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The first „Angelical System“ – Only with compass

CHAPTER 1

The first „Angelical System“ – Only with compass

Since I have brought the first year (book) to a close, where we followed the basics of sacred (natural) geometry that only indicated the immeasurable extent of this discipline, logic tells us that there is nothing impossible in it and it can solve everything only with a compass and unmarked straightedge. The sole requirement is researching, discovering data and grouping it, systematizing it and using it appropriately in all the branches of natural science (in spite of the fact that I primarily took upon myself the mission to simply make such information known to the public and to make random comments only here and there just to entice the readers’ “curiosity to explore” with the aid of this percept of geometry, as well as others beyond it). I am sure that such “others” are already present in this time, and that before long civilization will not need to wait any further for the egression of new results and forward movements even though at present they are not visible anywhere except amidst the solitude of thousands of youngsters “of an alternative spirit” in the pursuit of newness. And that newness will certainly come sooner or later whether we want it or not.. For if we were to stay where we are now, we would “stagnate” and a “stand-still” is not in the nature of the human spirit because in that case, as ancient books say, “man would be signing a contract with Death”. No matter how impossible this may now seem, it will suffice to look back just a hundred years ago and compare that time with today and we have the answer, for at that time all that we now have seemed impossible. Even though we live on Earth, „we are travelers through time”. If we are to judge by motility then, in a roundabout way we are returning in a roundabout way (from a geometrical aspect) to the starting point from which we were launched into this journey to learn what is presently just something portended. I’m certain that I have initiated a theme that is not exactly in the spirit, or does not suit the spirit, of present-day man, for it is easier not to exert ones effort and to simply surrender to the enjoyment of what is now; or as the case may be. Centuries-old experience (history) tells us that civilizations declined precisely due to stagnation or surrendering to pleasure (but again, as the case may be and at whose expense and efforts?), thus our contemporary civilization will surely disappear (not people) just like the ancient civilizations of Egypt, Persia, Greece, Rome, as well as those of mediaeval times and times closer to our own; and the cornerstone of all those disappearances are in the new knowledge and endeavors of those “restless pioneer spirits in search of wisdom or matters close to wisdom,” who in point of fact understood its principles that led to new exacting possibilities – to deeds – not philosophizing. Although I myself am aware of the mentioned long journey, on basis of just this geometrical initiation I too have become a part of that journey. In the first chapter of this new book (or new year) I start out by giving it the apparently strange title “Angelical Geometry”, in so doing, I had in mind the spherical drawing of lines (that differs from the rectilinear drawing) and the series and series of progressions that were presented to me during my research for the first book but were not shown, in addition to my daily new observances of nature (seen and unseen) as well as on the basis of cognitions by others who have access to technology to observe micro-nature (and it is in that domain that I comprehended the existence of an initial and genuine geometry). Its first dimension emerges because in nature there is no such thing as rectilinear movement for umpteen and umpteen reasons and conditions that science no doubt envisions slowly but surely. However, looking into spherical geometry, I comprehended that this that I will try to present is just the “primeval” beginning that will not become a reality until the new technologies are incorporated into the verification procedure of this kind of geometry. In order to get clear and exact results in the future, it will arrive from individuals possessing (in our opinion) undreamed of possibilities. They will come from the future like “angels”, but nonetheless subjected, or more appropriately, in harmony with the universe and everything in it.

I realize that I am going on this new journey “like David armed with a slingshot”, in other words with a primitive and ancient tool, i.e. only with a compass. But in this way it will be possible to crystallize the umpteen omissions of contemporary geometry, though some may say that this is not of any particular importance? So my question is, why the persistence on something being impossible if it isn’t of any importance? And wherefore such resistance over something that is not essential (which I am confronted with on my road)? For me, it is not important because I have no obligations towards institutions nor anybody else, but of my own free will I am undertaking as much as I can of an obligation to inspire the forthcoming generations to reflect on whether cognition is important or not. Therefore it may happen that this book will also fail in getting a review from relevant institutions, but I am pleased that from time to time I receive reactions from many persons who agree with my deliberations, for I am but a man, thus an encouraging word even when it comes from some irrelevant unofficial person is to me like “bread and a glass of water” on this journey that I am taking and that I may even call “the way of the cross”, Thus, may nobody bear a grudge against me if I sometimes make an error especially in the “angelical geometry” in which just a second of a degree can change the product (result), so lets direct our attention to the “how and why” principle, and the future computer technology will come out with the exact results.

