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Star of David

CHAPTER 12

STAR OF DAVID

UNIVERSAL TRISECTION OF AN ANGLE

As I have mentioned several times before, if there is no pain there is no gain, especially in the field of ancient times that are realities still unknown to us; a great spiritual reality, regardless of how such a reality falls into place, like a golden thread, throughout the history of mankind. Actually, we simply find it hard to believe because to us it seems incomprehensible because our dimension of comprehension is materialized energy of a twofold condition hence we are in collision with ourselves while a third dimension is in the domain of “faith”, but then again everything gets so entangled in stories that the common man can no longer “make head or tail” of it, as the folk saying goes.(and this applies to all peoples regardless of nationality or race). Thus a big Gordian knot was created, so big that even Alexander the Great could not cut through it with his sword. Why then do we not radically clear this situation? At least those who pride themselves as being the descendants of the patriarch Abraham should remember his prayers for Sodom and Gomorrah; or we Christians should recall Christ’s metaphor about separating the wheat from the chaff.

Therefore, we have no alternative left than to wait and on the way try to at least find a “better”, “more humane way of waiting”. However, some of us are “relentless”, individuals who live or die in searching, who believe that all things are not the way they should be, nor should they be; men who simply say “this is not my world”, men fated to suffer being called various names such as godless, heretics, scientific crackpots, unreasonable, ignoramuses – men who don’t belong anywhere, but go their own way, relying on that invisible cane that leads them along some path that has no pity for them but perhaps that is precisely so that they find their own world, the world that the ancients spoke of, saying that in the beginning it should have been or was or will be. That is why I am an adherent of Ezekiel’s Third Temple, because this contemporary Gordian knot can, it seems, only be unknotted by “a truth for all” (or the One that we call God – for all, as the “primordial” prophet Daniel said). But, nevertheless, we are only human beings. Half or even more of mankind prays for the coming of God, however, with us humans there is always a “but”… but give us just a little more time to do this or that, but just a little time to build this or that, but just a while more to enjoy this or that kind of life, and so on and so on, without realizing that this “but” is the wall between the dimensions of faltering knowledge and knowledge, of many truths and one truth, etc., and we will show that there is a universal truth by way of the geometrical example, step-by-step (only with a compass and unmarked straightedge) firstly, how I came to it and then confirm it by examples. That’s what in the previous chapter I promised to share with my readers and therefore I will briefly drop out of spherical (only with compass) geometry.

* * *

Standard setting of circle of a radius divided by 6 circles of same radius (divisional circles of the basic circle).

* * *

Division with full circles generates intersections of the divisional circles. We circumscribe them with a circle of that radius.

* * *

If we connect those intersections with line segments we acquire a star-shaped horizontal polygon and an inscribed rectilinear polygon of the basic circle – a hexagon.

* * *

And if through the poles of the basic circle (subtended through center to the perimeter of the divisional circles) we once again inscribe line segments, we will divide the basic circle into six equilateral triangles and parts of the circle – a cube of the radius sides of the basic circle under an angle of 30°.

* * *

But we won’t follow them, but the cube (or hexagon) of the circle’s intersection radius of the divisional circles – we inscribe its star polygon (the crosses divide the arc of the basic circle into six more parts).

* * *

These crosses serve to divide the cube into the cube of 3 (system of division shown in Book One – division of cube– chapter for kids) – the start of cube division is perpendicular in line with the center in order to “open” new points of division.

* * *

The cube is in line with the center or 1/27 (one twenty-seventh). Circumscribe it with a circle.

* * *

Divide it with circles of its radius and continue division with same radius all the way to the arc of the divisional circles radius of the basic circle (acquiring a Flower of Life perimeter of the Star of David) – just confirming the cube’s division of the cube of 3. Logically, this constellation is always the same. * * *

The circumscribed circle of the cube, and its radius, is divided with semicircles in six parts  (floral pattern – petals – division with semicircles). Inscribing of star polygon of basic circle (not necessary) but only confirms the 1/27 division – its full angle – its inscribed circles and its division into 12 parts – but we will disregard this for now (it is another theme). …

