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Angle trisection Part 1

ANGLE TRISECTION

PART 1
CONSTRUCTION OF SQUARE AND THE DODECAGON

If anybody thinks that I’ve lost the thread that links me to kids (and I myself am a big child of infinity), they are wrong because this book is coming into existence within a short period of time and we are in the summer season and I myself know that this is the season of freedom from study, the season dedicated to child’s play, to the sun and the sea (and in so doing, I have in mind the incoming generations in accordance with ancient norms, which will certainly be the inheritors of this age-old newness). But since the beginning of the new school year is nigh, I have decided to slowly bring them into that circumstance, and to once again, in the most elementary manner, refresh their memories with regard to the so-called “most widely known” enigma that the ancient mathematicians “handed down” to us, although I do not agree with such a definition because from the aspect of pure geometry it is actually the simplest to solve. Why? Because it is “one-dimensional”, and that would also mean – a divided – “two-dimensionality” and belonging to the two-dimensional world of geometry we can include the squaring of a circle (enigma) that has already been shown to the kids and adults but we will simplify it even further, whereas the enigma of doubling the cube has the reputation of being a three-dimensional enigma. For each of the three enigmas certain preliminary procedures are necessary – of course always in accordance with the rule of the “compass and unmarked straightedge” – in order to bring it somewhat closer to contemporary geometry, thus serious thought must be given to which “pattern” can be shortened to the extent that it does not, on the one hand, upset the equilibrium of ancient geometry and yet points to the modern era’s simplification that we come upon on the pages of present-day text books on mathematics (geometry). Therefore we will start with the preliminary, even more simplified constructions of the square and its circumscribed circle and its division into 12 segments. It will then become our “tool” – our “pattern” for trisecting the angle, namely a quarter of it (otherwise this pattern was used for ground-plan determination of facility measurements on building sites back when construction workers, besides measuring tapes, used plumb lines, which are rarely used nowadays – yet there are records of their use as far back as in biblical times.)

So, let’s begin with the construction of the square in the simplest way.

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The given length divided by the axis of symmetry into two equal parts (partially perceivable circles from the endpoints of the length   – where they intersect – the axis of symmetry.

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Line of symmetry bisects (divides) the line segment into two parts. From that midpoint and with the radius of the endpoints of the line segment we draw a semicircle. The intersection of its arc and line of symmetry is the center. From the ends of the line segment we “draw out” rays through that point.

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From that center we circumscribe circle radius-center right ends of the given line segment and connect the intersections of the rays on the circle – the circumference of the square to the given line segment, thus the rays are the diagonals of the square.

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Described circle has its radius. With same radius and from one of the vertices of the square we divide it into 6 parts.

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Thereafter, from the other vertex of the given line segment we divided the described circumference of the square into 12 parts – a 12-sided polygon.

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For trisecting we will use a quadrant, an even shorter procedure – more simplified – thus bringing it closer to present-day plotting in geometry – because angle trisection is our goal.

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ANGLE TRISECTION (SIMPLEX)

When this generation of ours passes away and when some future ones comes into existence, they will certainly find it funny that we thought it “impossible” to trisect an angle only with a compass and an unmarked straightedge and still funnier will be the ridiculous discussion about calculation without any outline, so that in most cases it turns out that the division with a prime number is more or less an irrational number hence every result is sealed in advance as “irrational” or just “approximately accurate” or simply “impossible” without supplemental tools aside from the compass and unmarked straightedge. It would be funny to those who lived long before us and who applied sectioning, and not only in construction engineering.

Nevertheless, ancient geometry is a demanding knowledge if it is “full”, which means depicted without shortening, be it spherical or rectilinear, and that is why I endeavor to shorten it only to the extent that this “transition” from the ancient to the contemporary is made easier – apart from my retaining the rule to do away with letter or numeral markings, because markers would introduce confusion and diminish our brain’s perception like any contour does, be it in a religious or an artistic-scientific context, hence our inability to imagine many things is logical, thereby it produced in us a long-lasting traditional epitome of the average – the worst possible invention that always gives faulty conclusions and results because it isn’t precise – equal for each unit. But let’s go on with our aim and deal with the angle trisection – first of all with the “simplex” for angles up to 90°.

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If the given angle is less than 90° it has its arc, its chord, its line of symmetry (the line of symmetry is the start of the division because it divides the angle into two equal parts, the chord and the arc.

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As referred to during construction of the square, one half of the given line segment and the semicircle of the radius of half the line segment of the line of symmetry is the center of the described circle’s square whose side is its line segment. Therefore we apply same procedure to the angle’s chord (take it as the side of the square).

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From this center we circumscribe a circle of the radius center of the ends of the chord. (The arc of the square’s circle is above the arc of the given angle, and this confirms that the given angle is smaller than 90°.