Spherical geometry certainly rests on an x-series of progressions or systems, but it is necessary to start with one, exploring it and its products. Certainly my intention is to systematize all these cognitions, but it also seems more important to present as many results as possible, step by step, only with a compass, and how they come forth and to adhere closely to the basics of Sacred Geometry – in full circles, but nevertheless truncated into semicircles or by presenting some of the results only within the basic or initial circle, in various ways, partially for easier reference and partially for better layout of targeted and important results.

However one should always bear in mind that the basic (initial) circle is the mirror of all radii outside and inside of the circle which, in addition to the basic radius divides it into a various number of sections and thus into diverse magnitudes of peak (midpoint) angles. Each division by any radius starts from one point – usually from the point of the peak pole and ends in it.

That is the basis of spherical geometry. Therefore, let’s only take the compass and proceed to, from the start, create the first “angelical” system in order to immediately see how much has been omitted by not drawing in full circles (or at least in semicircles) so that in this first chapter we can begin with the initial “products”, their combinatorial possibilities, their new products (partially), so that from the start we have an insight into “how it goes”. So, here we go with the first “angelical” system – only with a compass.

* * *

Consequently, we begin as usual: radius – circle – division of circle with circles of same radius. Circumscribe the intersections on the divisional circles. Hence, another radius.

* * *

With said radius we divide the circle into its 6 sections by circles (semicircles) only up to the perimeter of the basic circle.

* * *

We circumscribe the intersections of these circles with their own respective circles…

* * *

… and from the poles of the basic circle we have divided the basic circle with that radius. The division is into 12 sections (once again with semicircles).  

* * *

With same radius we divide the basic circle from its newly issued poles (now 12 of them).

* * *

And then with radius we go back to the first radius that came forth from the division of the basic circle, its divisional circles and with that radius we divide the circle from six new poles (therefore from a total of 12).

* * *

With this concept of division we have acquired progressions of new intersections within the basic circle.

* * *

And likewise, outside of the basic circle – a situation that we will successively and systematically explore, due to the diverseness of radii that divide the basic circle in versatile ways, thus forming diverse polygons, but for a starter let’s go back to the basics.

* * *

The division of divisional intersections by the first radius and thereafter by the second radius forms new external intersections – a synthesis of two basic radii – a new radius.

* * *

Starting from the peak pole from divisional point to divisional point the arc of the basic circle is divided into 7 equal sections – a heptagon.  

* * *

Now the same procedure is done from the subtended pole of the basic circle – the division of the basic circle into 14 equal sections (always by only following the divisional points).

* * *

And now, from three poles of the basic circle we have a 21-sided polygon.

* * *

And from all six poles of the basic circle – a division into 42 equal sections is performed. We now have progressions of intersections outside and inside the circle, and once we will separately explore this system.

* * *

Now we will proceed in another way, with the heptagon and external radius, only from the 6 poles of the basic circle – with semicircles to the perimeter of the divisional points of the basic circle from the first divisional point.

* * *

Externally we acquire the radius that is the product of the first (divisional intersection) and the heptagonal radius from the 6 poles (only from them).

* * *

From peak pole of the basic circle this radius divides the basic circle into 5 sections – circumscribed (segment of star-shaped polygon) pentagon.

* * *

We enter the radius of the same pentagon only from the 6 poles of the basic circle – semicircles to the perimeter of the divisional circles of the basic circle.

* * *

Enter into compass the radius of one of the intersections that have come forth. It divides the basic circle into 44 sections (2 x 22 or 4 x 11), thus only the product on the arc of the basic circle.

