Talk:Compass and straightedge constructions

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"First notice that we can write n = 2^c \cdot d, where d is an odd number and 2^c is a power of two."

This needs rewording, since it is only about some values of n. Even ones at the very least. Lithograph (talk) 02:14, 7 December 2010 (UTC)

Well, it's true for all n, it just might be that c=0 (so 2^c=1), which is the case for odd n. But the point is that we can reduce to the question of which n-gons with n odd are constructible, so rephrase it as you see fit. Great work on the edits by the way -- sorry this page is such a mess. --MarkGall (talk) 02:29, 7 December 2010 (UTC)
D'oh, I failed on that, you are correct. The trouble I had with the page is that from the n-gon section on, it really has nothing to do with RW. It's just some abstract math that I doubt people come here for. The first few sections are awesome, though. Lithograph (talk) 04:19, 7 December 2010 (UTC)
Good point, got carried away, I'll move it to the talk page. What do you think of the "technical" section? Axe that too? --MarkGall (talk) 05:59, 7 December 2010 (UTC)
I dunno, although it's a bit hard-core, it does make a nice closing to the article. Let it stay for now? Lithograph (talk) 06:12, 7 December 2010 (UTC)
Sounds good for now. I'll try to expand the crankery-related sections later this month when I'm at home and I have the Dudley book. I think quite a bit more could go in there... --MarkGall (talk) 06:18, 7 December 2010 (UTC)

Deleted Section[edit]

A related problem: regular n-gons[edit]

Animation showing how to construct a regular hexagon.

Another old problem is to construct regular n-gons for various values of n. For example, it's easy (and classical) to construct equilateral triangles, squares, regular pentagons, regular octagons, and a few others. It reasonable to expect the above characterization of the constructible numbers to help us figure out exactly for which n it's possible to construct the n-gon. The amazing answer to this question was provided by Gauss. First notice that we can write n = 2^c \cdot d, where d is an odd number and 2^c is a power of two. If it's possible to construct a regular d-gon, then it's possible to construct a regular n-gon just by repeatedly bisecting the angles out from the center of the d-gon. So an equivalent question is to figure out for which odd numbers d it is possible to construct a d-gon.

Gauss's theorem is that the d-gon is constructible if and only if d is the product of a collection of distinct Fermat primes, i.e. primes of the form F_n = 2^{2^n}+1. The first five Fermat numbers are all prime, but it's an open problem whether there exist infinitely many of them. Those in the know think probably not, and it's been verified that numbers 6 through 33 are not prime, and indeed no further prime Fermat numbers are known.[1]

This answer has one surprising implication: it's possible to construct the 17-gon, since 17 = 2^{2^2}+1. This was the first new advance in this field since ancient times. Gauss's proof was purely algebraic and so it didn't actually give a way to construct the 17-gon, but a contemporary of Gauss actually gave the construction a few years later. Later on another mathematician constructed the 257-gon, which had been proved possible by Gauss since 257=2^{2^3}+1. In 1894 Johann Gustav Hermes completed a heroic 10-year effort to construct a regular 65537-gon (the possibility of which is guaranteed by Gauss's work: 65537=2^{2^4}+1), which would be basically indistinguishable from a circle. This culminated in a 200 page book describing the construction. John Conway has suggested that the construction is probably incorrect,[2] though these days nobody is too worried about how to actually carry out such constructions.

Footnotes[edit]

  1. Fermat factoring status
  2. So says Wikipedia, anyway

Just An Observation[edit]

Is it just me or is this the perfect example of the difference between mathematicians and engineers? The mathematician says it's impossible to square a circle and the engineer says "Maybe, but we can easily come within acceptable tolerance". Perceptron (talk) 06:18, 8 March 2011 (UTC)

It is just you, as it was the mathematician who proved you could come within acceptable tolerance as well. Mcxz (talk) 16:11, 10 June 2011 (UTC)

"The result involves cube roots"[edit]

