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The Cube part 2

The CUBE (part 2)

Examples of column of measurement of digit 5

As I mentioned in the previous chapter, we will surely be going back to series of digits 2 and 3 since they comprise an enormous opus of products. That opus is associated with various angles and their construction only with the use of the compass and unmarked straightedge so that it is “impossible” to present all the many essential factors on the modest number of these pages in this chapter. Therefore we will demonstrate the most important ones and they will be beneficial for mathematical (read geometrical) enigmas about which the ancient Greek mathematicians, regrettably, had no approximation and this has been the case to this very day. No matter how hard they tried, they had no success since the discipline of geometry was by far older and was considered sacred. The relatedness of ancient peoples towards the crescent was indeed “godlike” because it comprised series of elements, the meanings of which we can only foretell, but “for the time being” that is not my assignment. Present-day scientists seem to “run away” as soon as they hear the word “sacred” or “divine”, who knows what motivates them and why. Nevertheless, such an attitude is “avenged” by ignorance hence such unexplored regions have remained taboo subjects. And what is the situation today? If scientists would take in hand at least one of the books of the Old Testament from the Pentateuch or the Prophet by Moses, and not only in the segment of applied geometry but also to tackle other possibilities of importance for people (starting with biological impact to  ingredients in nutrition). Once again I’ll take the liberty to point this out with another example. Near the end of the last century when I was scrutinizing another of my “passions” (physics), dealing with the problem of energy and water that are of vital importance, I realized that the Earth’s depressions are an ailment that can be healed by human activity and intelligence. In Egypt there is a folk legend about a man called Ben Yusuf (it seems as if this is a reference to the biblical Joshua). He noticed that after floods the Nile stays put in a given area. So he dug a channel to allow the water to flow. Thus 3500 years ago, one the the most beautiful oases in Egypt (Al Faiyum) came to be. Depressions on earth are located in the desert region of Qatar, but the largest is in Egypt. At that time, I made a proposal to the Egyptian Embassy in Switzerland (since I was then living there) to, by means of the simple physical process of the siphon (a rubber tube filled with water) from a number of vessels (the Nile) from a higher level run to a lower level into a barrel full of water at the bottom of the depression. The water is in that way forced through the tube until the atmospheric pressure equates their levels (this differ5s from the geyser process that functions by the access of air into the outflow). However, if the problem is in tube diameter being too small, I proposed application of a cluster of tubes, and all this without any additional consumption of energy. Nature alone will “do” the job. But my proposal was rejected, and what’s worse, it was a long-term rejection. The Qatar project is under way. The expected building construction is ten years. At what cost? The geyser principle is as follows: underground tubes are embedded with accompanying turbines to produce the power. Where does the water come from? From the Mediterranean Sea. Thus a saltwater lake will emerge. And electricity? Maintenance? These are just some examples of human irrationality. The depression in Qatar in the middle of the desert from the Libyan border to one-half of Egypt widthwise has plenty of sunlight (like the solar fields in America) and the Nile is as far away as the Mediterranean. But the Nile flows out into the Mediterranean, to ne end. And what about fresh water that nourishes people and plants? But this is not the only example of human shortsightedness and escapement from natural simplicity. Another example of this is Sacred Geometry; so we will carry on with the regular division of the cube on 5 and column of measurement (fifths) with examples that emanate products of the system of fifths (partially because there are so many).

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So, the circle is divided by its divisible circles of same radius as the basic (central) circle. 

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Hexagon (the intersections of the divisible circles are control points of correctness the drawing). See chapter in Geometry for Kids. 

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Division of central (basic) circle into parts (pole – subtended pole – diameters – extended to perimeter of divisible circles).

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Radius of intersections of divisible circles – center of basic circle (the so-called triangular radius)… 

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… because the poles of the central circle create in it a star-shaped – rounded hexagonal polygon that we circumscribe with a circle. 

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Thereafter we circumscribe from each of the 6 poles of the central (basic) circle. 

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… and at the points where these circles intersect the sides of the hexagon of the basic circle (this time only inside of the basic circle we being with the connecting – first vertically alongside the diameter

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… and then along the left and right diameter, then along the side of the hexagon of the basic circle in the same manner (vertical – slanted – slanted) in the center is the cube as a part – one fifth of a cube. 

