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And I quote from the first subsection "compass and straightedge tools", first bullet point:
Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle.
I think these two sentences are contradictory. The first seems to imply a collapsing compass, as you must pre-define the radius and center point in order to draw the circle there. If a rigid compass were permissible then you need only a center point in order to draw a circle whose radius youve already defined with the compass.
At some time in the past I Heard the commentators for professional wrestling refer to the ring in which activity takes place as the "squared circle". This is an excellent example of pseudo-science in the service of pseudo-sport. Eclecticology
We need more on doubling the cube and angle trisection. The Anome
Mathematicians are notoriously incompetent historians. Gauss NEVER gave a proof of the necessity of the constructibility of the regular n-gon. WANTZEL proved this in 1837. If you read otherwise, it's because mathematicians are sloppy historians. Revolver
The article mixes the two words compass and compasses with obviously the same meaning. Both words are acceptable according to
Webster's, but for the sake of consistency I'm changing all to compass. Why not compasses? Because that way I don't have to move the article, and anyway "ruler and compasses construction" is less agreeable to the ear.
—
Herbee 18:43, 2004 Mar 13 (UTC)
MERGE: This needs to be merged with Constructible number and all the other articles on specific impossibility proofs. These overlap a lot, yet none of them should try to cover every topic. Revolver 13:08, 14 Nov 2004 (UTC)
The regular polygon with the largest number of sides that was ever actually drawn with a ruler and compass had 1,024 sides. It was drawn by graduate student Mr. Sam Bronstein at the University of Kentucky in 1963, a feat that took him 33 days and a sheet of paper 9 meters square.
Is it really as imprtant to be here? Tosha 03:37, 18 October 2005 (UTC)
technically, using a compass and a straightedge to draw a hexagon is impossible, since the circumference is 2Πr. not 2(3)r.
there is a slight distortion. the only true way to draw a real hexagon is to 1. bisect the circle at any point 1/2 way between the center[o] and an edge[e] to form point [a] 2. using point [a] as a center, draw a line [b] perpendicular to the angle formed between [e] and [o] 3. mark the two points where [b] intersects the circles radius 4. repeat 180 from [e] across [o]
above unsigned comment 17:39, 8 December 2005 Kargoneth
For the record, a regular hexagon is constructible. John Reid 23:36, 30 March 2006 (UTC)
A ruler is commonly understood to be marked; this type of construction specifically forbids markings. John Reid 19:18, 26 March 2006 (UTC)
A ruler is a marked straightedge; this is explicitly forbidden in compass and straightedge construction. For practical purposes the ruler is the more generally useful tool, certainly the more popular; thus it's not hard to see why the term is also more popular. But this is one of those times that the Google test gives a false indication.
To the quibble: Yes, strictly speaking, a "ruler" is a straightedge only, a tool for ruling (drawing straight lines). A tool for measuring is a "scale"; thus the commonplace object found in home and office is truly a ruler with a scale printed upon it -- a dual-purpose tool. It is, however, ambiguously called a ruler; and the less-common "straightedge" is clearly and unambiguously unmarked. John Reid 23:55, 28 March 2006 (UTC)
The distinction is essential, which is why I am involved. You can, indeed, create a scale of any constructible numbers along a line. If you were to look at that, it would appear to be a ruler, in common speech. In order to use it as a new tool, though, you would have to cut it out from the paper on which it was drawn and translate it. If translation by a non-constructible distance is permitted -- and we generally understand that two objects may be offset, relative to one another, by any real-number distance -- then this opens the door to all three classic forbidden constructions.
If cut-and-translate is permitted, then it's not even necessary to produce a scale. See Tomahawk.
The issue of the marked straightedge is of a class distinct from that of the non-collapsing compass. Euclid I.2 and .3 justify the non-collapsing compass and show that any construction possible with the non-collapsing compass can be performed with the collapsing. The definition of the compass never changes throughout Euclid; but later constructions are simplified. It is understood that any of them could be expanded on demand. This is important; the actual process of duplicating a given line segment is hellishly complex compared to the ease of simply drawing a new circle with the old radius held in the compass. But the latter operation is only justified by the proof of the former.
It is exactly this kind of subtle distinction between permitted and forbidden operations that invites so many cranks to waste their lives on fallacy. And this is why I feel so strongly about the issue of term used to describe this subject. John Reid 23:32, 30 March 2006 (UTC)
The result of the debate was move, by a 13:7 margin. — Nightst a llion (?) Seen this already? 07:07, 3 April 2006 (UTC)
Oppose
Google hits
For all that's worth, the google returns 110,000 answers to Straightedge and Compass construction and 378,000 to ruler and compass construction. Not scientific by any measure, obviously, but worth keeping in mind. Oleg Alexandrov ( talk) 21:46, 29 March 2006 (UTC)
Google hits are misleading here. If you click search results to the very end, you will notice that both give nearly the same number of unique google hits. `' mikka (t) 18:12, 31 March 2006 (UTC)
Whatever the outcome be, IMO it would be bad idea to chase each and every reference in wikipedia articles and replace by one and the same. This would be imposing a POV bias unto the readers, since in real world both terms are used interchangeably. `' mikka (t) 18:16, 31 March 2006 (UTC)
I don't agree with the move of the page to ""Compass and straightedge". As i wrote above I think that "ruler and compass constructions" is the most common name for the subject of this article. In any case if we were to replace "ruler" by "straightedge", the article should be rewritten to reflect that change, and the title should be "Compass and straightedge constructions". I am going to move the article back until we can arrive at a consensus. Paul August ☎ 17:08, 28 March 2006 (UTC)
I must agree that the "compass and straightedge" has its points. For one thing, it permits the marked, or markable, object to be called a ruler within the article, and then we can discuss the neusis constructions which permit the ssolution of quartics.
Common usage should decide; Wikipedia is no place for "correctness" - but only if it is -er- decisive. I am not convinced that "ruler and compass" dominates the field.
