## Angle trisection (internal)

**ANGLE TRISECTION (internal)**

This is a chapter on angle trisection (dividing a randomly given angle into three equal parts) only with a compass and unmarked straightedge. A chapter that consolidates series of rules hitherto exhibited in this ancient geometry “reader” that is, logically, based on the column of measurement of series 3 or thirds, in a combination of straight-line and curved drawing and the code system (flower-of-life) of the ancient artifact and its elements – thirds of hexagonal circles and code system of the “Star of David” – a hexagonal star-shaped polygon, with reference to the third six-sided polygon whose circumscribed arch is the floral pattern that divides the basic circle. It divides the circle into 18 parts and that is why we say it is an internal trisection that on the described circle also manifests itself as a division into eighteen parts, thus as an eighteen-sided polygon. We therefore take one-third of the arc of the circle above the side of the equilateral triangle as an adequate division for the magnitude of arbitrary angles up to 120°. The vertices of all these angles are located on the centerline side of the equilateral triangle from the center of its described circle (which is 120°) and onward all the way to 1° if we insisted on it. All that is described may seem demanding, but if we accept that the division of the circle’s arc never happens directly but through the assistance of divisional codes whose arcs are always above the arbitrary arc of the randomly given angle. We reiterated this in the basics of trisecting (or sectioning in general). Then it is plainly understandable, although this method calls for exceptionally precise drawing of the hexagon and its flower-like pattern, which should not be difficult to verify today by means of undertaking a precise computer simulation. Such simulators have been around for quite a number of years (University of Zurich) and certainly elsewhere. Unfortunately, alone and without anybody else’s support, I will not be able to provide the money for such simulations; in fact I can’t even say how long I can withstand the present-day circumstances, but that is another story. However I don’t know where it is best to align this presentation (in the chapter for kids or for adults?).

And so: the circle, with its radius is divided into six parts – a flower-like pattern and star-shaped, straight-line drawing, six-sided polygon

The star-shaped polygon is the “creator” of the smaller hexagon within it – we inscribe its hexagonal star-shaped polygon and once again it makes another smaller hexagon.

We circumscribe it with a circle.

The flower-like pattern of the first arbitrary circle divides the circle of the third hexagon (at the center) into 18 equal parts in tandem with the six poles (12 + 6), and the peak angles at the center are 20° (18 x 20° = 360°).

For example let’s take an angle visibly smaller than the right angle (90°). Its arc – chord and centerline that divides it in half.

The chord of the arbitrary angle is the side of the equilateral triangle and one of the poles (when plotted) is on the centerline, hence with centerlines we look for center and circumscribe its circle (the arc of the full angle over the arc of the arbitrary angle).

We have now acquired conditions for the code system so, from the circle of the equilateral triangle (from its peak angle), we inscribe the division into six parts – the flower-like pattern – and the star-shaped hexagonal polygon.

The hexagon of the star-shaped polygon and its star-shaped linear polygon – we acquire its hexagonal polygon.

We circumscribe it with a circle.

The whole “petal” divides that circle into parts (a 120° into six or 3 equal parts if we take every other intersection point and project it onto the circle’s arc of one side of the equilateral triangle).

From the projected points in the direction of the peak of the arbitrary angle we trace lines. On their way towards the peak of the arbitrary angle they intersect the arc of the arbitrary angle into three equal parts. We have trisected the arbitrary angle. The size of the angle and size of its trisected parts can only be determined with a compass if we more attentively study chapter 7 (news in geometry).

And this is what the explanation would look like. The demonstrated method of ancient geometry described in the foreword to this chapter shown on one arbitrary angle with its arc (every angle has its own arc but also the same starting points – the poles) as an example of performance of the code system on its division into three equal parts on the principal of application of the compass and unmarked straightedge.

* * * *

**MEASUREMENT COLUMNS (part two)**

**– SERIES (PROGESSIONS) OF NUMBER 5 –**

**FIFTHS, TENTHS and onward …**

Everywhere in the cradles of ancient cultures there is factual evidence of systems of measurements. They are referred to in legends, from Ancient Egypt to the Bible and among the ancient peoples of South America. Even to the present day the system of tenths has in many respects been retained. From a geometrical viewpoint the system as a depicted component is connected with many unknowns since its products (besides their calculating simplicity) connected with the first two columns of measurements (progressions 2 and 3) and numerous artifacts contain within them the measurements of fifths and tenths of the “pyramidal” code, the trunk with the tablets of the Commandments, the doubling of the cube, as well as of the trisection (its connection of number 5 with square 3 i.e. with 9). An example: 3.5 fifths of a radius divide the arc of a basic circle into 9 parts (namely 7 tenths of a diameter). Therefore we won’t deal with the products of measurements column number 5, but with the construction of measurement and the correct construction of cube 5 so as to acquire the fifth at the center of the circle, albeit abridged, and in so doing ignore the products outside of the circle, without much describing. The correct construction of cube 5 (4 + 1) is of importance since the circle is divided by poles:

- every other pole with semi-circles, and their intersections determine the radius of the smaller circle by means of whose division we construct the cube
- cube 5 (4 + 1) and if with that radius we describe the semi.circles and from them the 6 poles of the basic circle, we then acquire the intersections of those circles and sides of the hexagon of the basic circle that we associate with the principle so that alongside the centerline in the center of the circle a hexagon of one-fifth is formed, and at the poles 1.5 fifths make a hexagon and the radius of the inscribed circle divides the arc of the basic circle into 9 parts from one of the poles (and 18 from the subtended pole, etc.)

