free hit counters

Special trisection of angle

SPECIAL TRISECTION OF ANGLE

Sometimes I find it hard when friends of mine (admittedly unfamiliar with geometry) tell me that journalists searching the internet come across comments by youngsters (nonetheless this was before I began publishing this site) who assert that geometrically it isn’t possible to solve trisection only with a compass and unmarked straightedge. Of course it’s hard, when even professors of mathematics keep on repeating that it isn’t possible without additional tools, and they don’t even give any thought to at least check or try to find a solution; indeed hard, when academics and top mathematicians today don’t try to tackle the problem but circumvent it, because they find it easier to be a “parrot” that constantly goes on repeating “impossible, impossible”, until somebody actually appears and solves the problem and is then recognized by the civilized community, everybody praises him with cries of approval; and I find it even harder because I then comprehend that the superiority of knowledge of our human nature suddenly gives the impression of being more backward than the civilization of a couple of millenniums ago. However, the cognition can also be comforting by the fact that there are individuals of a different spirit (and it seems that there have always been such individuals), withdrawn, unknown idealists doing work for the welfare of civilization – for its generations, for new knowledge, but then again, instead of getting support they come upon great resistance, confronting a modern inquisition whose attitude is irresponsible for its word. To say that something is impossible, without determining the accuracy of the statement, is the talker’s foolishness and an attack against the person striving for a solution. Still I find it a consolation that many the world over are calling for a repetition, and so by repetition I too am endeavoring, step-by-step, to make the solution simpler, but always taking care more or less not to violate the principles of the new ancient geometry. Otherwise, how would anybody be able to understand ancient geometry and “read it” if they encounter it anywhere and anytime? I will therefore repeat the principle of angle trisection of any unknown angle and I will tend to get closer to childhood simplicity, which is by no means all that easy for a 60 year-old man. So, let’s give it a try once again.

* * *

So, if any unknown arbitrary angle is smaller than a right angle, we inscribe its arc, its chord, and by drawing a line of symmetry divide it into two equal parts (something we learned in primary school).

* * *

We now treat the chord as a side of the square. We know that half the side of the square is on the centerline of the center of the circle that it circumscribes. That’s why we take into the compass half of the chord and with its radius we describe a circle from the midpoint where it intersects the centerline and from the endpoints through the center we inscribe straight lines.

* * *

From the halfway point between the chord’s endpoints we circumscribe a circle. We then notice that the straight lines from the chord’s endpoints that pass through the midpoint are actually diagonals of the square. The points where straight lines intersect the circle are the vertices of the square – a regular square. Now we just follow the circumscribed circle of the square – the given arbitrary angle is put aside.

* * *

Now, from one end of the chord we circumscribe a circle of same radius as the described circle of the square so as to acquire the intersections that will divide the circle into 12 parts.

* * *

After dividing the circumference of the square, above the chord and arc of the given arbitrary angle, we get the points of division that help us divide the arc of the given angle into three equal parts.

* * *

From there, we now draw lines in the direction of the vertex of the given arbitrary angle inscribing half-lines. They pass through the arc of the given arbitrary angle and divide it into three equal parts. This is angle trisection of given arbitrary angles up to 90° and smaller.

* * *

However if the angle is more than 90° and less than 180°, we divide it into two parts with the centerline. The difference is that each half has its own centerline instead of a common one.

* * *

Now it suffices to trisect only a half by applying the principle of angles less than 90°. Then with the centerline we divide the half into two equal parts.

* * *

We focus only on said half and divide it into three equal parts in accordance with the pattern used for angles up to 90°.

* * *

When this is solved, it will suffice to take the pole’s radius (the centerline intersection of the arc of the given arbitrary angle) into the compass and the first point of division (trisection of the half) and circumscribe a circle so as to transmit it to the other half of the given arbitrary angle’s arc.

* * *

And precisely there, where the circle intersects the arc of the given arbitrary angle, lie the angle trisection points of the given angle larger than 90° and smaller than 180° (extended angle). This is angle trisection of given arbitrary angles up to 180°.

And we need not more than 180°. But if we do, then we would use other somewhat more sophisticated methods – the nonagon or the octadecagon (single – double, for angles up to 240°; triple is also possible). In addition to this, in the history of human cultures there also exist ancient geometric artifacts, man-made symbols. One such artifact is the “flower of life”. Nobody knows its meaning, who drew it and when, but all around the world since time immemorial it has been held sacred to this very day. Another artifact is the Star of David or the six-sided star polygon; then there is the star octagon of Arabia, or the system of the cross, etc. From the aspect of geometry, in addition to other meanings, each and of itself they all have the trisection code. There are also other artifacts, but they are all to such an extent “ornamented” by artistic add-ons that they are difficult to “read”. Nevertheless, the division of the circle into 12 segments is the simplest way in which we can solve trisection – single up to 90° or double up to 180° and that is the basis that we should bear in mind.
The arc of the circle with which we divide must be larger than the arc that is being divided.
Hereby I finally hope enough has now been said about angle trisection.

One Response to “Special trisection of angle”

  1. K N Rao says:

    Dear Sir,
    I was studying the drawing method of SRI YANTRA. I came across problem of ‘trisecting of any angle’ more so to divide 40 degrees. The moon transits nearly 13.3 degrees in each of the 27 stars. The classical methods say that it is impossible, specially so in the old past when geometry was not so advanced. Then I chanced in your site Sacred Geometry. Ancient Indians were surely believed in sacred oneness of the universe and Human being. It pains to read your farewell letter. I am fortunate to read your article before you closed your site. God Bless you and people like you. May your tribe flourish. Thanks.

Leave a Reply

Powered by WordPress | Designed by: suv | Thanks to trucks, infiniti suv and toyota suv