I would oppose any effort to make ruler and compass a different article, however. If it redirects here, is there really a problem? Septentrionalis 23:55, 28 March 2006 (UTC)
There are three issues:
To answer these questions someone should really survey the literature for the opinions of "experts" (by the way, there are roughly zero references in the article). In lieu of doing that tedious work, I vote for what I see as correctness: name the article "compass and straightedge", mention all of the variations in the intro, and then redirect all of them here.
Apparently the larger issue is how to refer to these constructions in other articles; I submit that anyone reading the phrase "compass and straightedge construction" will automatically translate it into "ruler and compass construction" or whatever their favorite variant is, so accomodating popular usage is not really so crucial. Joshua Davis 14:23, 29 March 2006 (UTC)
I do not see consensus here, but I have added the normal move vote format above, this page should also be listed on WP:RM, I suppose, but those of us talking about it now should probably get their chance to vote first. Septentrionalis 16:46, 29 March 2006 (UTC)
I find David Kernow's preference for construction/s tempting. Of the two, construction is better; this is about the process of construction in general, more than a list of constructions. On the whole, I prefer the shorter name: it is also a direct link in phrases like can be drawn with compass and straightedge.
I am also somewhat concerned by Nightstallion's never having heard of straightedge. How general is this, do you suppose? (I don't think this should decide the issue by itself, but it is an issue.) Septentrionalis 15:13, 30 March 2006 (UTC)
As someone who learned about this (twice) in undergraduate and graduate studies within the past several years, I would note that it was introduced both times as "straightedge and compass" with both professors, at different institutions, commenting on the difference between a straightedge and a ruler specifically to eliminate confusion.
I know that I am jumping in late here, but I support the more accurate title, and I think that historical precedence is an absurd reason to continue the propagation of ambiguous terminology. Marc Harper 19:54, 3 April 2006 (UTC)
It's important that we sign our own comments, but I don't think it's appropriate for us to take credit for public references in this section. John Reid 04:28, 30 March 2006 (UTC)
You missed nothing, friend. I just put it there. Thought it made sense.
I agree with you that it's wise to annotate each ref with the term chosen by the ref's author. I don't agree that web refs are automatically uncitable. I do agree that a random smattering of web refs is substandard. I chose refs such as an article published in AMM and the highly-respected and popular Math Forum at Drexel. Certainly some nutball crank in a high school in Outer Okeefenokee who happens to have a page on cube doubling is uncitable. John Reid 05:02, 30 March 2006 (UTC)
I tabulated, retaining your division into three classes. We may argue about it, but at least we will do so neatly. John Reid 05:31, 30 March 2006 (UTC)
citation | preference | note | ||||||
---|---|---|---|---|---|---|---|---|
Books | ||||||||
I Stewart: Galois Theory | Ruler and Compass | Had an interesting note about how in ancient Greece, they did not have access to marked rulers, so for them a ruler was just a straight edge. | ||||||
Dummit and Foote, Abstract Algebra, 2nd edition | Straightedge and Compass | explicitly mentions the distinction between a marked ruler and plain straightedge in explaining preference for "straightedge" | ||||||
Ellis: Rings and Fields | ruler and compass | with the text saying unmarked ruler | ||||||
Isaacs: Algebra: A Graduate Course | compass and straightedge. | |||||||
E. Artin: Galois Theory (2nd ed) | ruler and compass | |||||||
Michael Artin's Algebra (1st ed) | constructions with ruler and compass | however the text in the section sez "Note that our ruler may be used only to draw straight lines through constructed points. We are not allowed to use it for measurement. Sometimes it is referred to as a "straight-edge" to make this point clear." | ||||||
Hardy and Wright, Introduction to the theory of numbers (5th ed.) | 'Euclidean' constructions, by ruler and compass | discussing Gauss' construction of the 17-sided regular polygon. | ||||||
Morris Kline, Mathematical thought: from ancient to modern times | straightedge and compass only | repeated reference | ||||||
I.N. Herstein, Topics in algebra (2nd ed.), | construction by straightedge and compass | section with this title; repeated use of straightedge and no mention of "ruler". | ||||||
M.Anderson,T.Feil: A First Course in Abstract Algebra:Rings, Groups and Fields, second edition, Chapman & Hall/CRC, Boca Raton, 2005, 673 pp., USD 89,95, ISBN 1-58488-515-7 | straightedge and compass | cited in Newsletter of the European Mathematical Society 59 Mar 2006 review PDF | ||||||
Diggins, Julia, String, Straightedge and Shadow, Viking Press, 1965 | straightedge | cited by
The Early Greeks Contribution to Geometry by Joseph A. Montagna of Yale-New Haven Teachers Institute. "This book was written for students of middle school age."
| ||||||
Journal articles | ||||||||
Compass and straightedge in the Poincare disk American Mathematical Monthly 108 Jan 2001, Chaim Goodman-Strauss PDF | Compass and straightedge | in title
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Websites | ||||||||
The Geometry Junkyard David Eppstein UC Irvine | Compass and Straightedge | in subtitle; notable in that this is a broad cross-section of related pages, presumably collected by author without a narrow outlook | ||||||
Geometry Construction Reference Paul Kunkel | Compass and straightedge | follow links to individual constructions to see each word in context | ||||||
Geometric Construction MathWorld | Geometric Construction | in title; text includes straightedge and compass and Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. | ||||||
Geometry Constructions with Compass and Straightedge Dr. Math, The Math Forum Drexel University | Compass and Straightedge | in title; note consistent use of this term across dozens of related Math Forum pages
|
The poll seems to have stalled, with about a 2 to 1 preference for Compass and straightedge. This is not a clear consensus but neither is it a ringing endorsement of Ruler and compass constructions. Let's see if we can get together behind something.
I'd like to ask 2 questions, very different. John Reid 23:00, 2 April 2006 (UTC)
1. If your support or opposition was marginal or indecisive, can you imagine any circumstance under which you might consider reversing your position or becoming fully neutral? What might that be?
