# Talk:Compass-and-straightedge construction

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## older entries

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

Thanks, Pierre! 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

Or is it the historians who are sloppy, notoriously incompetent mathematicians?  ;-)
Herbee 18:43, 2004 Mar 13 (UTC)
No, the people doing the history are mathematicians specialising in history of math, not the other way around. (True for most sciences...you have to have some understanding of the material.) Revolver 13:08, 14 Nov 2004 (UTC)
It is definitely the fault of mathematicians here. Some VERY famous names aided and abetted this myth, without checking into it themselves. You'd think being math people, they would question things. Revolver 13:09, 14 Nov 2004 (UTC)

## 'Compass' vs 'compasses'

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

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)

I think there's room for a lot of reorganization here. Some pages should be merged; some new ones should be split off. I just ran into a problem with Regular polytope; somebody got carried away with the difference between real and ideal construction. That's a whole subject in itself; but it's not math. John Reid 22:15, 27 March 2006 (UTC)

## largest regular polygon

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)

No, but it might go to Polygon. John Reid 23:33, 30 March 2006 (UTC)

## Drawing A Regular Hexagon

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

I'm not a mathematician, but I don't see why the hexagon construction doesn't work. Is it because pi is irrational? Can someone clarify this?
(Also, this proposed construction seems rather vague. How do you draw a line perpendicular to an angle? What does "180 from [e] across [o]" mean? It is decipherable, but it takes some work.) Imaginaryoctopus 05:30, 10 December 2005 (UTC)
Addition: I did a little research, and it seems to me that the hexagon construction is fine. From the "Constructible polygon" page:
"A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes."
We let n=6 (representing a hexagon). We know that 2^1=2 and that 3 is a Fermat prime, so we can say 3*2=6.
Even if you don't buy this, the construction itself comes straight from Proposition 15 in Euclid's Elements, Book IV. Imaginaryoctopus 05:56, 10 December 2005 (UTC)

For the record, a regular hexagon is constructible. John Reid 23:36, 30 March 2006 (UTC)

## "Ruler" or "Straightedge"

A ruler is commonly understood to be marked; this type of construction specifically forbids markings. John Reid 19:18, 26 March 2006 (UTC)

As you say most people commonly think of a ruler as being a "marked" one (technically called a "measure"). However a ruler can also be unmarked. See ruler) for a fuller discussion of all this. Paul August 16:47, 28 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)

Being a quibbler by nature, I cannot resist pointing out that the "unambiguous umarkedness" of the straightedge may not be straightforward: [1][2][3]. LambiamTalk 05:29, 9 April 2006 (UTC)
The verb "rule" is sometimes synonymous with "measure" or "judge". Therefore, I suggest that all wording in this article be changed to "straightedge" instead of "ruler". SharkD (talk) 07:39, 18 December 2007 (UTC)

## Does it actually matter if a ruler is marked or not?

Does it actually matter if a ruler is marked or not. It is a trivial matter to produce a marked straight edge from an un marked straight edge and a compass. Step 1: draw a small circle, call the radius of the circle your unit. Next draw a straight line with the ruler, pick some point on the line to be zero, using the compass set to the unit radius mark off one unit from zero. Move the point of the compass to the first unit, mark of the second unit, .... The same procedure can be repeated for any straight edge in the diagram, effectivly giving the same construction possible as with a marked streight edge and compass. --Salix alba (talk) 00:09, 29 March 2006 (UTC)
See Doubling the cube for a "ruler (straightedge with two marks)" and compass construction which cannot be simulated by a "ruler and compass" construction. — Arthur Rubin | (talk) 00:46, 29 March 2006 (UTC)
One can indeed create a Cartesian grid (as described in the article) with equally spaced out points along the axes. However, although it may look like you've effectively constructed a replacement for a marked straightedge, the point is that you are still only allowed certain operations with your straightedge and compass. So you are still restricted as to how you can use elements of this "grid" and you will not be able to double the cube, etc. Now if you were to actually violate the rules for a Euclidean construction somehow, you can double the cube. For example, see the particular example Arthur gave. In that example, one creates a point of intersection between two lines by measuring along a marked straightedge, which is disallowed under the normal rules. --C S (Talk) 09:10, 30 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 following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the debate was move, by a 13:7 margin. —Nightstallion (?) Seen this already? 07:07, 3 April 2006 (UTC)