* * *

Let’s go back a little. We had a divisional radius of the basic circle, an external heptagon as well as another one that came forth from the first divisional intersection of the basic circle’s divisional intersections.

* * *

The said radius divides the basic circle into 4 sections, starting from the circle’s peak pole.

* * *

Thereby a new intersection came forth in combination between that radius and the heptagon alongside the perimeter of the basic circle of the 16-sided polygon on the basic circle.

* * *

And once more, between the 12 divisional semicircles and basic circle a new radius comes forth (one redundant) that divides the basic circle into 72 sections (6 x 12 or 9 x 8 etc).

* * *

And as an example let’s take the first internal semicircle. The synthesis of radius seven of 6 poles and division of 12.

* * *

That radius divides the basic circle into 20 equal sections (4 x 5 or 2 x 10 = 360° ÷ 20 = 18°), in other words, a new product.

* * * *

We have of course skipped series and series of radii (intersections) that came forth inside and outside of the basic circle but only as an introduction to geometry solely with a compass it was essential to demonstrate a small-scale insight into the possibilities of this kind of geometry, even when we omitted new products – radii of some presentations only to the perimeters of the basic circle – one can still sense how much data was left out only by observing the internal divisional intersections. Anyhow, I consider that we have barely scratched the surface of the first system of this geometry as conceptualized at the start of this chapter, but instead we “deflected” from our course to get somewhat better acquainted with the system’s diversity as well as the concordance and relationship between systems. However, with regard to the products of various divisions, here the sole help of computer animation would be needed in most cases, so our focus on the radius products will be limited to a certain degree that can be safely recognized. At the same time we can imagine the volume of this kind of geometry, which by appearances might not be as visually beautiful as some other systems of geometry, but it is exact and relevant. Moreover, didn’t we learn that whaz is beautiful does not necessarily mean that it has to be good?

Our analysis of the first “angelical” system – with compass only – will be continued in the second chapter.

* * * * * *

5 Responses to “The first „Angelical System“ – Only with compass”

  1. Brodic says:

    Pokušavam dobiti isti rezultat kao u zadnjem crtežu – podjelu osnovnog kruga na 20 dijelova – latica ali ma koliko precizno radila rezultat je uvijek drukčiji. Pogreška u razlici radijusa osnovnog i prvog kruga od +1 mm rezultira u puno manjem broju latica a -1 mm u puno većem broju latica od ovih 20. Radim s penkalama 0,4. Možda treba raditi s rapidografima još manjeg promjera?

  2. Brodic says:

    Ispravak – već razmak između kružnica rezultira većim brojem latica i obratno.

  3. Brodic says:

    Proučavanje lekcije o peterokutu (samo sa šestarom) pomoglo mi je u razumijevanju kako dobiti 20 latica – iz svakog četvrtog pola. Do tada mi nije ni bila jasna svrha plavog radijusa – a on ustvari označava novih 6 polova osnovne kružnice. Upisujući radijuse unutarnje kružnice iz svakog 4. pola dobije se 20 latica :-).

  4. Brodic says:

    Ispravak- Iz svakog 3. pola (jer ih ima 12 a nama treba 4 pola).

  5. Antonio says:

    En verdad no sé ni como saludarle. Hago usted una reverencia.

    Lamento no poder comunicarme en su idioma.

    Como usted he dedicado parte de mi tiempo a observar, mas no dibujar lo que se conoce como la flor de la vida; es cierto que he utilizado algunas veces un compas, y otras he intentado hacerlo con mis basicos conocimientos de ese accesorio de los computadores llamado paint, que en realidad es para que los niños dibujen. pero estoy a añoz luz de poder presentar a usted un trabajo tan exquisito como el que usted generosamente comparte y explica.

    Esto es lo primero. quiero agradecerle el hacerlo. tambien agradecerle de manera especial que piense en los niños y en los jovenes. yo ya no lo soy tengo 46 y un infarto encima.

    La segunda es que quiero compartir algunas de las cosas que en estas noches de desvelo he creido reconocer en la flor. por ahora este saludo

    respetuosamente

    Antonio Galvis tovar

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