2. No matter your position on the poll, please propose alternate titles for the article that satisfy your concerns.
At the end of the move discussion, Elroch wrote (in part): Hopefully...all redirects (of which a large number are needed) will be changed to being direct. By the way, I hope my slightly more formal section on the interpretation of geometric constructions as operations in the complex field is helpful. Further work needed includes removing some of the repetitions in other sections, and a description with a diagram (or alternatively a link) for the geometric construction for each arithmetic operation.
Great job to everyone who has been working on all the article. I will try to get some time to help out also. -- C S (Talk) 16:34, 4 April 2006 (UTC)
Currently the intro mentions that the proofs of impossibility of the famous classical constructions rely on Galois theory; even the Galois theory article makes this assertion. However, the proofs just rely on basic field theory with no need of Galois groups, etc. The current phrasing would even seem to take credit away from Pierre Wantzel, giving it to Galois. I think a change is needed. -- C S (Talk) 16:42, 4 April 2006 (UTC)
It might be worth mentioning that it _is possible to solve these problems to arbitrary accuracy (just not exactly). Simlply draw a cartesian grid of whatever scale is required (if you make it fine/big enough you can get any desired level of accuracy). Only mathematicians make a distiction between "as close as you like" and exact. -- Pog 15:21, 25 April 2006 (UTC)
I disagree - I think there should be some mention of this, to inform the casual reader, (without being overly technical) that compass and straight edge are at least as powerful as the rational numbers, and can be used to solve any problem to arbitrary precision. The important point is that they cannot represent all of the real numbers - so one can produce an infinitely good approximation, given infinite time, but can never produce an exact answer in the strictly mathematical sense (because they lack the expressive power in the same way as the rational numbers cannot express the exact solution).
Shouldn't this be moved to the "Constructible Number" article, since they have little to do with compass and straightedge directly?
Shouldn't this define what is meant by an angle of finite order or link to a definition? -- OinkOink 13:52, 18 July 2006 (UTC)
A further generalisation of this theorem (due to K. Venkatachala Iyengar) gives constructions using only a ruler given a point and five distinct points equidistant from it.
This is false - for example, given (0,0), (0,1), (0,-1), (1,0), (-1,0), and (1/√2,1/√2), the points that can be constructed with only a straightedge have coordinates in Q[√2]. Even supposing specific special points (x1,y1), ..., (x6,y6) (rather than arbitrary points), all points constructible with only a straightedge would have coordinates in Q[x1,y1,...,x6,y6], which cannot capture the closure of Q under square roots (which is what ruler and compass can construct).
I suspect that this false generalisation is due to a misinterpretation of another result - given those 6 points, one can construct arbitrarily many points of the circle given by those 5 equidistant points, as well as its center. If one actually had the circle and its center, that would be enough, but having arbitrarily many points on it isn't.
David.applegate 18:33, 9 August 2006 (UTC)
The brief plug to Mathematics of paper folding under impossible constructions is poorly worded. It needs to be brought up to the level normally present on Wikipedia. -- Whiteknox 01:39, 17 November 2006 (UTC)
Unsourced; and largely false. Archimedes' more "advanced analytical method" was the Method of exhaustion, which Newton knew about, and used; it's in Euclid. Newton published his results with Euclidean proofs because these proofs were, and were understood to be, rigorous; Newton's fluxions were not, and would become so for another two centuries. Septentrionalis PMAnderson 19:03, 20 February 2007 (UTC)
Do you think that the article is ready to become a featured article? Tomer T 18:44, 23 March 2007 (UTC)
These two section appear to have almost identical content. Should they be merged? -- Salix alba ( talk) 23:37, 10 May 2007 (UTC)
IMO, this article spends most of its time reviewing (a) what can't be done and (b) complicated proofs of this and complicated proofs of some thinghs that can be done. It appears to lack almost completey a review of what can be done and of how to do (some) of these constructions. I was tempted to add the following (next para) but thought I'd put it here for comment first.
"An example of a simple achievable construction is to divide a line into two sections of ratio 1 : N (integer). First construct perpendicular lines at each end of the target line, next construct, with the compass, one target line length 'down' on the 'left' and N target line lengths 'up' on the 'right'. Join the two resultant points for a 1:N intersection."
-- SGBailey ( talk) 23:10, 31 May 2008 (UTC)
"Infinite in length and only having one side." Isn't that physically impossible? 207.62.186.233 ( talk) 02:59, 3 June 2008 (UTC)
The fourth paragraph of the introductory section reads as follows:
"The most famous ruler-and-compass problems have been proven impossible in several cases by Pierre Wantzel, using the mathematical theory of fields. In spite of these impossibility proofs, some mathematical novices persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using ruler and compass alone.."
Regarding the end of this last sentence: If an example is offered, it should be given, not withheld. Doubling the cube is "possible" using what geometric constructions? Daqu ( talk) 23:01, 26 March 2009 (UTC)
The discussion on moving the article seems to have stalled a while ago with no decision. As a purely practical matter, it's usually much easier to make a link to the article when the article name is singular.-- RDBury ( talk) 05:40, 27 June 2009 (UTC)
Moved this here from the article:
This sounds nonsensical to me. A finite (but unbounded) number of operations can certainly produce an infinite number of possible points. All rational points on the plane can be constructed, even though there are infinitely many of them. The real impossibility proof is that the number of constructible points is countably infinite but there are uncountably infinitely many reals. If anyone wants to write it up, go for it.
