Why is trisection of an angle impossible

There is a rich theory of algebraic fields, but we will not require anything here more than this definition and two basic lemmas Lemmas 2 and 3 , which we prove here. If the number of iterations performed is odd, the sign of the result is changed not necessary in above example. Definitions: Linear independence and degrees. In other words, the notions of algebraic degree and field extension degree coincide when the field is generated by an algebraic number.

We now prove an important fact about the degrees of field extensions, which will be key to our main impossibility proofs below:. Definitions: Constructible numbers. The next step is to establish what kinds of numbers can be constructed by ruler and compass.

First, it is clear that addition can be easily done with ruler and compass by marking the two distances on a straight line, then the combined distance is the sum. Subtraction can be done similarly. Multiplication can also be done by ruler and compass, as illustrated in the figure to the right taken from here , and division is just the inverse of this process.

One such letter even missed out several pages of the "proof" on the grounds of security of the writer's copyright! Needless to say, all these so-called proofs contain flaws and are worthless. If the writers really wanted to convince anyone that it is possible to trisect the angle, their time would be better spent trying to find an error in Wantzel's proof. The real mystery here is why people keep trying to solve the problem in the face of a proof of its impossibility.

How would you reply to them? Mathematicians prove the impossible all the time. One such example is the proff that it is impossible to find the ratio of two integers that equal the square root of two. That is NOT proving the impossible. Rather, its proving impossible something huh? As in, its "un"-proving something. Actually the aphorism is that it's not possible to prove a negative and even that is sometimes wrong. Proving that things are impossible is done all the time In my case I am looking at it as a way to keep my mind and thinking active.

The easiest thing to do is to prove that it is impossible to trisect any angle precisely. However there is still the question of is there any method to effectively trisect any angle to an acceptable accuracy in a limited number of steps using only a compass and unmarked straight edge. This is a different issue and is a valid question. I recomend to you my comment at the top ofthe column of comments: I claim that it works: Straightforward in angles between 0 and , higher angles need to be partitioned into a straight angle tri is 60 easy and an acute or obtuse angle the smaller or residual angle is then trisected using my described technique asnd then the trisect copied to augment the angle of 60 derived from the straight angle, the composite angle being one third of the original angle.

We divide any angle by this method into the ratio of x:y:x but not x:x:x so the trisection method you proposed is actually a method for partitioning an angle into ratio x:y:x where values of x and y can be calculated for general case. You can try it and show that x is not equal to y!!!

Talking generally, we can divide any arbitrary angle into a paliandromic ratio by this chord sectioned method. There are different auxilliary curves like hyperbola for trisection only , spiral for n section , sinusoidal wave for n section , etc which can trisect or nsect any given angle.

Sometimes only intuition can give wrong results bro so read more and derive more. First, you may not draw a circle with an arbitrary radius "any radius you like". The only thing you can do with a Euclidean compass is to draw a circle with the center at a known point, and the circumference containing a known point. I will try it again and get back to you later.

I have managed to generate a 7 steps algorithm for trisecting an arbitrary angle. In this draft, i have presented the results based on accuracy. My algorithm is purely compass-straightedge construction, no arithmetics nor measurements involved. It works for all angles, whole number angles and non-whole number angles trisection.

Alex, I understand that it appears tiresome. Plus the algebraic proof of why this work. Slide 26 presents the results when the graphic method is compared to the algebraic method for various values of the angle.

I do abide to the rules of straightedge and compass I you can provide an email address I will be glad to send you a PDF copy of the book. Thanks for your interest Florentino. I got something else on cube duplication. I got a very interesting approach of constructing a line of magnitude 1. In the work i have rounded up this factor to 1.

I do not believe it is good we confuse an inability with impossibility. I think the most difficult thing under the sun is to proof something to be impossible. As a physicist, i understand what it means by something being impossible. For one to proof an impossibility, we have to take into consideration all the laws of nature. Not just acceptance of statements that something is impossible. Any way, in this article, i have addressed my solution as an approximate construction because of the proof i employed.

