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The Nonagon – Trisection Code

CHAPTER 12a

Although it is possible to create the nonagon (as we have seen in a series of chapters and in various ways) and its “lineage”, as well, the first star-shaped nine-sided polygon outside of the base circle (emerging from a peak pole of the base circle) emerged from the concept of the dodecagon of the basic circle only with a compass. It is the most prolific with regard to data, among which we will find the solutions for the three widely-known enigmas: doubling of the cube (as we saw in the previous chapter), angle trisection of whatever angle, and the area of a circle. In any case, the “cause” is number 12 and its groundwork structure only with a compass, or one might say, a New Testament concept. Nevertheless, it is not! Number 12 is a sacred line of legends since recorded history from time immemorial and belongs to the roots on which our civilization rests. That is why the concept of groundwork is important not only as a fundament for enigmas but as a fundament for angles in the order of magnitudes difficult to solve only with a compass, such as numbers five, seven, thirteen. And as for nine, we have had our say. Whoever can draw up an outline of number nine, i.e. a nonagon, can say that he knows how to solve trisection of any randomly given angle, i.e. divide the arc of a random angle into three equal parts only with a compass and unmarked straightedge. It is a folly of all those who strive to question the validity of this with the help of numerals or numerical one-hundredths, failing to grasp that one-hundredths as a unit of measure may emerge in the process of depiction as a human error of ours, but they disregard the essence, which is the concept or manner that “tells” us how. Even the ancient Egyptians would ridicule such “hairsplitting”. But, let’s not bother ourselves with such futile discussions and focus our attention on geometry. In the last chapter we “contemplated” the concept of division of the circle into 12 parts only with a compass, so let’s just “fit” it in by dividing any random angle into three parts, for example.

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Now we have a system of dividing the circle into 12 parts only with a compass and unmarked straightedge by delineating all the radii (6, 3 and 4) from 12 poles. The third intersections outside of the circle are the intersections of radius 9 of the base circle.

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The nonagon solely from the vertex pole. (18 from two opposite poles, 36 from four opposite poles, etc.) Nevertheless, a warning! It is important to delineate all this as precisely as possible, otherwise you will be challenged by advocates of “taboo themes” who will no doubt find a hundredth more or a hundredth less.

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Thus we have a randomly given angle and its arc. We enter its scope into the compass so as to get the magnitude of the bisectors and triangle sides of the given angle’s arc.

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For the time being we “forget” the given angle but we use its two poles to acquire with bisectors an equilateral triangle, i.e. to get its center point.

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We also could have delineated a chord to then present it as a triangle, but why when in the end we are not dividing the chord into three parts, but the arc of the given angle.

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As outlined, the bisecting straight lines circumscribe the center point of the circle of the acquired triangle.

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From the vertex pole that is at the same time the vertex pole of the arc of the given angle we can now outline the “groundwork” system that we were familiarized with on doubling the cube and that we repeated in the beginning of this chapter. System 12.

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I accentuate the exceptional importance of the initial part: the symmetry of the center of the circle that should, as precisely as possible, pass through the poles of the arc of the given angle since the concept originally starts from its vertex pole. Eventual errors (here they are intentional) must be rectified.

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A Few Words Before Trisecting

It is no surprise that the mathematicians of Ancient Greece (even Archimedes) tried to solve geometrical problems by rectilinear drawing. Yet even today, the compass is by far the basic instrument of geometry, and like any other instrument it demands the skill of a “master”. Although the knowledge of this geometry has been given me, I cannot say that I am a “master” of the tool because my skill is only what I learned during my schooldays and, once again, it was within the framework of a non-geometric trend, hence it cannot be considered as high “exactness”. Luckily, schools of preciseness still exist, schools for draftsmen in the various segments of engineering and construction building where youngsters are still taught precise drawing by hand regardless of the fact that nowadays such drawing is looked upon as backward. Unfortunately, such schools have no knowledge about geometry of this kind. The information processing system, unfortunately, is still close to my level of cognition, which I call “rough preciseness”. Nevertheless, it is good enough since the principles of this geometry give us the answers on how to solve problems. Even more important, as you have noticed, they are grounded on several “attestants” (intersections, crossings), never on just two identical points which then “speak” about preciseness or “acceptable impreciseness”. I only sympathize with the “grade school” kids because instead of a compass they were given a gum-labeled “sticker” compass, hence rarely any kid succeeds to draw a circle on the blackboard because the “sticker” moves. But, let’s continue with the trisection of this seemingly complicated system with a compass and unmarked straightedge, that we may, step by step, learn more.

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Delineate the nonagon on the base circle.

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Nonagon on base circle or opposite division into nine parts through center into 18 parts; interior intersections of the nonagon must always be 18 (the third from the perimeter of the base circle).

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Now we return to the randomly given angle. From two poles of the nonagon and in the line leading to the vertex of the given angle the line segments divide the arc of the randomly given angle into three equal parts (though the depicted triangles are not necessary because the division is of the arc, but just for the sake of order as ordained in the enigma.

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If the given angle is in excess of 120° – we simply split it with a bisector and subject one of its halves to trisection and that only with the compass from the intersection of the bisector and arc of the angle and transfer one of its thirds to the other side where there was no trisection, etc. Why did we choose this, seemingly complicated, groundwork when angle trisection as I remarked can be carried out in a simpler way? The nonagon construction only with a compass contains the “answer” to all three enigmas, and the analysis of its intersection, i.e. sequences of radiuses on the “mirror” of its base or initial circle carries within itself sequences of data, not to mention the data that would emerge if we had “turned” inversely towards diminution of the cube. Nothing more needs to be said as regards this chapter but to pass on to the perimeter and area of the circle only with a compass and unmarked straightedge about which we have often “spoken”, but not in this way that arises from this “groundwork” and from the enigma of duplication and is connected to the nine-sided polygon and its star-shaped polygon outside of the circle.

Croatia – Rijeka, April 24, 2014
Author : T. Periša
Web : Slim
English : S.F. Drenovac

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