Arvindn (
talk) 18:17, 1 September 2009 (UTC)
Is it mentioned anywhere how to construct 60 degree angles? It is one of the simpler constructions. SharkD Talk 04:03, 28 November 2009 (UTC)
thumbnail doesn't seem to animate, tho the gif itself does. -- Arkelweis ( talk) 07:54, 28 February 2011 (UTC)
I would like to call into question the value of this section. First, the paper referred to is now 14 years old ... not particularly recent in my mind, so I find the section title a bit misleading. The section has been in the article since Sept. 2002, with no change in content (but there have been minor changes in how the reference is given). The single sentence is pure hype and does not add anything to the article. On the basis of Wikipedia:Notability I would say that the section should be struck. In support of this I note that according to Math Reviews, the paper that this section references has never been cited by any other mathematical article. As a Wikipedia editor I should not be commenting on the quality of referenced sources, but as a working mathematician I feel the need to point out that there are professional standards for published mathematical works and the paper being talked about fails to meet those standards in many ways. The fact that the author of the paper is also a founding editor of the journal in which this paper appears may explain how it got published in the first place. Comments? Bill Cherowitzo ( talk) 03:33, 1 April 2012 (UTC)
Is there any analogous model where instead of limiting the compass and straightedge constructions to a single 2D plane, that we instead use them on a number of intersecting 2D planes in 3 dimensional space?
siNkarma86—Expert Sectioneer of Wikipedia
86 = 19+9+14 + karma = 19+9+14 +
talk 05:25, 19 December 2012 (UTC)
I've checked Google books and only found two instances of the name with hyphens in instead of spaces. Therefore this new hyphens name is not a common name and the article should be move back to its old title. Dmcq ( talk) 22:43, 23 May 2013 (UTC)
I did the checking by putting "compass-and-straightedge" with the quotes into Google Books. I did not do any special checking that hyphens were inappropriate and therefore an entry should be ignored. If some demonstration can be made that there is some criteria I should use which is better and changes the statistics enough to favour the hyphens from the 2% I found I'd like to see it. Dmcq ( talk) 14:52, 24 May 2013 (UTC)
Well I think that established 'Straightedge and compass construction' as a preferable title for Wikipedia, but should we even include the 'construction' bit? It seems a bit unnecessary to me and people more often just say something like 'using straightedge and compass'. Dmcq ( talk) 13:38, 27 May 2013 (UTC)
An alternate method for constructing polygons is by taking any two sides of a right-angle triangle as radii of a circle & its sectors,some of the results will be apeiroga,but most give good results. AptitudeDesign ( talk) 08:53, 21 April 2014 (UTC)
Unless I missed something, the article currently contains no history prior to Gauss, except a brief mention that a proposition of Euclid's implies that it doesn't matter whether a compass is collapsible, a brief statement that the requirements can be expressed in terms of Euclid's first three postulates, and the dubious statement
(I thought they kept trying rather than assuming impossibility--is that not right? The above assertion needs a citation.)
I think it would be worthwhile to have a history section that answers:
Loraof ( talk) 19:17, 11 January 2015 (UTC)
Could we make a section with much used constructions, so that readers have some directory linking to much used (but not bascic) constructions, I know the list should not be to long or maybe a seperate page for a longer list. but on this page a list refering to "midpoint of segment", "angle bisector" , "mirror point in line" and maybe 7 more is in order. WillemienH ( talk) 08:55, 8 June 2015 (UTC)
I have just removed an addition dealing with unit distance graphs. The quoted result says that for any algebraic number there is a graph (in the Euclidean plane) with a pair of vertices at this distance for any unit distance graph representation. If this was related to compass and straightedge constructions, it would be saying that all algebraic numbers are constructible - which is clearly false. If I am somehow mistaken, the section would have to indicate how it is related to the topic of this article before being replaced. Bill Cherowitzo ( talk) 04:01, 26 November 2015 (UTC)
I did like the link to [www.euclidea.xyz/game] Euclidea an online compass and straightedge construction game that was added 18 march 2016, but was removed soon after with as reason (revert - rm promotional link; this is not a web directory) I did like the link and would like it reinstated, because it is related , I like it (but that is personal and is not within the scope of Wikipedia:NOTDIRECTORY. Also I would like to add a link to [6] but that would have I guess the same result. WillemienH ( talk) 18:36, 18 March 2016 (UTC)
Euclid seems to have had a broader view of what a "construction" is than this. For example, proposition 3.1 constructs the center of a circle from the circle. That can't be done with the basic constructions because there are no intersections to make a point from and no points to make a line or circle from. However, Euclid does it by drawing a chord through the circle at random, relying on the fact that the construction works no matter which chord one draws. Perhaps the definition of construction has changed? — Preceding unsigned comment added by Mrperson59 ( talk • contribs) 23:04, 7 September 2018 (UTC)
No move was done after the discussion at #Bad move where it seems 'Straightedge and compass construction' is more common. Also the version without hyphens seems more common but seems to occur about a third of the time. Any objections to a move now? Prefer hyphens? Dmcq ( talk) 09:54, 8 September 2018 (UTC)
The History section doesn't explain the origins of the compass and straightedge restriction, nor does it distinguish between eras in which the restriction was considered absolute and eras where constructions using other tools were considered legitimate but less aesthetic, less fundamental or otherwise second class. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 13:35, 20 May 2021 (UTC)
== Regarding Doubling the cube: Is it still impossible? ==
This is the first time I have contributed anything to Wikipedia besides donations, so I'm not sure what exactly I'm doing here. And I am not a mathematician for sure, but I wanted to at least point out something in this article that may no longer reflect current knowledge. The latest Scientific American Magazine (October 2021 Volume 325, Issue 4
[7]) features an article on math entitled "Infinity Category Theory Offers a Bird's-Eye View of Mathematics"]
[8] by Emily Riehl which seems to indicate that it is now possible to construct "with an imaginary straightedge and compass, of a cube with a volume twice that of a different, given cube".
Just trying to make Wikipedia better in my own diminutive way. I'll check back to see which part of subterranean garbage this post has been banished to bc that's how I learn. Thanks. Glenn Wiens
GAWiens (
talk) 06:25, 20 October 2021 (UTC)
Article could use some cleanup, many external links have gone bad. I'd do it myself but I'm not a member so I don't know how to mark them and I don't want to start removing unsourced material if it can just be redirected to a different source. 2601:14F:8000:B3B0:0:0:0:5733 ( talk) 20:45, 1 June 2024 (UTC)
Straightedge and compass construction is a former featured article candidate. Please view the links under Article milestones below to see why the nomination failed. For older candidates, please check the archive. | ||||||||||
| ||||||||||
A fact from this article appeared on Wikipedia's Main Page in the " Did you know?" column on March 20, 2004. |
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||
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And I quote from the first subsection "compass and straightedge tools", first bullet point:
Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle.