## Move of page to "Compass and straightedge"

### Opinions

Add *Support or *Oppose followed by an optional one-sentence explanation, then sign your opinion with ~~~~
• Weak Support See discussion. Septentrionalis 16:46, 29 March 2006 (UTC)
• Support For the reasons that Joshua Davis gives. Ryan Reich 17:45, 29 March 2006 (UTC)
• Weak Support Since "straigtedge and compass" is more precise but should not confuse anyone used to "ruler and compass", etc. Joshua Davis 21:29, 29 March 2006 (UTC)
• Support I think the article itself says it best, "The ruler to be used has no markings on it and only one edge, and is known as a straightedge." In my humble experience, I have always heard such constuctions referred to as "compass and straight-edge constructions." IMHO, I think straightedge is clearer and less likely to cause confusion.Lunch 22:30, 29 March 2006 (UTC)
• Support rename to Compass and straightedge constructions (or singular "construction" if preferred). David Kernow 23:37, 29 March 2006 (UTC)
• Support Compass and straightedge as the traditional wording. Thumbelina 23:48, 29 March 2006 (UTC)
• Support: "ruler" can mislead, "straightedge" is a better term here. Jonathunder 01:23, 30 March 2006 (UTC)
• Support for reasons mentioned above. Additionally, one should not use Google hits, but rather limit the references to authoritative sources (cf list of books below). My perusal of that list so far shows that neither term is overwhelmingly preferred by the authors of advanced mathematics books. In fact, I very much suspect that "ruler and compass" is something of an artifact and not preferred by modern authors. --C S (Talk) 04:07, 30 March 2006 (UTC)
• Support Compass and straightedge. John Reid 23:07, 30 March 2006 (UTC)
• Abstain I can live with either title and there are good arguments on both sides. Whatever you do, it's crucially important that the reader be made aware of the fact that, even though rulers sometimes have marks, the technical term "ruler and compass construction" always refers to unmarked rulers. AxelBoldt 16:23, 31 March 2006 (UTC)
• Support move to straightedge, after reading the table below. -lethe talk + 15:34, 31 March 2006 (UTC)
• Support This is the more colloquilal name, and you don't need marks on the stick, i.e. 'a ruler' per se. Kevin Baastalk 17:22, 31 March 2006 (UTC)
• Support I believe using a marked ruler allows essentially the same constructions (unless the markings are in unconstructible ratios), but the suggested term more clearly describes the intended meaning. A google search shows "straight edge and compass" OR "straightedge and compass" OR "compass and straightedge" OR "compass and straight edge" beats "Ruler and compass" OR "compass and ruler" by 100,000 to 90,000 so a few redirect pages are going to be necessary. Fortunately, the term "Compasses" seems to be hardly used in this context. Elroch 17:48, 31 March 2006 (UTC)
I'd hate you to think I was criticizing your vote; I'm not. But you do suggest that a ruler marked in constructible ratios could not be employed to perform impossible constructions; it's my understanding that it can indeed. Any marked ruler -- even one with but two arbitrary marks -- is sufficient. John Reid 22:56, 31 March 2006 (UTC)
I think I must not have expressed myself clearly enough. An infinite ruler with two marks on it only provides a single unit of length which can be used anywhere in a construction. As we are only interested in ratios, this cannot allow any new constructions (any construction with compass and straightedge can start with the creation of an arbitrary distance using the compass). Likewise, no new constructions are possible using a ruler with any number of marks on it as long as the ratio of any two distances between marks is constructible using compass and straightedge (i.e. any ratio must be a solution to a quadratic equation). A ruler would only permit new constructions if a ratio of distances between marks is not constructible (for example, a ruler with three marks at 0, 1 and $\pi\;$). Would you agree with this view of the situation? Elroch 15:50, 1 April 2006 (UTC)
It's not just a matter of being able to construct the length between the marked segments; it's the fact that you can construct it in an unconstructable position. Have a look at Archimedes' trisection of the angle. The key step is to produce a line segment which a) passes through a given point on a circle, and b) has a length of 1 between the other intersection with the circle and the intersection with an extended diameter. You can easily do this by sliding around a marked ruler whose marking has length 1, but it's impossible otherwise (since it trisects the angle, for one). Another way to look at it is: no particular distance is unconstructable, because the plane doesn't have a preferred coordinate system. Only once you draw your first circle do you establish a metric. So if you have any marked ruler you can trivially make the marking a constructible length by making it the unit distance in your coordinates. Ryan Reich 16:30, 1 April 2006 (UTC)
Thank you for pointing this out. I was assuming neusis was not a permitted construction (it is far less direct than normal compass and straightedge constructions), but including it leads to an important superset of the compass and straightedge constructions. In light of this, I change my vote back to "Support". Elroch 17:28, 1 April 2006 (UTC)
Of course, neusis is not a permitted construction, but I understand what you mean. Anyway, glad I could help. Ryan Reich 18:59, 1 April 2006 (UTC)
• Very weak support, almost neutral - Straight edge is more accurate and the googling is inconclusive abakharev 20:18, 31 March 2006 (UTC)

