Trisection Of Angles !!!

[Domayele]

I know there is a proof that angles can not be TRISECTED,but does it really mean we should give up the search for any possible method?

Below is a TRISECTION procedure(I ever sent to a Board of Discussion ).

And I need the comments of this Board's members:either in FAVOUR or AGAINST with REASONS:

 

I need to reserve the calculations involved in the proof of the Trisection Procedure and go straight to an Undisputable(Practical) proof:

It is not by chance/guess that an(y) angle can be TRISECTED ; but has a Concrete Proof from a familiar but hidden Rule/Law I have Discovered which Scientists/Mathematecians have failed to Notice/Apply since the Dawn of maths to Trisect angles.

It States: The radius of any circle divides its own circumference into six(6) equal parts,when turned/rotated around it.In the TRISECTION sense:"The radius of any semi-circle divides its arc(i.e.,semi-circle) into three(3) equal parts"and hence the angle it sub-tends.

Did you know this? Because whoever knew this(Theorem) and would still argue that an angle can not be TRISECTED is denying himsef of knowledge.

 

TRISECTION OF ANGLES 2ND VERSION:where the angle is 180 degrees or less

This procedure is Similar to the first version but gives a very CLEAR PICTURE of the THEOREM stated above:

To Trisect angle OAB:

With O as a centre use any reasonable radius to mark an arc accross angle OAB.

With a straight edge join the arc intersects with line OA and OB,name this line alpha(line A).

Now bisect angle OAB and name the bisector line beta(line B),preferably short-dashes line.

Use the point of intersection of the two(2) lines as a centre and adjust the compass to the radius of half(1/2) line alpha and construct a semi-circle/circle between the lines OA & OB.

With the same radius use line alpha intersection line OA as a centre and mark arc alpha along the circle and next,arc alpha as a centre mark another arc beta towards line OB.

[you can decide to repeat this procedure from line OB to obtain arc gamma(arc G) and arc omega(arc W) to ascertain accuracy].

Now join the arc(s) intersects along the circle/semi- circle to centre O with a straight edge and measure the three(3) seperate angles.

[if Construction/Drawing were possible on this board,my illustrations would have cleared all the doubts].

Now your comments.

NOTE:

There is SCARCELY perfection in DIVISIONS(esp.,Angles):

For the division of an angle into "n" equal parts;beside Human Error,a deviation of ±[n/n1/n]° is an avoidable is some special Cases; where n=No. of divisions.

And for multiple/repeated divisions,

deviation=±[(n/n1/n)x (n1/n)r],where r=No. of repetitions.

eg for trisection deviation maximum=±[3/31/3]=2.08°.

If an angle is bisected and again (bisected)a deviation of [(2/21/2)x(21/2)11]°=2° is possible.

Dark Orange Text means index or raised to the power

Thank you.

Its Domayele,

Greeting(s) from Ghana.

[Guest plusunim]

>>

The radius of any circle divides its own circumference into six(6) equal parts,when turned/rotated around it

>>

 

Isn't this the basic priciple to construct a hexagon within a circle?

 

cheers

[bloodhound]

It depends what means u trisect an angle by. If ur using a compass and a straight edge, then its impossible to trisect a general angle, but certain angles like pi/2 and pi are.

 

In addition, trisection of an arbitrary angle can be accomplished using a marked ruler

 

 

check

http://mathworld.wolfram.com/AngleTrisection.html

[Guest HaReLdNkUmAr]

.

[123rock]

You can trisect an angle in curved space.

[TheMathGuy]

I know there is a proof that angles can not be TRISECTED,but does it really mean we should give up the search for any possible method?

 

Yes. (at least when it comes to methods that only involve a compass and unmarked straightedge)

 

...in exactly the same way as we should give up looking for two integers whose ratio is exactly the square root of 2. A mathematical proof is very different than a scientific fact, which has been established by empirical observation and could always potentially be disproven by further empirical observation (however unlikely). A mathematical proof, once it has been confirmed to be valid, can never and will never change. Two plus two will always equal four, never five.

 

That's not to say there won't be constructions with an unmarked straight-edge and compass that will come very close to trisecting an angle, but there can never be such a construction that gets it exactly right for every angle. In particular, an angle of 60 degrees cannot be trisected in such a way, since this would imply that the cosine of 20 degrees is the root of some quadratic polynomial with rational coefficients, and one can prove using the laws of trigonometry and algebra (particularly field theory) that this is not the case (see http://en.wikipedia.org/wiki/Angle_trisection). However, notice that the Wikipedia article also points out that there are other methods of trisecting an angle. it's not like the problem is impossible by any means.

[Bignose]

Here's hoping the OP wasn't waiting 4 1/2 years until he got this last post (or in other words, why bring up this ancient thread?)

[John Cuthber]

Waiting 4 1/2 years for a reply is slightly less of a waste of time than trying to trisect an angle with a straightedge and compass.

[K.A.Alex]

To all those still cant believe angle trisection is possible.

[Klaynos]

Here's hoping the OP wasn't waiting 4 1/2 years until he got this last post (or in other words, why bring up this ancient thread?)

Or even 12 and a half years...

[DrP]

I am assuming special cases do not count.... you could do it for a 180 degree and a 90 degree angle.. (and 45, 22.5, 12.25 degrees etc..) by constructing/bisecting 60 degree angles with a compass and straight edge.

 

I will assume this doesn't count though. ;-)

[Strange]

There is a great article by a mathematician who decided to engage with the trisectors who contacted him with their ideas:

http://web.mst.edu/~lmhall/WhatToDoWhenTrisectorComes.pdf

[DrKrettin]

Excellent article, thank you. I love diagram 1. It's a pity that he seems to place trisectors in the same category as those proving Fermat's conjecture. So you never know...

[studiot]

You can, of course, construct a marked straight edge, using a compass and an unmarked one.

 

But that would be against the spirit of the game.

 

Edited November 17, 2016 by studiot