But, i think if one is able to provably construct the third root of 2, what importantly matters is the logic followed towards the solution, and not much of the generated value, though the results should agree with the set claim. The method works for both the compass-straightedge construction and the CAD methods. I used GeiGebra in generating the results. Any feedback is highly welcome. Correspond through the provided e-mail in the paper. With humility, I am convince that many problem may have undiscovered solutions.

By freeing one of the algebraic constraints of the trisection of an arbitrary angle, one can reduce it from a cubic root to a square root approach. I do not think it will be a good thing to bend any algebraic condition, based on the classical geometric view of things. No one can deny that; Euclidean geometry has its initial shape in nature and sharply shine than any other new development in academia.

It will remain in existence. However, i believe, we need to appreciate the fact that, the traditional geometers solely worked out their findings using only compass and straightedge. The analytical interpretation of geometrical findings came later, due to interest by humans to work with specific quantities. Number systems was created.

I deeper look at the ancient times work would reveal that, Euclid did not work with numbers, and instead, he used the term magnitude. He did not have nationalities such as root 2. Instead, he worked with incomparable. But he was able to construct a line of magnitude root of 2. I Do agree with you that the most difficult thing in the universe for humans is to proof something to be completely impossible. No matter the set conditions for a particular problem, we need to be sure and clear that impossibility does not mean the inability to solve a particular problem.

For example, if we consider a quantity power, in mechanics, power is defined by energy per unit time. When someone say; it is not possible to create not destroy power, i think it is quite sensible. The nature agree with such a claim. It is possible, physically, to have almost two identical people competing to run up and down a system of stair cases, and to have one of them wining the race.

Therefore agreeing with the statement that it is impossible to create power. What importantly the matter should be is the analogs employed in disproving something.

Statements based on incomprehensible algebraic proofs make no sense. I too use the CAD for precision in the graphic method for precision. I tested it using really arbitrary angle value, like Just straightedge and compass. The algebraic analysis using cos 3a showed that the solutions for the trisections are at the intersects of arc circle when you consider an axe translation for your coordinates systems. No rule broken.

Please provide an email address so I can forward you a PDF file. Please use the email above. Got issues with my network connection. Try it, you will be amazed. Dont get stuck to the idea that it was proven to be impossible to achieve.

In this site, the provided tutorials would help one learn not only the classical construction of all whole number angles, but also the construction of the infinitely the many regular polygons, the geometric trisection of all whole number trisectible angles, and much more. Elephant Rock. Maybe someone can explain…. Now draw two lines connecting each third of the opposite trisected line segments such that both lines pass through the vertex.

The center angle will be different than the two sides. Hi Elephant Rock I looked at your construction results and i could not connect the figures well. However, i must say that, the angle trisection problem is still an open challenge. What importantly people need to put into consideration is the conditions governing the solutions to the problem.

The compass straightedge construction is such a serious restriction, and from this stem other constraints that: no measurements are allowed, and any geometric procedure should not involve arithmetic. In other words, to proof any geometric problem, one needs to consider that, the solution is deeply inherent from the construction. It has to be geometrical. The angle trisection solution is possible.

I cannot, though not a mathematician, call myself a crank. Any one with passion in geometry can solve this problem provided he has the basic classical geometric knowledge. This is the only weapon. In this paper, i have discussed much on classical geometric constructions, the construction of all whole number angles, and the relationship between the classical construction of whole number angles and the angle trisection problem. I also have, in brief, shown the relation between classical geometry and the analytical geometry.

We have to understand why one has to be careful in saying it is generally impossible to solve a certain problem. This is the most difficult task to do on earth. This among other problems in the proof make it geometrically invalid. Hi guys! Here is my angle trisection solution. In my view, the angle trisection problem is solvable. In-fact in a very simple way. But, before i provide the solution, i think it is good scientists, consider the conditions governing the angle trisection problem.

In classical geometry, the use of trigonometry has no space. This only make sense in analytical geometry. Therefore, any solution involving the use of trigonometry, and the other mechanical methods aimed at solving the angle trisection problem are generally wrong. In my view, what maters as Euclid did in his works, is the analogs followed towards the solution. Here, any solution should reasonably agree with the set conventions. Your comments and constructive views are welcomed. Hi I would like to point out some few things about some of my earlier posts on this blog as follows:.

