Talk:Constructible polygon

Incomplete

We do not show how to construct n-gons where n is a power of a prime number!

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)

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)

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)

I recommend constructing simultaneously a regular pentagon and a regular trigon, centered at the same point and sharing one vertex. The angle spanning between some secondary vertex of the pentagon to some secondary vertex of the trigon can be used to properly space vertices on the circumscribed circle. CogitoErgoCogitoSum (talk) 15:49, 21 October 2023 (UTC)

Ambiguousness

"Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in 1837." Does "which was proved by Pierre..." refer to the proof of the sufficient, or to the necessary? — metaprimer (talk) 01:58, 28 October 2007 (UTC)

List

Perhaps you should list which polygons can be constructed. At least up to lets say a 100 sided figure. —Preceding unsigned comment added by Tailsfan2 (talk • contribs) 00:17, 11 May 2009 (UTC)

Listing the first few and linking the OEIS seems enough to me. Algebraist 19:00, 13 June 2009 (UTC)

The location of the list and the general wording of the article make it sound as though the only constructible n-gons are those that fit the (2^k)*F model. However, the n-gons that are powers of 2 (n = 1,2,4,8...) can't be calculated this way. Unless I'm completely missing something here (and that is entirely possible) there should be some mention that power of 2 n-gons come from the degenerate 2-sided polygon.Djibouti (talk) 21:43, 17 February 2010 (UTC)

Construction of a regular pentagon

The moving diagram is pretty, but inefficient. After the first side is constructed, the other corners can easily be stepped off by compass. 82.163.24.100 (talk) 16:27, 10 March 2010 (UTC)

Hermes criticism

The article currently says "Conway has cast doubt on the validity of Hermes' construction, however." The citation is a forum post which quotes an email from John Conway. The guts of Conway's criticism is "I can hardly believe that he really got anywhere, because the size of the problem is much bigger than one might suppose." Conway ends the email with "I would very much like to see this calculation done, since there might well be some patterns that would enable one to write down the answer more simply than the way sketched above." This criticism is very weak, and he professes his suspicion that his method is inefficient (so perhaps the problem is smaller than he thinks). It's an offhand comment qualified by "I can hardly believe", and it seems the only reason the criticism is mentioned is because Conway is famous. I think the reference and comment in the article should be removed, or replaced by a real source of criticism if any exist. The Hermes article doesn't include this information either, and it's a disservice to him to call into question his most famous achievement with such poor supporting evidence. 67.158.43.41 (talk) 02:23, 21 February 2011 (UTC)

I agree. I am removing the statement. --kundor (talk) 20:50, 16 November 2014 (UTC)

Animation

There's an animation of the construction of a 257-gon on the Italian Wikipedia, but i dont know how to put it in (im not good with computers). Somebody do it please! search 257-gono on the Italian Wikipedia. —Preceding unsigned comment added by 71.65.217.167 (talk) 02:35, 17 March 2011 (UTC)

What about constructible 3d polygons?

Imagine if instead of fixing one end of a compass on a point in a 2d surface, we fix that point in three dimensional space? In 2D we could a trace a circle, while in 3D, we can imagine tracing a sphere (though it would take an unimaginably long time). Is there such a thing as "constructible" 3d polygons, or whatever such might actually be called in geometry?siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 05:34, 19 December 2012 (UTC)

What's a "3d polygon"? John Baez (talk) 16:24, 26 February 2013 (UTC)

Sorry about the confusion. I meant polyhedra.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 12:38, 27 February 2013 (UTC)

33-, 65- and 129-sided polygons

Right now the article says:

Restating the Gauss-Wantzel theorem:

A regular n-gon is constructible with ruler and compass if and only if n = 2^k+1 (k ≥ 2) or if n = 2^kp₁p₂...p_t where k is a non-negative integer, t is a positive integer, and each p_i is a (distinct) Fermat prime.

There's no condition here saying that 2^k+1 needs to be a Fermat prime or even prime. So, according to this statement, regular polygons with 33, 65 and 129 sides are constructible. But that doesn't sound right at all. Could someone fix this? John Baez (talk) 16:24, 26 February 2013 (UTC)

I can't fix it. 33=3*11 but 11-gon is not constructible. 65=5*13 but 13-gon is not constructible. 14:17, 4 August 2014 (UTC)~ — Preceding unsigned comment added by 79.185.51.170 (talk)

360

Since construction of a 360-sided polygon is impossible using straightedge and compass alone, how wwre the Ancients able to divide the circle in 360 degrees? Actually… were they? CielProfond (talk) 02:33, 25 February 2017 (UTC)

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Gauss-Wantzel theorem

A regular n-gon can be constructed with compass and straightedge if and only if n is a power of 2 or the product of a power of 2 and any number of distinct Fermat primes. Is the use of the word "distinct" only encompassing the occurrence of more than 1 Fermat prime numbers ? If so , it will not encompass 1 Fermat prime as a distinct amount of Fermat primes and thus the statement of the theorem is incomplete. If it does encompass 1 Fermat prime as a distinct amount of Fermat primes , then this should be made more specific. EuclidIncarnated (talk) 16:54, 26 April 2024 (UTC)