* * *

… since the petals divide the circles of the intersection radius of the described 1/27th circle – its circumscribed arc divides into 18 parts in tandem with its 6 poles. Straight lines from the center through every second point of that 1/27th divide its basic circle and circle (if observed only as 1/3 (one third) into three equal parts. All this is known to us from the trisection chapter in the First Book (Flower of Life, the nonagon, etc.). But… 

* * * *

New discovery

“Almost” all this regarding trisection of an angle has already been said in the First Book, its various possibilities both rectilinear and spherical, along with the “flower of life” and the Star of David (the hexagonal star polygon of the basic circle), drawn either shortened or complete, thus the reader will wonder what is so new about all this; what is so simple that it is (apparently) worthwhile to be repeated? Are there other enigmas in question if we observe the radius of divisional circles of the basic circle and their intersections in which the division of the basic circle into 12 parts is visible? We will also disregard all that. Is it a matter of a radius that is of importance for some other enigma? That is so, but we will disregard that as well. Could it be that we can divide the basic circle by means of the internal part (around the center) the Flower of Life into 9, 18, 36 parts? Well, that is right but we saw all that in the first book so that is nothing new. So, what is it then? Does it derive from spherical “angelic” geometry – it apparently does not, since we are using an unmarked straightedge, a cube and its division of cube of 3. This we also had, yet this is different. Is the radius of divisional circles and their intersections so important? Yes, but only as the “window or impetus to cognition”. However, one “triviality” makes a different kind of trisection from all the others, and we will show this through examples both single (approximately 120) and double (up to 240), shortening them so obviously so as to be understandable with a assistance of “a key” – the Star of David (the hexagonal star polygon, its circumference and its “floral-life force” division into six parts). However, words can “talk” without being understood, but the eye is “light to the body” in the way that the “spirit” expresses itself, so let us now see what is it that this makes trisection of an angle “universal”.

* * *

Example One: Single trisection of angle up to 120° – randomly given angle with its arc, chord and and centerline that divides it into two parts.

* * *

The chord of the arc is the side of the equilateral triangle. We divide its sides with centerlines – center of triangle – circumscribe with circle and thereafter from same center and arbitrary radius we circumscribe with a circle. The centerlines of the equilateral triangle divide it into 6 parts. Therefore, we can inscribe into that arbitrary circle its hexagonal star polygon. We leave the randomly given angle “on hold”. Our focus is on the star polygon.

* * *

Inscribe its internal star polygon (it is horizontal) and inside of it is another  hexagon – 1/27th  (one twenty-seventh) cube of 3.

* * *

We circumscribe it with a circle, and the radius of the random circle is divided by its radius into 6 parts (semicircles – petals). The internal circle of 1/27th part, its arc of petals is divided in tandem with its poles into 18 parts. we focus on one third of the system from the center, and from the points where the petals intersect the circle of that central cube part of cube of 3, we inscribe straight lines that on the way divide the circumscribed circle of the equilateral triangle, the arc above the chord, into 3 equal parts.

* * *

Now we have acquired trisection points from which we reflect half-lines in the direction of the peak of the arbitrary angle, which on its way divides the arc of the arbitrary angle into 3 equal parts. This is trisecting of the randomly given angle only with a compass and an unmarked straightedge for angles up to 120°.

* * *

As regards angles in excess of 120° (example larger than 120°) the procedure is the same, but we first divide the angle into two parts and the process is carried out only on one half (we don’t touch the other half). A big error would take place if the chord of the angle was inscribed for the whole angle – its arc. The result would be wrong (the parts would not be of three equal sizes).

* * *

Therefore, leave the given angle “on hold” and focus on half of it in accordance with the scheme of trisection of angle up to 120°. We divide the half with a centerline, inscribe the equilateral triangle of the half of the arc’s chord – divide the sides with centerlines – center, circumscribed circle of half of equilateral triangle of the randomly given angle.

* * *

Inscribe arbitrary circle – inscribe its star polygon, the first, the second, and inscribe the third part (the 1/27th part of cube of 3) by circumscribing it with a circle.

* * *

Execute division of randomly described circles – flower-like pattern – since it divides with its petals and poles of the circle of 1/27th  part into 18 parts.

* * *

Now reflect the rays from the center through every second point of eighteen (in other words project them onto the arc of the basic circle, the described circle of the equilateral triangle sides of chord of arbitrary magnitude of randomly given angle) and from these points in reverse towards the peak of the randomly given angle. (It will suffice it we only reflect on the first point of ½ of the randomly given angle) and pass it on the subtended side.