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We now divide the cube into 6 more parts (a 12-sided polygon) and repeat the divisional procedure using the centers of the hexagon sides, arising along the sides of the basic center. 

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Therefore, we divide the sides of the internal hexagon and thereby make a cube of 5 of the measurement column of fifths, tenths on the diameters, and so on. 

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So, we inscribe the star-shaped polygon of the described circle of radius of 4 tenths. 

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It makes a smaller hexagon that we describe with a circle so as to see what its radius is doing on the arc of the central (basic) circle. 

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Starting from the peak pole of the basic circle until we come back to the starting point, we divide its arc (with radius 5) into 40 parts (360° ÷ 40) = 9°.

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Another example: We draw within the hexagon (one-fifth of it) an inscribed circle. 

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Now we inscribe a circle to the circle of the smaller hexagon and once again depict circles from the peak poles (six from the basic circle). Its radius divides the arc of the basic circle into nine equal parts. 

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From the subtended poles of same diameter we make an 18-sided polygon (depicted in full circles but we do not analyze the products outside of the basic circle, though there are many). 

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Third example: half a side of the star-shaped polygon. 

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This radius divides the arc of the basic circle into nine equal parts. We will use it for an angle trisection. 

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“On the way” we will cross the “bridge” to the cube of 7, so that a fifth of the internal circle is divided into 6 parts. We take into the compass the radius where the 5-sided circle intersects with the basic circle. 

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That radius is a product and it divides the basic circle – its arc into 7 equal parts – from only a single pole of the hexagon of the basic circle. 

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And now vice versa. The way the described circle – its radius – acts upon the inscribed – radius of the peak pole.

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It divides it into 14 parts. We have “quartered” the system of circles (if we were to divide from these points we would acquire the squaring of the circle  or partitioning of the circle into 28 parts. 

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TRISECTION OF ANGLES UP TO 120°
(SINGLE)

As promised, I will quote a good example of another “escape from simplicity”. How do things stand today, with all the existing technological possibilies? Since I have a friend who is exploring pyramids and seeing the torments he is going through delving of their foundations, I made a proposal that he ought to apply a system (with the aid of another friend who is an applied scientist in geology and former director of a gas & oil corporation) of surgical drilling of unknown pyramidal structures with a hollow bore of 10 cm diameter. I am not a geologist and so I kindly ask my readers to have an understanding for my layman’s narrative. What I do now about geometrical and pyramidal structures, in addition to the central chamber there should be a peak chamber or a space that I prefer to call “a database chamber”. The answer is not at the bottom of the pyramid. By drilling from the top we would get a series of answers: about the material, for when we are talking about the pyramid we should expect to come upon an empty space or a kind of chamber. And today it would be no problem to send a camera to record whether there is anything in those chambers. However, this idea was rejected and instead of being accepted it was circumvented to the esoteric level (because the pyramid is “sacred” and a 10 cm opening would be inappropriate. On the other hand, people like secrecy, something about which one can spin a yarn without any consequences, or perhaps economy itself plays a certain role here. Something of long duration, also grants a long-term job, and then there is the media and so on, with a funny explanation that “maybe the chambers have collapsed”, as if this can’t be seen from the outside. Hence, the use of a camera was also rejected which is strange and hard to believe that a big gas and oil corporation would not be glad to finance such an endeavor, at least because of the publicity. They probably would not end up like the renowned Japanese firm that wanted to build a pyramid like the Egyptians – but nothing came of that! It is apparent that simplicity is undesirable, but mystery and making up stories – yes indeed!

As the Lord once said: „The writer’s mendacious writing implement has distanced me from you”. However, I promise that I will soon disclose “the pyramidal code with data – by means of a geometrical method.