The above posted video links are specifically based in the construction of all whole number angles, and the trisection of the infinitely many trisectible angles. They are not meant for the angle trisection solution. It the earlier post, on page 6 of the paper, figures 3 and 4 were interchanged during the copy-editing process, which we did not get off during the proofreading stage.

In annex-1, the line trisection procedure was misleading and a very important correction has been made. Thank you for those who contacted me on these corrections.

Any academician would acknowledge that, paper production comes after the review process. And that, a paper going though a peer-review process does not make it a perfect one. So such basic errors are common. Gary E. A call to all geometers of the 3, year, Trisect the Angle grand challenge. The dogma of this ages old, false impossibility, is greatly exaggerated.

And the lowly sixty degree triangle. Accordingly, it is my great pleasure to introduce to you and the world and hidden in plain sight all this time since a man worthy of surely nothing less than the Fields Medal, Mr. Harry Cohen:. Click to access Scan So we have five thumbs-down pinheads, huh? Wantzel did not use Galois theory.

For a detailed history and elementary proofs of non-constructibility theorems apart form the transcendence of pi I recommend:.

I can think of an iterative approach to trisect an angle. It is as follows. This implies first bisect the angle. Then bisect the bisected angle. Then bisect the first bisected angle and the second bisected angle. Repeat the process few times till required accuracy based on the series and we can trisect the given angle.

One can repeat the same procedure for the other half of the angle. Nebojsa Mitic. Dear Terence, why are you not deleting all the useless crank comments? Sometimes, freedom of speech gets us nowhere… Besides, they could freely post their junk on their own blog. Dear Joc With the highest degree of respect, allow me to respond to this your post. I am glad to have read your thinking.

I understand it is not wise to hijack someones blog, and above all, a respectful blog like this of Prof. Why let someone else do it for you? I think there is nothing wrong with freedom of speech. Any information of help? So between the other posts and the contents of your comment, which one is junk? Is this your blog so as to post your feelings and not information in here? Please Note: I got nothing personal with you. I only have problem with how you cannot control your emotions.

I find it healthy when one develops the courage to share a concept in a forum as this. If there is any junk that should never cross the eyes and in the minds of the future generations, through this blog, is your post. You just discourage others without any constructive contribution! How would you feel if you were told you cannot do anything from the capacity of your thinking? This is the harm you are causing the future generations.

You are among those robbing them the freedom to think. You are among those who just read and believe, and this is destructive i suppose! Your thinking is self defeating in simple terms. Of course, we can do all the constructions the Greeks did, in many cases a lot more simply, by using a compass that stays fixed. To use something as error-prone as a spring-loaded compass, then worry about possible imperfections in constructing a tool like the hatchet, or positioning a marked straightedge, is completely arbitrary.

Also, if you think about it, the business of lining a straightedge up through two points to draw a line also has a lot of trial and error about it - as much as the hatchet tool. You line the straightedge up against one point, then position it against the other, then go back and correct any shifting at the first point, and so on.

Then, actually to draw the line, you need to take into account the fact that any marking tool has finite width, so that as often as not the drawn line doesn't pass exactly through the two points. Renaissance instrument makers soon discovered this problem. They found out that markings plotted using only compasses were more accurate than those made using straightedges. They began devising alternative constructions that eliminated use of the straightedge.

Although the constructions were often more complex, they were still more accurate than those that required straightedges. Mathematicians finally showed that every construction that can be done with a compass and straightedge can be done with a compass alone. The only qualification is that we define constructing two points on a given line as equivalent to constructing the line itself.

In other words, trisecting an angle amounts to solving a cubic equation. That's why nothing has been said about doubling the cube. Doubling the cube amounts to finding the cube root of two, that is, also solving a cubic equation.