* * *

That is how we divided the randomly given angle into three equal parts – its arc with a compass and unmarked straightedge. Now on basis of these two examples we can explain the key difference and why we can say that it is universal.

EXPLANATION

Surely it has not yet been noticed in spite of the two examples (division into three parts if the angle is up to 120°). All this was demonstrated previously. We did notice the use of the “floral-life force” central circle. We also noticed that it derives from the Star of David or the hexagonal star polygon of a circle or as one twenty-seventh of a cube divided 3 x 3 x 3, once more with the aid of the Star of David, respectively the described circle of that cube (a twenty-seventh part). We made use of the chord of the circle of the equilateral triangle and found its centerline and circumscribed its center with a circle – and indeed it is a circle that “reflects” everything. Hence, it is always of importance to form points on it from which we can perform the trisection of any angle. Thus dividing it on the ninth, eighteenth, thirtysixth, is not important whether the division is from the outside or the inside. (We have seen the division of “the flower of life”, namely the division of its circumscribed arc into 6 parts, the radius of its circumscribed circle is formed by its “divisional petals”). Parts of the divisional circles are of the same radius, and otherwise they divide the radius into 18 parts or 20° x 18 = 360°.

So, my dear Archimedes, I’m sorry, but if you were to live in our modern times, your lever would for a long time be spinning the machine that emits a great deal of current. But if some hear this some of them will laugh and say did we not already do this with the lever law of Archimedes! I know we did, since I saw this in Switzerland by utilizing it, but in which way? What you give is what you get (or less – in mechanics of course). But… these are not just “variations” on the same theme. I’m sorry if Archimedes said this at all, namely that it is not possible to construct an angle magnitude of 20° only with a compass and unmarked straightedge. But the observations of who was by far ahead of his own if he had the peace and time for researching and creating. But, in order not to go too far, I had better explain this interpretation of geometry. We have adhered to the rules presented in the first book, and now the circle of any random radius, its star-shaped hexagonal polygon (a divided cube – again of any radius – arbitrary, cube of 3). Thereafter its division into 6 parts, its floral pattern divides the arc of the circle into 18 parts. These points reflect the straight lines from the center to the perimeter of the circumscribed circles of the hexagonal star polygon (two equilateral triangles of sides the magnitude of the chord of the randomly chosen angle). Thereby we acquired on top of the arc of the arbitrary angle points through which we draw the rays and in the measure of the peak of the arbitrary angle, and they again divide the arc of the arbitrary angle into three equal parts because as we have said – trisecting of angle only with compass and unmarked straightedge is done indirectly.

Thus, there is a difference. Any radius, randomly given, renders the possibility with its Star of David – hexagonal star polygon or cube of 3, a formation of trisection points from which we can carry out the division of the randomly given angle into 3 equal parts only with a compass and unmarked straightedge, and I repeat: of any radius, arbitrarily (without intersections).

Thus we can conclude something else as well. The “Star of David” or hexagonal star polygon is the key to the enigma (not only to the one presented). We would have come to this conclusion long ago, if it had not been fended off. If we made a closer study of the Bible we would know more. Many non-Christians do not know that in the Revelation one can read: “he will be given David’s key, and when one opens with it, it will nevermore be closed and vice versa” – (my paraphrasis). I know that an esteemed American Nobel prize winner established an institute for study of the Bible in order to get a series of data that would be useful to mankind in the various branches of natural science. Unfortunately I don’t know the name or whereabouts of this scientist but I remember seeing a documentary film about him on the German TV, but at the time I didn’t know that I was to become interested in geometry and therefore I appeal to my readers in America to further pass these pages on to him if they know who he is and where he lives. I do hope that he is alive and that his institute for research of the Bible exists.

This lunge forward from spherical geometry can only show how utilitarian just the one dimension or just the other (spherical or rectilinear) can be, but there is still a host of other data of importance to be found in spherical geometry, but now and then I have to jump out of the “framework” – (with compass only) because I know that many are eager for knowledge, although to be frank, I don’t know to what extent I will be able to follow the “spirit” that prompts me to continue (for after all I am just a man and – a poet).

HR – RIJEKA, 14.01.2012.
Tomo Periša

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