But, allow me to make the most of this knowledge and present some more trisections in the “manner of fifths”, column of digit five or the star-shaped radius of its 4/10 circle. I must note that the expression called trisection is still not comprehended. Trisecting is a division that requires “a tool” or “a code”, and only then can we divide the arcs of any randomly given angle. Any type of acrobatic geometry is unacceptable, said a professor of mathematics concluding that “trisection can only be solved through the art of origami.” If anybody wants to learn how to calculate the magnitude of an arbitrary angle only with a compass, he should go to the supplement of chapter 7. However, right now we are going to trisection of angles up to 120° (single), for if the angle is 200° then we will apply the double code (comparing each per se).

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The arbitrary angle (larger than 90°) with its arc, chord – divided by line of symmetry into two parts. 

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We take the chord of the arbitrary angle as the side of the equilateral triangle, and determine its center with the lines of symmetry. 

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We circumscribe its circle (we can slearly see arc of the circle passing through the chord ends of the arbitrary angle. 

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Since the centerlines of the equilateral triangle divide the described circle into six parts, we will shorten them by not inscribing the divisible system of circles and instead we take radius 3, and that means every other pole so as to get the intersections and circumscribe them with a circle of its radius and line segment of one side of the star-shaped polygon – its half. 

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That line segment is half the side of the star-shaped polygon’s fifth of the circle. The arbitrary angle with its arc “waits” to form “the tool” for trisecting. 

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That radius divides the arc of the chord of the circumscribed circle of the equilateral triangle into 3 x 3 parts  – the nonagon (three parts above the arc of the arbitrary angle). 

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From these bisecting points in the direction of the peak of the arbitrary angle, the rays divide the arc of the arbitrary angle into 3 equal parts (this goes for all angle up to 120° – (single) and that is the trisection of an angle in one more way – by means of column of measurement of digit 5. 

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Frankly speaking, this method of “teaching” is pretty exhausting since each week it is necessary to draw some 30 pages, write the texts by hand, hand them over to my friend, an excellent translator for translation into English (since English is the most representative language in internet communication). That will be the case until scientists invent a way to shift letters and syllables so as to come to the primal language by permutations that are in various forms “scattered” throughout all the tongues of the world. (For example: let’s take Taurus – a mountain range in Turkey – the rudiments of civilization; the German Taus – garden – ur taus – the primordial garden – which corresponds with the legend of a primeval garden, source of rivers that forks into four branches: Tigris and Euphrates still flow overground, a third river most probably flows underground, under present-day Israel, and that is why when discussing depressions I deliberately made no mention of Israel, where the depression is deepest; and then there is that unusual project – the system of artesian wells from the Red Sea. Nor did I mention the Death Valley national park in the United States, the hottest national park in America. This area is considered sacred since it prevents the clouds from the ocean “to proceed onward” across the mountain range due to the lack of an atmospheric water bridge that would form favorable evaporation if the depression was filled with fresh water.

Well, I’ve been somewhat carried away and deviated from the field of geometry. Thence, after being translated these texts go to another friend of mine whose specialty is their computer preparation as web pages, scanning, etc., because I am in that respect “a dinosaur”, a layman concerning computers. I can hardly switch it on. And then there is a certain cost in all this, and for the time being I am willing to bear it because the results of speculative cognition make me happy and the interest of kids that will one day apply all this in a different and better world brings joy to my heart.

Therefore I will simply abandon myself to the “spirit” that leads me, for he surely knows better than I do. All I request from my readers is that they don’t hold it against me if I sometimes inadvertently make a mistake, because in addition to all my obligations towards my family, the children, the house, the writing of poetry that I intensively keep up – in the past ten years I’ve written ten thousand poems or 250,000 lines of verse, not counting the poems written in the last century, although it seems that a time of shortsightedness has come in which there is no room for poetry (at least not in this country). As an example, I published a book of 1000 pages (format A4), distributed the book throughout Croatia and among the diaspora that I myself am still a part of, as well as a member of the literary organization Matica Croatia. No answer, not even a polite thank you! Therefore I would ask those in Croatia who are interested to get in touch with me by e-mail by sending me their address and I will give them a copy of the book as a gift. I would gladly send the book worldwide, since it is bi-lingual in Croatian and English, along with an additional 100-page supplement of trisection diagrams, but I am afraid that I cannot bear the financial cost of such an undertaking, since that would be at the expense of discontinuation of the work on these cognitions.

We will continue in the chapter that follows.

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