I think these two sentences are contradictory. The first seems to imply a collapsing compass, as you must pre-define the radius and center point in order to draw the circle there. If a rigid compass were permissible then you need only a center point in order to draw a circle whose radius youve already defined with the compass.
At some time in the past I Heard the commentators for professional wrestling refer to the ring in which activity takes place as the "squared circle". This is an excellent example of pseudo-science in the service of pseudo-sport. Eclecticology
We need more on doubling the cube and angle trisection. The Anome
Mathematicians are notoriously incompetent historians. Gauss NEVER gave a proof of the necessity of the constructibility of the regular n-gon. WANTZEL proved this in 1837. If you read otherwise, it's because mathematicians are sloppy historians. Revolver
The article mixes the two words compass and compasses with obviously the same meaning. Both words are acceptable according to
Webster's, but for the sake of consistency I'm changing all to compass. Why not compasses? Because that way I don't have to move the article, and anyway "ruler and compasses construction" is less agreeable to the ear.
—
Herbee 18:43, 2004 Mar 13 (UTC)
MERGE: This needs to be merged with Constructible number and all the other articles on specific impossibility proofs. These overlap a lot, yet none of them should try to cover every topic. Revolver 13:08, 14 Nov 2004 (UTC)
The regular polygon with the largest number of sides that was ever actually drawn with a ruler and compass had 1,024 sides. It was drawn by graduate student Mr. Sam Bronstein at the University of Kentucky in 1963, a feat that took him 33 days and a sheet of paper 9 meters square.
Is it really as imprtant to be here? Tosha 03:37, 18 October 2005 (UTC)
technically, using a compass and a straightedge to draw a hexagon is impossible, since the circumference is 2Πr. not 2(3)r.
there is a slight distortion. the only true way to draw a real hexagon is to 1. bisect the circle at any point 1/2 way between the center[o] and an edge[e] to form point [a] 2. using point [a] as a center, draw a line [b] perpendicular to the angle formed between [e] and [o] 3. mark the two points where [b] intersects the circles radius 4. repeat 180 from [e] across [o]
above unsigned comment 17:39, 8 December 2005 Kargoneth
For the record, a regular hexagon is constructible. John Reid 23:36, 30 March 2006 (UTC)
A ruler is commonly understood to be marked; this type of construction specifically forbids markings. John Reid 19:18, 26 March 2006 (UTC)
A ruler is a marked straightedge; this is explicitly forbidden in compass and straightedge construction. For practical purposes the ruler is the more generally useful tool, certainly the more popular; thus it's not hard to see why the term is also more popular. But this is one of those times that the Google test gives a false indication.
To the quibble: Yes, strictly speaking, a "ruler" is a straightedge only, a tool for ruling (drawing straight lines). A tool for measuring is a "scale"; thus the commonplace object found in home and office is truly a ruler with a scale printed upon it -- a dual-purpose tool. It is, however, ambiguously called a ruler; and the less-common "straightedge" is clearly and unambiguously unmarked. John Reid 23:55, 28 March 2006 (UTC)
The distinction is essential, which is why I am involved. You can, indeed, create a scale of any constructible numbers along a line. If you were to look at that, it would appear to be a ruler, in common speech. In order to use it as a new tool, though, you would have to cut it out from the paper on which it was drawn and translate it. If translation by a non-constructible distance is permitted -- and we generally understand that two objects may be offset, relative to one another, by any real-number distance -- then this opens the door to all three classic forbidden constructions.
If cut-and-translate is permitted, then it's not even necessary to produce a scale. See Tomahawk.
The issue of the marked straightedge is of a class distinct from that of the non-collapsing compass. Euclid I.2 and .3 justify the non-collapsing compass and show that any construction possible with the non-collapsing compass can be performed with the collapsing. The definition of the compass never changes throughout Euclid; but later constructions are simplified. It is understood that any of them could be expanded on demand. This is important; the actual process of duplicating a given line segment is hellishly complex compared to the ease of simply drawing a new circle with the old radius held in the compass. But the latter operation is only justified by the proof of the former.
It is exactly this kind of subtle distinction between permitted and forbidden operations that invites so many cranks to waste their lives on fallacy. And this is why I feel so strongly about the issue of term used to describe this subject. John Reid 23:32, 30 March 2006 (UTC)
The result of the debate was move, by a 13:7 margin. — Nightst a llion (?) Seen this already? 07:07, 3 April 2006 (UTC)
Oppose
Google hits
For all that's worth, the google returns 110,000 answers to Straightedge and Compass construction and 378,000 to ruler and compass construction. Not scientific by any measure, obviously, but worth keeping in mind. Oleg Alexandrov ( talk) 21:46, 29 March 2006 (UTC)
Google hits are misleading here. If you click search results to the very end, you will notice that both give nearly the same number of unique google hits. `' mikka (t) 18:12, 31 March 2006 (UTC)
Whatever the outcome be, IMO it would be bad idea to chase each and every reference in wikipedia articles and replace by one and the same. This would be imposing a POV bias unto the readers, since in real world both terms are used interchangeably. `' mikka (t) 18:16, 31 March 2006 (UTC)
I don't agree with the move of the page to ""Compass and straightedge". As i wrote above I think that "ruler and compass constructions" is the most common name for the subject of this article. In any case if we were to replace "ruler" by "straightedge", the article should be rewritten to reflect that change, and the title should be "Compass and straightedge constructions". I am going to move the article back until we can arrive at a consensus. Paul August ☎ 17:08, 28 March 2006 (UTC)
I must agree that the "compass and straightedge" has its points. For one thing, it permits the marked, or markable, object to be called a ruler within the article, and then we can discuss the neusis constructions which permit the ssolution of quartics.
Common usage should decide; Wikipedia is no place for "correctness" - but only if it is -er- decisive. I am not convinced that "ruler and compass" dominates the field.
