# Impossible Trisection

#### okieinexile

##### Well-Known Member
I was given this same problem by my dad when I was young. He said the same thing, that it was 'proven' impossible. I invested quite a few hours trying different ways, because it seemed intuitively wrong. I have disagreed with the conclusion of math. The 'proof' is in error:

The angle of any circle can be tri-sected and bi-sected to within the precision of the angle, the compass, and the ruler. I am afraid this is where this world and the math depart. Math is a language within the world, made to describe the world, and the language is fallible in that regard. The math is producing a FALSE statement. It IS possible to tri-sect an angle using a ruler and a compass to any precision desired with the correspondingly available precise tools. The procedure is this:

1. Bi-sect the angle in half. angle(Ray 1) = angle(X) / 2
2. Bi-sect the angle in half again. angle(Ray 2) = angle(Ray 1) / 2
3. Bi-sect the angle between step 1 and 2. angle(Ray 3) = (angle (Ray 1) + angle (Ray 2)) / 2
4. Repeat step 3 until the desired accuracy is reached.

The resulting sequence is:
Angle = 1/3 angle(X) = angle(X) * (1/2 -1/4 +1/8 -1/16 +1/32 -1/64 +1/128 -1/256 +1/512 -1/1024, etc….)

Where is the math error? The mathematician errors by extrapolating or interpolating what appears to be true at sizes that people are familiar with, to the sizes on the quantum and allegedly infinite precision.

The mathematician may note that it is impossible to tri-sect the angle with infinite precision. Infinite precision impossible? That is true. However, it is also impossible to bi-sect any angle with infinite precision. Furthermore, there is no ruler with infinite precision, and there is no compass with infinite precision, and if there were it would take an infinite length of time and an infinite amount of energy to carry out the bi-section… just as it would take an infinite length of time to carry out the tri-section to an infinite precision.

The mathematician may note that the math is considering a perfect circle, with a perfect compass, and a perfect ruler. Which is it… is the world imperfect, or is the language of math imperfect? It is impossible to produce a perfect circle, because there is no such thing as a perfect compass, and there is no such thing as a perfect ruler, in this perfect world.

The angle of any circle can be bi-sected, and tri-sected, to within the precision of the angle, any compass, and any ruler. At some level, math becomes a language for fiction. Either it is true to say that it is possible to tri-sect an angle with a ruler and compass, or it is false to say that it is possible to bi-sect an angle with a ruler and compass. Math wrongly suggests otherwise.

Perhaps another example of this fallibility of math will help. A distance is divided into two. Divide the resulting distance into two again. Repeat. The math says that the distance is always greater than zero. Prove it… DO IT. If you have not done it, then you have not proven it. The quantum physicist who has tried these things will report that it is impossible to measure the distance between two objects without affecting their position. The step ‘repeat’ at some level of precision is impossible. Impossible. Furthermore, the forces between the particles at close distances increases so that the proposed steps rapidly become impossible and imprecise. Furthermore, any known quantity, anything measured, any trait, cannot be divided as the mathematician suggests. There is a quantum limit.

As I use the word ruler, I intend the word straight edge. It is considered cheating to use a ruler. If you could make marks on the straight edge then Archimedes' method would be less effort.

Cola,
Your error is in step four. No mathematician would disagree that one can get to within any preassigned level of accuracy. It's been known for years, maybe even thousands of years. When we say it is impossible, we mean to do it exactly and with a finite number of operations.

You run into the same problem with your other example. I will agree that for practical purposes it doesn't matter, but we are not talking about practicality here. This is about exact mathematical reasoning. It is a theoretical exercise, and nobody cares but a mathematician, but it is true and no amount of whining about the rules is going to change it.

Au contraire: The trisection is perfectly finite. It is as finite and precise as taking 3 atomic particles and dividing them into 3 individual atomic particles. Whereas, the fantasy of infinite precision in division is a mathematician's fantasy.

A serious Star Trekker has perfectly reasonable rules, logic, and math for teleporters, warp speed, inertial dampeners, holodecks, time travel, and replicators. While perhaps there is hope for one of those technologies in the future, the math presented by the geometry problem will never be constructed in this world. The math has left reality further behind than Star Trek.

A story problem is based on fabrications and invites a thought experiment. The story problem submitted here even suggests that the experiment can be carried out, and suggests that infinite precision is just around the corner. Like any fabrication and thought experiment, unproven errors and false assumptions will enter in. The extrapolations may seem reasonable but they can be entirely false. Presumptions about the world will factor in, and those presumptions may be entirely false.

Here are a few more math story problems to consider, with implied but questionable belief systems about the world we live in:

A. A hot air balloon on Earth sets off to find the edge of the world. The balloon travels 40,075 kilometers in one direction. How far away is the balloon from the starting location?

B. An unmanned rocket accelerates towards a star 13 billion light years away, and decelerates to arrive. The rate of acceleration is 30,000 m/s/s, and the rate of deceleration is the same. What is the maximum velocity obtained and how long is the travel time?

C. 4 electrons are divided into 3 portions. How much of an electron is in each portion?

Au contraire: The trisection is perfectly finite. It is as finite and precise as taking 3 atomic particles and dividing them into 3 individual atomic particles. Whereas, the fantasy of infinite precision in division is a mathematician's fantasy.

The fantasy of infinite precision might seem silly to you, but it is the strength of mathematics. Mathematicians don't need to be sloppy; engineers and physicists are doing that already. The precision of mathematics forces it to follow discipline. If that's not your cup of tea, that's fine. However, to say that you've solved this problem that mathematicians aren't able to and scoff at how silly they are makes you look like an asshole to anyone who has a clue.

The only reason this problem is interesting is that is demonstrates that given a set of premises it is possible to prove the impossibility of solution within those premises. Getting an answer in a finite set of moves within a preset margin of error has been known for a long time. It can be done, it is easy, and doing it is not original. This has been known for a long, long, long time.