Therefore, we will construct the basis itself – the (most important) measurements column number 5 with the aid of the cube – in other words the division of cube of 5 (geometry for kids).

The circle with its diameters – its hexagonal polygon – the cube – intersections of triangle’s radius – every other pole (curved drawing forms the circle of the smaller radius). Semi-circles of the same radius from all six poles of the basic circle intersect the sides of the hexagon.

We connect those intersections subtended along the diameters, and thereafter along the sides of the hexagon. A fifth is formed in the center, and at the poles 1.5 of a tenth.

We divide the cube of 5. We have acquired a diameter of five fifths (ten tenths).

If we divide the tenths into two parts, we have acquired twentieths.

If we apply fifths on the hexagon the principle of division into three parts, we have acquired fifteenths and if we split them in half we have – thirtieths.

In that series we apply the division of 20 into halves, so that the fortieths and so on are in conformance with the rules of measurement columns numbers 2 and 3, but this is not basically necessary since the most utilitarian are the fifths, tenths, twentieths. Therefore that is the third column of measurements – column number five.

* * * *

**COLUMNS OF MEASUREMENTS (Part two)**

**SEVENTHS AND SERIES OF SEVENTHS**

**Progression of number seven**

In this series we also come across the construction of sevenths with the help of the cube of 7 in combination with series of 2 and 3. Thus we acquire series of seven multiplied with 2 and 3 (abridged – and inside the circle). Therein lies the basic assertion: the radius of the inscribed circle of the hexagon divides the arc of of the circumscribed circle into seven parts. A brief explanation and verification follows, once again adhering to the procedure of sacred geometry. I never concealed the spot from which I started on my road to acquire knowledge of ancient geometry from what the Man on Mount Horeb who gave Moses the Ten Commandments – the same soul as in the prophecy of prophet Jeremiah – in the words of the one that we human beings call God (even though we do not know what that is). I also recall my acquaintance of a long time ago with the mythological book entitled “Moses” by E. Flug in which Moses, at the same time when he was awaiting to receive the writings of the Commandments on Mount Sinai, he went to school with the “angels” (we also don’t know what they are), so a logical conclusion would be that he was taught ancient geometry because after 40 days when he came down from the Mount he gave clear instructions of how to build a “habitat” – a tent for meetings, issuing very clear-cut measurements, proportions, dimensions, etc.; hence I myself have become a kind of “pupil” by checking and using ancient measurements (minimized) – ancient tenths to allow me to draw all this on an A4 format and an interior (inside) circle presentation: 3 human cubits 45 cm (tenths) 4.5 x 3 = diameter. For a full ancient-geometry presentation of 1 cubit (tenth) radius of a human cubit is (4.5 cm = r) according to legend. When anything seems undecipherable I seek for the help of the angelical cubit – tenth (5.25 cm) – radius. It may sound strange but it helps, since somewhere deep down in the cerebral structure a door of cognition opens. On basis of that I would finally like to “eliminate” the suspicions regarding the heptagon. Anyway, 4 years ago I sent this to the profession for publishing, but never received a word of reply (who knows where it ended?). It seems that “the deputies of knowledge on earth” don’t give a damn. Logically, checking is also performed by calculating and drawing is geometrical – only with a compass and unmarked straightedge. Therefore, here is an example: diameter 13.5 cm (circle r = 6.75 cm) 13.5 : 7 = 1.9285714 > circumference = Pi () 3 integers and 1/7 d > 40.5 + 1.9285714 cm > = 42.428571 = circumference of described circle of hexagon radius 6.75 cm, and the inscribed: 13.5 – 1.9285714 = 11.571429 (diameter) : 7 = 1.6530612 (seventh) circumference = 3 x 11.571429 = 34.714287 + 1.6530612 = 36.367348 : 6 = 6,0612246 (inscribed radius) > circumference of described circle divided by inscribed radius > 42.428571 : 6.0612246 = 7 (hence, it is clear) and at the measurement column number 7 we will show the square of the circle’s radius, square of the circle’s circumference and the squaring of the circle – rectangle with a depicted equalization of the square construction of squaring the circle, whereas in the next chapter a series of essential examples of all four columns of measurement awaits you.

Circle divided by radius – hexagon with its diameters, inscribed with the radius and semicircles from all 6 poles of the basic circle.

The semicircles intersect the sides of the hexagon. We connect the intersections with line segments – alongside the diameters – alongside the center – one-seventh of a diameter at the poles 1.5 fourteenths.

The full division of the hexagon – cube on the cube of 7 – is formed on diameter seven of the hexagon – one-seventh of the diameter.

The fourteenths of the diameter (when halved = 28) thus divided into halves – by series (progression) 2.

If we use the cube of 7 of the circle’s star-shaped polygon as the outline of sevenths (thus, number 3) we acquire 21 parts of a division, and so forth.

The square of the circle’s radius or the square of the side of the cube of 7 = 7/14 x 7/14 parts of the diameter (with the help of division of the circle into 12 parts, and the 12 parts with the aid of the star-shaped polygon of the circle.

The circumscribed square or Pi () 14 – 3 = 11 fourteenths x 11 fourteenths.

Squaring the circle – rectangle

11/14 diameter x 14/14 diameter (circumference divided by 4 times the diameter) and plotted root of squared circle – square of the quadrature of the circle.

* * * * * *

A solution to the geometric trisection of any angle using straight edge and compass may have been found. Please check my blog.