I would oppose any effort to make ruler and compass a different article, however. If it redirects here, is there really a problem? Septentrionalis 23:55, 28 March 2006 (UTC)
There are three issues:
To answer these questions someone should really survey the literature for the opinions of "experts" (by the way, there are roughly zero references in the article). In lieu of doing that tedious work, I vote for what I see as correctness: name the article "compass and straightedge", mention all of the variations in the intro, and then redirect all of them here.
Apparently the larger issue is how to refer to these constructions in other articles; I submit that anyone reading the phrase "compass and straightedge construction" will automatically translate it into "ruler and compass construction" or whatever their favorite variant is, so accomodating popular usage is not really so crucial. Joshua Davis 14:23, 29 March 2006 (UTC)
I do not see consensus here, but I have added the normal move vote format above, this page should also be listed on WP:RM, I suppose, but those of us talking about it now should probably get their chance to vote first. Septentrionalis 16:46, 29 March 2006 (UTC)
I find David Kernow's preference for construction/s tempting. Of the two, construction is better; this is about the process of construction in general, more than a list of constructions. On the whole, I prefer the shorter name: it is also a direct link in phrases like can be drawn with compass and straightedge.
I am also somewhat concerned by Nightstallion's never having heard of straightedge. How general is this, do you suppose? (I don't think this should decide the issue by itself, but it is an issue.) Septentrionalis 15:13, 30 March 2006 (UTC)
As someone who learned about this (twice) in undergraduate and graduate studies within the past several years, I would note that it was introduced both times as "straightedge and compass" with both professors, at different institutions, commenting on the difference between a straightedge and a ruler specifically to eliminate confusion.
I know that I am jumping in late here, but I support the more accurate title, and I think that historical precedence is an absurd reason to continue the propagation of ambiguous terminology. Marc Harper 19:54, 3 April 2006 (UTC)
It's important that we sign our own comments, but I don't think it's appropriate for us to take credit for public references in this section. John Reid 04:28, 30 March 2006 (UTC)
You missed nothing, friend. I just put it there. Thought it made sense.
I agree with you that it's wise to annotate each ref with the term chosen by the ref's author. I don't agree that web refs are automatically uncitable. I do agree that a random smattering of web refs is substandard. I chose refs such as an article published in AMM and the highly-respected and popular Math Forum at Drexel. Certainly some nutball crank in a high school in Outer Okeefenokee who happens to have a page on cube doubling is uncitable. John Reid 05:02, 30 March 2006 (UTC)
I tabulated, retaining your division into three classes. We may argue about it, but at least we will do so neatly. John Reid 05:31, 30 March 2006 (UTC)
citation | preference | note | ||||||
---|---|---|---|---|---|---|---|---|
Books | ||||||||
I Stewart: Galois Theory | Ruler and Compass | Had an interesting note about how in ancient Greece, they did not have access to marked rulers, so for them a ruler was just a straight edge. | ||||||
Dummit and Foote, Abstract Algebra, 2nd edition | Straightedge and Compass | explicitly mentions the distinction between a marked ruler and plain straightedge in explaining preference for "straightedge" | ||||||
Ellis: Rings and Fields | ruler and compass | with the text saying unmarked ruler | ||||||
Isaacs: Algebra: A Graduate Course | compass and straightedge. | |||||||
E. Artin: Galois Theory (2nd ed) | ruler and compass | |||||||
Michael Artin's Algebra (1st ed) | constructions with ruler and compass | however the text in the section sez "Note that our ruler may be used only to draw straight lines through constructed points. We are not allowed to use it for measurement. Sometimes it is referred to as a "straight-edge" to make this point clear." | ||||||
Hardy and Wright, Introduction to the theory of numbers (5th ed.) | 'Euclidean' constructions, by ruler and compass | discussing Gauss' construction of the 17-sided regular polygon. | ||||||
Morris Kline, Mathematical thought: from ancient to modern times | straightedge and compass only | repeated reference | ||||||
I.N. Herstein, Topics in algebra (2nd ed.), | construction by straightedge and compass | section with this title; repeated use of straightedge and no mention of "ruler". | ||||||
M.Anderson,T.Feil: A First Course in Abstract Algebra:Rings, Groups and Fields, second edition, Chapman & Hall/CRC, Boca Raton, 2005, 673 pp., USD 89,95, ISBN 1-58488-515-7 | straightedge and compass | cited in Newsletter of the European Mathematical Society 59 Mar 2006 review PDF | ||||||
Diggins, Julia, String, Straightedge and Shadow, Viking Press, 1965 | straightedge | cited by
The Early Greeks Contribution to Geometry by Joseph A. Montagna of Yale-New Haven Teachers Institute. "This book was written for students of middle school age."
| ||||||
Journal articles | ||||||||
Compass and straightedge in the Poincare disk American Mathematical Monthly 108 Jan 2001, Chaim Goodman-Strauss PDF | Compass and straightedge | in title
| ||||||
Websites | ||||||||
The Geometry Junkyard David Eppstein UC Irvine | Compass and Straightedge | in subtitle; notable in that this is a broad cross-section of related pages, presumably collected by author without a narrow outlook | ||||||
Geometry Construction Reference Paul Kunkel | Compass and straightedge | follow links to individual constructions to see each word in context | ||||||
Geometric Construction MathWorld | Geometric Construction | in title; text includes straightedge and compass and Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. | ||||||
Geometry Constructions with Compass and Straightedge Dr. Math, The Math Forum Drexel University | Compass and Straightedge | in title; note consistent use of this term across dozens of related Math Forum pages
|
The poll seems to have stalled, with about a 2 to 1 preference for Compass and straightedge. This is not a clear consensus but neither is it a ringing endorsement of Ruler and compass constructions. Let's see if we can get together behind something.
I'd like to ask 2 questions, very different. John Reid 23:00, 2 April 2006 (UTC)
1. If your support or opposition was marginal or indecisive, can you imagine any circumstance under which you might consider reversing your position or becoming fully neutral? What might that be?