So algebraically, the two constructions are equivalent. Squaring the circle is a bit more complicated. Recall how the Pythagoreans, to their horror, found out that there are other kinds of numbers than integers whole numbers and rational fractions. The process of discovering all the types of numbers that exist turns out to be directly related to the proof that the three classic problems are unsolvable. Since the time of Pythagoras, mathematicians have discovered that there are many types of numbers:.

Proof: Triangle ACD is a right triangle since all angles inscribed in a semicircle are right angles. Hence BD equals the square root of X. One of the first fruits of these studies was the discovery by the young Karl Friedrich Gauss that it was possible, using a ruler and straightedge, to construct a polygon of 17 sides. This was something completely unsuspected. He also found that polygons of and sides could be constructed. Strictly speaking, Gauss only discovered it was possible; other mathematicians devised constructions and showed that no other polygons were possible, but Gauss made the pivotal discovery.

A number of books give constructions for the sided polygon; constructions for the other two have been devised but are hardly worth the effort. Numbers that can be expressed as combinations of rational numbers and square roots, however complicated the combination, are called surds. Only surds can be constructed using a compass and straightedge. We can now pose and answer without proof the following questions:.

Gauss did more than just find new polygons. Building on his results, other mathematicians showed that polygons of 2, 3, 5, 17, Of course you can also construct polygons by repeatedly bisecting the angles of these polygons to construct polygons with 4, 6, 8, 10, 12, 16, etc. You can also construct polygons of 15 sides by combining the constructions for 3 and 5 sides, etc. By enumerating what was possible, he ruled out many other things as impossible.

In particular, 7 and 9-sided polygons cannot be constructed using straightedge and compass. Constructing a 9-sided polygon requires trisecting a degree angle. Since this can't be done, obviously trisecting any desired angle is impossible.

The numbers are all primes, and equal to some power of 2 plus one, and the exponents are all powers of 2. Most people who still send solutions to the three classic problems to mathematics departments don't have a clue how the problems were finally solved.

Many seriously think mathematicians just gave up and decreed the problems unsolvable. The problems actually can't be solved because they require properties that a straightedge and compass simply do not have.

You can't draw an ellipse with a straightedge and compass although you can construct as many points on the ellipse as you like , so why is it a shock that you can't trisect the angle, duplicate the cube, or square the circle? You can't tighten nuts with a saw or cut a board with a wrench, and expecting a straightedge and compass to do something beyond their capabilities is equally futile. Here's another analogy: if you spend your entire life driving a truck, you might eventually be lulled into thinking you can see the entire country from the highway.

It's only if you get out and walk or fly over the landscape that you discover there are a lot of other places as well. Geometry with straightedge and compass creates a similar illusion; eventually we believe the points we can construct are all the points that exist.

It was only when mathematicians began studying the properties of numbers that they found out it wasn't so. Just as you have to get off the highway to see that other places exist, to find the limitations of geometry you have to get outside of geometry.

More mathematically literate angle trisectors are sometimes aware of the number-theory approach, but reject it because they think a geometrical problem can only be properly solved geometrically. But if the problem is that the solution requires capabilities beyond those of a straightedge and compass, how in the world can that be discovered from within geometry?

Anyway, who says geometrical problems can only be properly solved geometrically? The only thing that would justify that rule is some demonstration that the geometrical solution to a problem and the algebraic solution yielded different results - and then you'd have to prove the geometrical approach was the correct one. But there are no cases where this has ever happened, so there is no justification for rejecting the algebraic solution to the three classic problems.

Indeed, it has been shown that if there is an inconsistency anywhere in mathematics, it is possible to prove any proposition whatsoever. In fact, one of the most famous mathematical proofs of all times, Kurt Godel's Incompleteness Theorem, showed that there is always an "outside" to mathematics. Once a rule system gets complex enough and Euclidean geometry is plenty complex enough it is always possible to make true statements that cannot be proven using only the rules of that system.

Thus is possible to make true geometrical statements like "angles cannot be trisected using ruler and compass" that cannot be proven using the rules of geometry alone. You can construct a sided polygon with central angles of 18 degrees. You can construct a sided polygon with central angles of 15 degrees. Obviously you can superimpose the constructions, and the difference between the two angles is three degrees.