Oppose

• Stick to the most widespread definition. Oleg Alexandrov (talk) 17:53, 29 March 2006 (UTC)
• . I Stewart: Galois Theory, referes to it as Ruler and Compass. Good enough for me. --Salix alba (talk) 19:09, 29 March 2006 (UTC)
Dummit and Foote, Abstract Algebra, 2nd edition, refers to it as Straightedge and Compass. Plus the book is really thick. Ryan Reich 20:38, 29 March 2006 (UTC)
• I'd never heard of "straightedge and compass" or "compass and straightedge" before this discussion. (And I was designated crank-reader at Caltech for a short period of time, so I saw a lot of bad "ruler-and-compass" angle trisections. This doesn't necessarily indicate the correct term, but it certainly indicates the term popular among crackpots.) — Arthur Rubin | (talk) 00:24, 30 March 2006 (UTC)
• Use the most common term AdamSmithee 07:07, 30 March 2006 (UTC)
• Never heard of straightedge. —Nightstallion (?) Seen this already? 11:38, 30 March 2006 (UTC)
• I was going to jump in and vote oppose when I first saw the vote started, but I thought I'd wait and see if any arguments swayed me, first. Having been in the 'neutral' camp for a day or two, I've now plumped for oppose. I've never heard of 'straightedge'; the "ruler and compass" words seemed familiar as soon as I saw them, and I didn't make an assumption that the ruler was used for length measurement; the resources identified below are by no means unambiguous. Noisy | Talk 10:06, 31 March 2006 (UTC)
• The negative statements - the things that can't be constructed with compass and straightedge - are simply not true in general for ruler and straightedge. Therefore the current name could be seen as misleading. On the other hand, it's naming policy to use the most widely familiar name, even where that name contains factual inaccuracies. Such inaccuracies should be clarified in the article text, ideally in the intro, not corrected in the title. If the "compass and straightedge" name were equally widespread, or even half as widespread, I might agree to prefer it. Deco 01:35, 2 April 2006 (UTC)

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)

How many does it return for compass and straightedge constrution? Kevin Baastalk 17:22, 31 March 2006 (UTC)
Woa, never mind, i did it and it's only 5 digits. weird. Kevin Baastalk 17:25, 31 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)

### Consequnces

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 agree with the recommendation, although I don't see it as a PoV issue. If we leave this alone, eventually the inward links should divide in proportion to all editor's feelings. This is the wiki process at work. Septentrionalis 20:48, 31 March 2006 (UTC)

### Discussion

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 think "Straightedge-and-compass constructions" would be better. "Ruler" is technically incorrect, as one can construct cube roots (not normally considered a ruler-and-compass construction) using a straightedge with two marks and a compass.Arthur Rubin | (talk) 18:37, 28 March 2006 (UTC)
Never mind. Already discussed at ruler. I recommend moving it back. — Arthur Rubin | (talk) 18:40, 28 March 2006 (UTC)
Agree with moving it back. "Ruler and compass" is by far the most common name. Oleg Alexandrov (talk) 18:53, 28 March 2006 (UTC)
What evidence is there that it's the most common? I myself never hear it, and never use it, and have been indoctrinated against its use. Ryan Reich 02:29, 29 March 2006 (UTC)
It took me over 7 hours to correct every link in every related article, fix every category tag, and rewrite several snippets of text. I'm really sorry I haven't gotten around to the rewrite of Compass and straightedge, but I intend to do so. Please note that the issue is not merely this article title; it is also the linking text and other references to the subject scattered throughout the project. I have corrected them all, with a few notable exceptions, each of which I'm prepared to defend.
I don't care a fig for what is the most common usage; only what is more correct. We're not in the business of perpetuating error. Fine to have redirects from all possible variations, including those with "ruler" in the title. Not okay to title the article page itself that way; that's just plain wrong -- and I think everyone tangent to this discussion knows why.
I'd like to make it clear that no consensus existed in the project before I did my work over these last few days. The subject was referred to by no less than 16 alternate, inconsistent titles. The actual title of the article is not the biggest deal but it's just plain silly to war over it. I think we all know what's right.
Now for the boring, point-by-point argument -- feel free to skip this if you already know better:
• "straightedge" over "ruler" -- see Ruler. Note that the "common usage" argument fails to endorse "ruler", because a ruler is marked in common usage. Adherence to strict interpretation fails as well: if "ruler" is strictly interpreted to be unmarked then this jerks the rug from under any claim to "common usage" for "ruler and compass". "Straightedge" is unambiguous, strict, and still commonly understood.
• "compass and straightedge" over "compass and straightedge construction" -- The preference is shorter, therefore presumably better barring some other argument. The former is quite unambiguous, in the sense that the pairing of these two tools is universally understood to apply equally to the set of tools, to the operations that may be performed with them, and to the theory which governs all. Thus there is no need to disambiguate by adding the needless word "construction".
Note that any argument supporting "construction" might apply to "method". No extra word is needed.
I have, in some cases, retained the word "construction" in the text of other articles, where appropriate.
• "compass and straightedge" over "straightedge and compass" -- All constructions can be performed with compass alone; thus it is the senior tool.
• "compass and straightedge" over "compass-and-straightedge" (and all hyphenated forms) -- In this case, hypenation is simply poor English style.
Until I started work, nobody had made any effort to untangle the rat's nest of links, descriptions, redirects, categories, and general blundering surrounding this subject. Now, with the exception of the rewrite of the main article (and perhaps one other task, that of an article for angle trisection), we have it all correct. Please, for heaven's sake, let's not screw it all up now. John Reid 21:08, 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)