2. No matter your position on the poll, please propose alternate titles for the article that satisfy your concerns.
At the end of the move discussion, Elroch wrote (in part): Hopefully...all redirects (of which a large number are needed) will be changed to being direct. By the way, I hope my slightly more formal section on the interpretation of geometric constructions as operations in the complex field is helpful. Further work needed includes removing some of the repetitions in other sections, and a description with a diagram (or alternatively a link) for the geometric construction for each arithmetic operation.
Great job to everyone who has been working on all the article. I will try to get some time to help out also. -- C S (Talk) 16:34, 4 April 2006 (UTC)
Currently the intro mentions that the proofs of impossibility of the famous classical constructions rely on Galois theory; even the Galois theory article makes this assertion. However, the proofs just rely on basic field theory with no need of Galois groups, etc. The current phrasing would even seem to take credit away from Pierre Wantzel, giving it to Galois. I think a change is needed. -- C S (Talk) 16:42, 4 April 2006 (UTC)
It might be worth mentioning that it _is possible to solve these problems to arbitrary accuracy (just not exactly). Simlply draw a cartesian grid of whatever scale is required (if you make it fine/big enough you can get any desired level of accuracy). Only mathematicians make a distiction between "as close as you like" and exact. -- Pog 15:21, 25 April 2006 (UTC)
I disagree - I think there should be some mention of this, to inform the casual reader, (without being overly technical) that compass and straight edge are at least as powerful as the rational numbers, and can be used to solve any problem to arbitrary precision. The important point is that they cannot represent all of the real numbers - so one can produce an infinitely good approximation, given infinite time, but can never produce an exact answer in the strictly mathematical sense (because they lack the expressive power in the same way as the rational numbers cannot express the exact solution).
Shouldn't this be moved to the "Constructible Number" article, since they have little to do with compass and straightedge directly?
Shouldn't this define what is meant by an angle of finite order or link to a definition? -- OinkOink 13:52, 18 July 2006 (UTC)
A further generalisation of this theorem (due to K. Venkatachala Iyengar) gives constructions using only a ruler given a point and five distinct points equidistant from it.
This is false - for example, given (0,0), (0,1), (0,-1), (1,0), (-1,0), and (1/√2,1/√2), the points that can be constructed with only a straightedge have coordinates in Q[√2]. Even supposing specific special points (x1,y1), ..., (x6,y6) (rather than arbitrary points), all points constructible with only a straightedge would have coordinates in Q[x1,y1,...,x6,y6], which cannot capture the closure of Q under square roots (which is what ruler and compass can construct).
I suspect that this false generalisation is due to a misinterpretation of another result - given those 6 points, one can construct arbitrarily many points of the circle given by those 5 equidistant points, as well as its center. If one actually had the circle and its center, that would be enough, but having arbitrarily many points on it isn't.
David.applegate 18:33, 9 August 2006 (UTC)
The brief plug to Mathematics of paper folding under impossible constructions is poorly worded. It needs to be brought up to the level normally present on Wikipedia. -- Whiteknox 01:39, 17 November 2006 (UTC)
Unsourced; and largely false. Archimedes' more "advanced analytical method" was the Method of exhaustion, which Newton knew about, and used; it's in Euclid. Newton published his results with Euclidean proofs because these proofs were, and were understood to be, rigorous; Newton's fluxions were not, and would become so for another two centuries. Septentrionalis PMAnderson 19:03, 20 February 2007 (UTC)
Do you think that the article is ready to become a featured article? Tomer T 18:44, 23 March 2007 (UTC)
These two section appear to have almost identical content. Should they be merged? -- Salix alba ( talk) 23:37, 10 May 2007 (UTC)
IMO, this article spends most of its time reviewing (a) what can't be done and (b) complicated proofs of this and complicated proofs of some thinghs that can be done. It appears to lack almost completey a review of what can be done and of how to do (some) of these constructions. I was tempted to add the following (next para) but thought I'd put it here for comment first.
"An example of a simple achievable construction is to divide a line into two sections of ratio 1 : N (integer). First construct perpendicular lines at each end of the target line, next construct, with the compass, one target line length 'down' on the 'left' and N target line lengths 'up' on the 'right'. Join the two resultant points for a 1:N intersection."
-- SGBailey ( talk) 23:10, 31 May 2008 (UTC)
"Infinite in length and only having one side." Isn't that physically impossible? 207.62.186.233 ( talk) 02:59, 3 June 2008 (UTC)
The fourth paragraph of the introductory section reads as follows:
"The most famous ruler-and-compass problems have been proven impossible in several cases by Pierre Wantzel, using the mathematical theory of fields. In spite of these impossibility proofs, some mathematical novices persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometric constructions, but not possible using ruler and compass alone.."
Regarding the end of this last sentence: If an example is offered, it should be given, not withheld. Doubling the cube is "possible" using what geometric constructions? Daqu ( talk) 23:01, 26 March 2009 (UTC)
The discussion on moving the article seems to have stalled a while ago with no decision. As a purely practical matter, it's usually much easier to make a link to the article when the article name is singular.-- RDBury ( talk) 05:40, 27 June 2009 (UTC)
Moved this here from the article:
This sounds nonsensical to me. A finite (but unbounded) number of operations can certainly produce an infinite number of possible points. All rational points on the plane can be constructed, even though there are infinitely many of them. The real impossibility proof is that the number of constructible points is countably infinite but there are uncountably infinitely many reals. If anyone wants to write it up, go for it.