I should point out that just because a number is written using cube roots, it doesn't necessarily mean it's cubic-irrational. As an example, consider the equation

x^3 + 3x - 4 = 0

Using Cardano's formula yields the solution

x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}

Since this polynomial is monotonically increasing, the above is its only zero. Yet from just looking at the equation, we can guess that x = 1. The above expression involving cubic roots cannot be simplified by elementary operations, but with a calculator or computer program, one can see that it's equal to 1 with any desired precision. To actually prove that the root of the equation 8l^3 - 6l - 1 = 0 cannot be constructed with a compass and straightedge, we would need to prove it has no rational roots, from which it follows it cannot be solved in square radicals either. - LucidFox (talk) 19:18, 10 August 2011 (UTC)

It's true; feel free to fix it if you see a way to do it without messing with the flow too much. I think there are some other little white lies in the article, though maybe I removed a few. Of course, this is hardly the biggest flaw in the "proof" -- nowhere does it really justify that constructible numbers are as claimed! --MarkGall (talk) 01:46, 11 August 2011 (UTC)

Incompleteness Theorems allowing proof?[edit]

Derision is expressed in regards to those attempting to square the circle. I am slightly confused, as my understanding of the incompleteness theorems meant that you could have a system that was inconsistent. My calculus professor would joke about the possibility of a contradiction in math when students found two solutions when there was only one ("Either one of us made a mistake, or we get to go home.") Can it not be the case that math is inconsistent and you can square the circle? Imarcuson (talk) 05:45, 22 December 2011 (UTC)

Perhaps. Perhaps pigs will start flying south for the winter. Perhaps Satan is busy putting in a large order for down comforters. Mjollnir.svgListenerXTalkerX 06:19, 22 December 2011 (UTC)
I wish to provide some context before responding directly to your response.
  • I often have trouble knowing when to put on my epistemology hat and when to put on my science hat. Statements about truth and falsity are very different when you can assume a brain-in-the vat scenario compared to those times when Occam's Razor applies.
  • I can certainly appreciate the need for ignoring cranks who do not understand the basic premises of a question, as demonstrated in the Squaring the circle article in the section on methods to approximate the solution.
Am I to interpret your response as "It's possible, but damn unlikely?" I think that is how to interpret it, but I am not sure. Imarcuson (talk) 06:42, 22 December 2011 (UTC)
You should interpret it as sarcasm, because ListenerX (and me) are dismissing your suggestion entirely. Star of David.png Radioactive afikomen Please ignore all my awful pre-2014 comments. 06:56, 22 December 2011 (UTC)
Indeed. Squaring the circle would require using a compass and straightedge to calculate the square root of π, and there is a proof that this cannot be done. LucidFox's point notwithstanding, the incompleteness theorems just state that there are some true, unprovable statements; they do not invalidate existing proofs. Mjollnir.svgListenerXTalkerX 07:05, 22 December 2011 (UTC)
I'm not sure why you invoke the incompleteness theorems here. They only apply to systems that include first-order arithmetic. Euclidean geometry doesn't; in fact, it has a complete axiom system (the Hilbert axioms). - LucidFox (talk) 06:53, 22 December 2011 (UTC)
"The Hilbert Axioms" sounds like a great name for a band. Star of David.png Radioactive afikomen Please ignore all my awful pre-2014 comments. 07:17, 22 December 2011 (UTC)
Dammit! I had just read that Euclidean Geometry had a complete axiom system in the incompleteness theorem article on this very wiki. As to why I bring up the incompleteness theorems, a) I forgot what I had just read, and b) I may have been conflating several things my calculus teacher said. I do recall her saying that we have to hope the system of mathematics is consistent; this was said in the same lecture as her mention of the incompleteness theorems. Further, she did joke a fair bit about possibly finding a contradiction in mathematics when someone calculated something wrong. I may have been taking some statements too literally. That said, I have heard, paraphrased, "in a complicated enough system, you can have completeness or you can consistency," and I gathered that inconsistency was a possibility. Imarcuson (talk) 07:49, 22 December 2011 (UTC)

Is this an error half way down the page?[edit]

It seems that the diagram half way down the page is in error, the second iteration of the spiral gives an hypotenuse of (1+(2)^1/2)^1/2 or root(1+root2), I believe this should be root 3 or 3^1/2 — Unsigned, by: 79.97.159.118 / talk / contribs

Hmm, that is correct. I have no idea how to fix the image though. Nobodydon't bother 15:09, 7 January 2013 (UTC)