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With this presentation we merely confirm the construction of the square’s sides of magnitude of the chords of the given angle, but in our continuation this will be omitted, however we should “keep this in mind”.

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. So, only the randomly given angle, its arc, chord, axis of symmetry, center (which is always on the line of symmetry) and the described circle of the radius-center of the chord’s endpoints.

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Therefore the circumference of the square has its radius and passes through the endpoints of the chord of the randomly given angle. Now with that same radius we divide the circle into 6 segments starting from one of the chord’s vertices.

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Then from the chord’s next vertex with same radius we divide circle into six more segments – a total division into 12.

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Three segments are on the arc of the circle above the arc and chord of the randomly given angle. We inscribe rays. They divide the arc of the randomly given angle into three equal parts (not its chord). We have acquired three isosceles triangles that are equal. That is the angle trisection of a given angle.

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And this is what the most succinct depiction of the same trisection shown on so far on these pages would look like. Randomly given angle – arc – chord – line of symmetry –semicircle’s radius of one half of the chord – center of circumference of the square with sides the length of the chord; division of circle from chord vertices with same radius. Divisional poles above the arc, and from them, rays in the direction of the peak of the given angle – that divide the arc of the given angle into 3 equal parts = angle trisection (simplex) of angles smaller than 90°.

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DUPLEX TRISECTION OF ANGLES
LARGER THAN 180°

We ought to know that, apart from some exceptions, we can never trisect a given angle directly but at all times indirectly by means of “exceptions”. In this case that is the 12-sided polygon – when the chord is taken as a side of the square. I call such “exceptions” “tools” of partitioning for in them lies one more law. The arc “tool” is always larger than the arc of the given angle – above – or outside.

Secondly, in trisecting the tool never exceeds 360°. Consequently, the “tool” times 3 (thus the angle of 180° is unsuitable for trisection (3 x 180 = 540°) – hence the ultimate tool is 120°. Therefore, besides simplex we apply duplex trisection, and if necessary even triplex, but each operation independently.

How? We will demonstrate this on an angle larger than 120° and smaller than 180°.

You should not be afraid of this method. Simply follow the instructions exemplified in the first part, but do them twice. Whether side by side at or each part independently (apart from the other), is all the same. And then it will suffice to carry out the division of the circles from the end points of the arc of the randomly given angle (altogether shortened – since „we kept this in mind”).

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Thus, we have the randomly given angle with its arc and line of symmetry.

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The difference here is – two chords (of same magnitude)

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Now we divide them also, each with their line of symmetry.

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For each we apply the simplex principle of the described construction up to now – twice.

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Each chord has its circumscribed circle (and each chord is a side of its square)

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Exclude the full circles and only leave each a part of its arc that is above its chord.

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We carry out the division with that radius of each part of the arc above the arc of the randomly given angle, and from the endpoints of the chords.

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Hence we have 3 and 3 segments on each semicircle above the arc of the angle that we are segmenting. From the first points (segmented poles) and alongside the line of symmetry the rays towards the vertex of the given angle divide the arc of the given angle into three equal segments. This is the trisection of angles larger than 90° and smaller than 180° (duplex).

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And this is just the presentation of the internal division of semicircles (each on its own) with its 12-sided polygon.

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And if we were to show the full external system (outside of the circles) it would be more than necessary for the beginning “and too rough on persons who are just entering the world of geometry, and we don’t want to get lost”).

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AFTERWORD

If you learn this procedure, and anyone anywhere tells you that it isn’t correct, then ask that person to give you an angle that you will solve on the principle of the chord as a side of the square and its circumscribed circle divided into 12 segments. If the person then says that the result is approximate, that would be the same as someone saying that the radius of a circle does not divide it into 6 parts and moreover their lines of symmetry into 12.

All one needs to remember is:
Trisection of a given angle is never performed directly, but indirectly on its arc with the aid of “exceptions”, i.e. “tools”, of proven partition of the randomly given angle and in the direction of the vertex (this goes for all sectioning – 3, 5, 7, etc.). How come nobody has discovered this a long time ago, I don’t know. But I surmise why nobody wants to accept this. Perhaps due to a statement that I won’t deviate from. In prayer I asked an entity who more than 3000 years ago gave the guidelines for a life of harmony with nature and of our rightful place within it, and more than 2000 years ago another entity was crucified because of this (I sometimes wonder whether this could have been one and the same entity?). Today, everybody seems to want to “run away” because that “entity” is no more, but whether it exists or does not exist, we don’t know so that one can believe or not believe in it. But there is a possibility to clarify all this since it is not esoteric or magic or theoretic, but real; a project that I promised to present in one of my following chapters. So, when you master this kind of trisection, all you can say is – I KNOW.

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