solving the impossible

[miriam_e]

I'm finally back home after looking after my folks' place and their lovely dog. I love being with my family, but it is great to be home again.

As you know, I like to relax with a video on my computer during dinner. Tonight I was watching a talk about the mathematics of Alan Turing -- one of the people I most admire in the world. During the talk the lecturer said something extraordinary. In saying that certain problems are impossible he mentioned that ancient Greek geometers looked for a way to trisect an arbitrary angle, and that finally in 1837 it was proved that there was no way to use a ruler and compass to do it.

This is a bit of a problem, because I can clearly see a very simple way to do exactly that: using just a straight-edge and compass I can divide any arbitrary angle, not only into thirds, but any number of equal parts. I don't even need the ruler. In fact I don't need the compass either if I'm allowed to simply fold the paper.

Perhaps I don't understand the problem. Surely large numbers of very smart people couldn't have puzzled over something so simple for thousands years.

So here is my solution using compass and straight-edge. It is a very simple iterative method that rapidly achieves any degree of precision desired simply by repetition. It rapidly homes in on the exact solution.

Additional: Okay, I should have read about it before going to the trouble of making this post. I found that the use of iteration is considered to be outside the rules. That strikes me as kind of silly. It makes sense to use iteration. It isn't high-tech and can achieve whatever accuracy is desired. The reason given for excluding it that it requires an infinite series of operations, but from a practical perspective that's not so. Does an irrational number cease to be useful because we must represent it to a limited accuracy? Do all real-world representations of geometric objects fail because they are all necessarily limited in accuracy?

This separation between the idealised and the practical is one of the greatest mistakes made by ancient thinkers, in my opinion. I think it is one of the reasons why we have the deeply flawed mind/body duality in so much of philosophy. It has caused people to overlook and denigrate much that is practical because it is "messy". Thankfully that has changed to some degree in recent decades with the exploration of fractals and chaos.

Comments

[thorfinn]

Close enough is often good enough, and the general method of successive approximations is certainly used quite a lot when it comes to the various applied sciences and engineering.

But when you're talking about pure mathematics, there is a categorical difference between an algorithm that produces an "exact" result and one that is "close enough". Iteration does not produce the exact result, in this case.

In some other cases, with the right problems, infinite iteration does produce an exact result, see Calculus for an entire field about doing that sort of thing.

[miriam_e]

I know what you mean, that pure maths is concerned with exact results, but I never really understood why. I mean, the whole reason for maths is the real world, and ignoring the limits of reality makes maths less relevant, not more.

Good point about calculus. It might be harnessed in a similar way... though to be truthful I've almost lost interest. The programmer feels simple iteration is good enough. :)

One thing that does niggle at me though, is that I can almost see a way to extrapolate the few angles (e.g. 90° and 27°) that can be trisected with compass and straightedge (immediately, exactly, without iterative series) and use their proportions to produce immediate, exact trisections for arbitrary angles... in a sense, piggybacking on their solutions.

But I have other, more pressing stuff at the moment, so I don't know if I'll actually get around to playing with it further.

[miriam_e]

Oops. I meant to write "the programmer in me feels simple iteration is good enough."

[thorfinn]

It's a trickle down thing. The whole point of pure mathematics is that it's tool building and pure thought research. The entire edifice of fields of mathematics are cloth of pure thought, spun with chains of logic from as few axioms as possible. And unlike pretty much any other endeavour, when you prove something true or false, it really is true or false, for all time (barring actual uncaught errors, which are pretty rare). (There are complications with that statement, but Douglas Hofstadter covers the subject much more comprehensively and interestingly in his book "Godel, Escher, Bach" than I will in a blog comment.)

That's why it's pure mathematics is nothing to do with the real world - the real world is messy, and scientific hypotheses never become definitely true, just proven true to the limits of our testing gear (and therefore good enough until new and better gear comes along).

Applied mathematicians take those pure mathematical tools and can then confidently use them in relation to real world problems... and there's nothing yet discovered/invented in pure math that some application hasn't been found for.