Wikipedia is no place for "correctness" - Soon to be quoted in a news story about Wikipedia, to be sure. Joshua Davis 14:23, 29 March 2006 (UTC)
Oh, I'm sure; but it is policy: See WP:UE. Equally quotable is WP:V:The criterion for inclusion is verifiability, not truth. Septentrionalis 16:40, 29 March 2006 (UTC)

There are three issues:

• Which is the correct (most accurate) term?
• Which is the common (most popular) term?
• Which of these should name the article?

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 perfectly agree with Paul August. It is not Wikipedia's purpose to decide which is the most correct way, rather, the most used way is chosen.
Writing an encyclopedia is hard enough without us spending time on aruging which terminology is the right one. Just use the existing one. Oleg Alexandrov (talk) 16:19, 29 March 2006 (UTC)
I'd like to ask Oleg: What do you mean by "existing (terminology)"? Do you mean the inconsistent text (at least 16 alternate forms) that existed before I standardized on one form? Do you mean the consistent text that existed before Paul's counter-move? Or do you mean the inconsistent text that exists now, wherein all linking articles use one term and the article itself uses another? Please don't take this as a confrontation; I'd like to know your preference. John Reid 23:06, 29 March 2006 (UTC)
Existing terminology is "ruler and compass" as far as I am concerned. I am not taking it as a confrontation, but please don't stress so much that "all linking articles use one term and the article itself uses another". That is easy to fix one way or another. Let us be patient for a few days and see which way the consensus goes. Oleg Alexandrov (talk) 23:34, 29 March 2006 (UTC)
At what time in the history of this project has "ruler and compass" been the standard term for this concept? It has been mentioned in many other articles, with a wide range of terms. The last time you (Oleg) edited the text of Euclidean geometry ([4]), you chose to retain the text compass and an unmarked straightedge. Did you think, then, that you were right but you think, now, you were wrong then? Or it just doesn't matter what we call it?
Don't stress the inconsistency? This is the root of the whole problem. Inconsistency makes us look like fools and amateurs. It was not easy to fix -- not one way, not at all. I spent at least 7 hours fixing links, text, and redirects. Now that I've paid for my concerns in sweat, yes, it's easy for you to run a bot -- because I brought order from chaos. Don't stress the inconsistency? If you don't think it's important what we call it, why oppose the effort to gain consistency? John Reid 04:53, 30 March 2006 (UTC)
John, you seem to be conflating the issue of what we title the article with how we describe or refer to these constructions in other articles. As I tried to explain in our discussions on my talk, whatever we choose to title this article, we can describe to these constructions in different ways, in different contexts. Here is what I wrote:
Understand that I'm only talking about titles of articles here. Elsewhere, there is more latitude. For example one may choose to describe things rather than name them. So for example if I wanted to describe these constructions I might write "constructions using only a straightedge and compass" for clarity, while if I wanted to refer to them by name I might write "traditionally called ruler and compass constructions", perhaps noting what is meant by "ruler". Or if I was writing for mathematicians I would just write "ruler and compass constructions".
I don't think such variations are "inconsistent", nor do I think this make us look like "fools and amateurs".
Paul August 16:41, 30 March 2006 (UTC)
To me the issues are one and the same. I think that, as a project, we should agree on one correct term for every concept and replicate it throughout all articles, much as we settle on one correct style for, say, taxoboxes. There may be exceptions, but each requires a local justification. You may not see it that way and I don't think this is the place for metaphysical argument. But I don't generally like misleading pipelinks. John Reid 03:21, 31 March 2006 (UTC)
You seem to be saying "I worked my ass fixing all of those articles and links, therefore, how dare you all people argue with me?" Maybe that's not what you mean, but if that is what you mean, that's a lame way of making a point. And we are not arguing about inconsistency here, we are arguing about the most appropriate name. In short, yes, you did a lot of work, but you are also making too much fuss about it. Oleg Alexandrov (talk) 05:03, 30 March 2006 (UTC)
Thank you, Oleg; you are exactly correct, although I might have put it more delicately. I paid for my preference in the only coin of value here. The more that you denigrate my effort, the less respect I have for your point of view. Sorry.
I would never have been so bold as to overturn a clear consensus of any kind if that is what I had seen. I did not. I saw a raggedy, mixed-up scatter of references to the subject, some clearly ignorant, others thoughtless. Regardless of the merits of my preference, at least I made a choice and did the work required. Now I find I'm asked to spend an equal amount of time debating the issue. My work is beginning to seem a poor investement; perhaps I should have permitted the mess to fester as it was. But if you will not do the work yourself, I ask that you allow me to do so. John Reid 23:05, 30 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)

