *

… in accordance with the concept that is presented throughout the pages of this chapter. 

* * *

Six-sided star-shaped polygon… 

* * *

… with division extended to the perimeter of the circles. 

* * *

We find the center and take the ABC range into the compass and the intersections internal to the triangle beside the center. 

* * *

The nonagon of the described circle (above the arc of the given arbitrary AB angle of the described circle of triangle divided into three segments) with their shared points A and B. 

* * *

From these points and in the direction of the center and vertex of the given arbitrary angle of trisection. The arc of the given arbitrary angle is divided into three equal parts but we will take the liberty to “verify” this rectilinearly, although we don’t need to because the angle trisection reads correctly and the division of the given peak arbitrary angle is also divided into three equal parts. 

* * * *

AFTERWORD TO THE CHAPTER

As we have so far been able to learn and grasp, if angles are larger than the “tools or codes” by which trisecting or sectioning is performed (e.g. a given arbitrary angle of 170°) we divide it in half and trisect that half and then we shift the third part to the arc of the second part and vice versa. If the angle is 40° for example (approximately or apparently) we double it in order to trisect it with the “tools” or 120° code, and then we return the trisected part with the ascertained – bisector. It somehow seems to me that the sectioning is not of any major importance or purpose in all the concepts of the ancient-geometry heritage, yet againit is the simplest and so I wonder wherefrom did the adjective “most famous” come from. It could be just one of our human afterthoughts. I merely do not see its purpose (application). I find knowing how to determine the given arbitrary angle only with a compass more meaningful in acquiring the magnitude of a peak angle, and with such information one can logically find the magnitude of the other angle in the equilateral triangle, and then the peak angle can algebraically be divided or sectionalized with any number we want. But that’s another subject.

We have somewhat reckoned with the theme of trisection for who knows how many times. But we are looking forward to go to “higher levels of perception” and small discoveries that will surely bring some light into the “darkness” of unknowns, dogmas and our human “wanderings”. Therefore, we look forward to the continuance of the spherical triangle – Solomon’s Lily.

HR – RIJEKA, June 7th, 2012
Author : Tomo Periša
English translation: S.F. Drenovac
Web Master: Slim

 

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