Talk:Constructible polygon

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Some comments.

Link to Gaussian period, which explains the cyclotomic technology Gauss used.

I'd like to see a page on Disquisitiones Arithmeticae, as it is.

The Wantzel part is fairly simple Galois theory, once one has identified the Galois group of the cyclotomic field. Something I'm not sure is always emphasised is the role of the totally real field inside the cyclotomic field, ie the field generated by the cos(θ) for the angles one wants. Here totally real means that the roots of the quadratics one wants to solve to do the constuction are all real (rather than imaginary).

Charles Matthews 10:59, 13 Mar 2004 (UTC)

[edit] question

Question: the article claims that specific concrete constructions are known for ALL constructible polygons. Yet, it then gets it down to the case for those associated to Fermat primes, and throws up its hands and seems to say, "we're not sure about 16,000-whatever, and we haven't told you how to do this in general". The article seems to contradict itself. Which is true? Do we know how, or not? Revolver 13:04, 14 Nov 2004 (UTC)

I don't think there's a mystery. Using Gaussian periods one can get an algorithm for producing the quadratic equations one needs to solve. In principle, getting from an explicit quadratic equation in terms of constructible reals, having real roots, to an explicit geometric construction of the length of a root, is nothing genuinely deep. Just effectively pointless in practice. But I suppose one could ask about it as a computer science project.

Charles Matthews 13:59, 14 Nov 2004 (UTC)

Let me see if I understand you correctly. It seems you're saying there's a definite algorithm to give you the construction, which I interpret to mean that explicit constructions are "known". In this case, the only "problem" with the 200-page example might be with the details, were any mistakes made...but not in the question of whether it is known HOW to do it (i.e. how to write it down). Revolver 20:33, 14 Nov 2004 (UTC)

Isn't it really all about handling nested radicals, as data structures? I have never looked deeply into it, though? Charles Matthews 20:47, 14 Nov 2004 (UTC)

[edit] how do you construct a 15-gon?

At the bottom the article says:

Thus an n-gon is constructible if

   n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ...

How do you construct a 15-gon with a compass and straight edge? (Its not a power of 2 co-prime.)

Desrosier 08:45, 29 August 2006 (UTC)

To construct an angle of 2π/15, if we can construct 2π/5 and 2π/3? Is that so hard? Charles Matthews 09:02, 29 August 2006 (UTC)