Arvindn (
talk) 18:17, 1 September 2009 (UTC)
Is it mentioned anywhere how to construct 60 degree angles? It is one of the simpler constructions. SharkD Talk 04:03, 28 November 2009 (UTC)
thumbnail doesn't seem to animate, tho the gif itself does. -- Arkelweis ( talk) 07:54, 28 February 2011 (UTC)
I would like to call into question the value of this section. First, the paper referred to is now 14 years old ... not particularly recent in my mind, so I find the section title a bit misleading. The section has been in the article since Sept. 2002, with no change in content (but there have been minor changes in how the reference is given). The single sentence is pure hype and does not add anything to the article. On the basis of Wikipedia:Notability I would say that the section should be struck. In support of this I note that according to Math Reviews, the paper that this section references has never been cited by any other mathematical article. As a Wikipedia editor I should not be commenting on the quality of referenced sources, but as a working mathematician I feel the need to point out that there are professional standards for published mathematical works and the paper being talked about fails to meet those standards in many ways. The fact that the author of the paper is also a founding editor of the journal in which this paper appears may explain how it got published in the first place. Comments? Bill Cherowitzo ( talk) 03:33, 1 April 2012 (UTC)
Is there any analogous model where instead of limiting the compass and straightedge constructions to a single 2D plane, that we instead use them on a number of intersecting 2D planes in 3 dimensional space?
siNkarma86—Expert Sectioneer of Wikipedia
86 = 19+9+14 + karma = 19+9+14 +
talk 05:25, 19 December 2012 (UTC)
I've checked Google books and only found two instances of the name with hyphens in instead of spaces. Therefore this new hyphens name is not a common name and the article should be move back to its old title. Dmcq ( talk) 22:43, 23 May 2013 (UTC)
I did the checking by putting "compass-and-straightedge" with the quotes into Google Books. I did not do any special checking that hyphens were inappropriate and therefore an entry should be ignored. If some demonstration can be made that there is some criteria I should use which is better and changes the statistics enough to favour the hyphens from the 2% I found I'd like to see it. Dmcq ( talk) 14:52, 24 May 2013 (UTC)
Well I think that established 'Straightedge and compass construction' as a preferable title for Wikipedia, but should we even include the 'construction' bit? It seems a bit unnecessary to me and people more often just say something like 'using straightedge and compass'. Dmcq ( talk) 13:38, 27 May 2013 (UTC)
An alternate method for constructing polygons is by taking any two sides of a right-angle triangle as radii of a circle & its sectors,some of the results will be apeiroga,but most give good results. AptitudeDesign ( talk) 08:53, 21 April 2014 (UTC)
Unless I missed something, the article currently contains no history prior to Gauss, except a brief mention that a proposition of Euclid's implies that it doesn't matter whether a compass is collapsible, a brief statement that the requirements can be expressed in terms of Euclid's first three postulates, and the dubious statement
(I thought they kept trying rather than assuming impossibility--is that not right? The above assertion needs a citation.)
I think it would be worthwhile to have a history section that answers:
Loraof ( talk) 19:17, 11 January 2015 (UTC)
Could we make a section with much used constructions, so that readers have some directory linking to much used (but not bascic) constructions, I know the list should not be to long or maybe a seperate page for a longer list. but on this page a list refering to "midpoint of segment", "angle bisector" , "mirror point in line" and maybe 7 more is in order. WillemienH ( talk) 08:55, 8 June 2015 (UTC)
I have just removed an addition dealing with unit distance graphs. The quoted result says that for any algebraic number there is a graph (in the Euclidean plane) with a pair of vertices at this distance for any unit distance graph representation. If this was related to compass and straightedge constructions, it would be saying that all algebraic numbers are constructible - which is clearly false. If I am somehow mistaken, the section would have to indicate how it is related to the topic of this article before being replaced. Bill Cherowitzo ( talk) 04:01, 26 November 2015 (UTC)
I did like the link to [www.euclidea.xyz/game] Euclidea an online compass and straightedge construction game that was added 18 march 2016, but was removed soon after with as reason (revert - rm promotional link; this is not a web directory) I did like the link and would like it reinstated, because it is related , I like it (but that is personal and is not within the scope of Wikipedia:NOTDIRECTORY. Also I would like to add a link to [6] but that would have I guess the same result. WillemienH ( talk) 18:36, 18 March 2016 (UTC)
Euclid seems to have had a broader view of what a "construction" is than this. For example, proposition 3.1 constructs the center of a circle from the circle. That can't be done with the basic constructions because there are no intersections to make a point from and no points to make a line or circle from. However, Euclid does it by drawing a chord through the circle at random, relying on the fact that the construction works no matter which chord one draws. Perhaps the definition of construction has changed? — Preceding unsigned comment added by Mrperson59 ( talk • contribs) 23:04, 7 September 2018 (UTC)
No move was done after the discussion at #Bad move where it seems 'Straightedge and compass construction' is more common. Also the version without hyphens seems more common but seems to occur about a third of the time. Any objections to a move now? Prefer hyphens? Dmcq ( talk) 09:54, 8 September 2018 (UTC)
The History section doesn't explain the origins of the compass and straightedge restriction, nor does it distinguish between eras in which the restriction was considered absolute and eras where constructions using other tools were considered legitimate but less aesthetic, less fundamental or otherwise second class. Shmuel (Seymour J.) Metz Username:Chatul ( talk) 13:35, 20 May 2021 (UTC)
== Regarding Doubling the cube: Is it still impossible? ==
This is the first time I have contributed anything to Wikipedia besides donations, so I'm not sure what exactly I'm doing here. And I am not a mathematician for sure, but I wanted to at least point out something in this article that may no longer reflect current knowledge. The latest Scientific American Magazine (October 2021 Volume 325, Issue 4
[7]) features an article on math entitled "Infinity Category Theory Offers a Bird's-Eye View of Mathematics"]
[8] by Emily Riehl which seems to indicate that it is now possible to construct "with an imaginary straightedge and compass, of a cube with a volume twice that of a different, given cube".
Just trying to make Wikipedia better in my own diminutive way. I'll check back to see which part of subterranean garbage this post has been banished to bc that's how I learn. Thanks. Glenn Wiens
GAWiens (
talk) 06:25, 20 October 2021 (UTC)
Article could use some cleanup, many external links have gone bad. I'd do it myself but I'm not a member so I don't know how to mark them and I don't want to start removing unsourced material if it can just be redirected to a different source. 2601:14F:8000:B3B0:0:0:0:5733 ( talk) 20:45, 1 June 2024 (UTC)