### Discussion of List of terminology references

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)

Ok, sorry, I missed this comment earlier. I think we should at least annotate the refs though so it's easy to see at a glance what the ref's position is. --C S (Talk) 04:37, 30 March 2006 (UTC)
John, I would prefer if we only added book references, preferably well-known textbooks and the like. I don't believe getting a cross section of Google hits will be nearly as helpful or representative of what may be considered "authoritative" usage. --C S (Talk) 04:43, 30 March 2006 (UTC)
Annotate if you like, but Mathworld and AMM are pretty authoritative. Septentrionalis 04:56, 30 March 2006 (UTC)
(edit conflict) Are we now to take this as endorsement by AMM? Or are you going to go through all AMM articles and see if one usage is preferred? A simple JSTOR search shows the Monthly lets authors use whatever term they like. --C S (Talk) 05:12, 30 March 2006 (UTC)
Not endorsement by AMM; but an AMM article is a credible source, just like somebody's algebra textbook. Septentrionalis 17:34, 31 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)

See my response right above; the Monthly definitely is not endorsing a particular terminology. What if someone else decides to put another Monthly ref indicating "ruler and compass" is preferred? Where will this end? Anyway, I'm going to create a "books" subsection then. I think there are several others besides me that would find a separate section useful. --C S (Talk) 05:12, 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)

### List of terminology references

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

### Seeking consensus

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.

A 2/3 majority is quite a strong endorsement for the suggested, less ambiguous title. On a related point, I have noticed that at present we have some double redirects to the article. Hopefully, if the name is changed (or if, as now seems unlikely, decided to be not changed) 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. Elroch 00:17, 3 April 2006 (UTC)
The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

## General improvement

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.

I went around to all linking articles and fixed all links to point directly to Compass and straightedge. I'll take a look and check to see that all are still correct and all rds direct and single.
Before the move discussion, I'd intended to finish up my work with a rewrite of this article. Now that the page is "hot", there have been plenty of edits so I'll hold off on a comprehensive rewrite; it may not be necessary with many hands on the page.
One particular plan of mine for this field is to deal with Trisecting the angle. Like Doubling the cube and Squaring the circle, this is one of the 3 classic impossible constructions. I think either one article Impossible constructions? should stand for all or each should have its own page. I intend to slim down description of these subjects in the main article, in proportion.
I can provide good graphics for any construction if someone will kindly indicate what's required. Let's get together in a corner and thrash it out. John Reid 05:00, 4 April 2006 (UTC)

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)

## Galois theory?

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)

Ok, I fixed it. Hopefully it is acceptable. The page on Galois theory though now needs to be fixed. --C S (Talk) 08:29, 7 April 2006 (UTC)

## Approximate constructions

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)

Firstly, this is an article about a topic in mathematics, not about practical drafting methods. That should really be clear from the article. Secondly, it is true but rather a minor point that approximate constructions for squaring the circle, etc, exist: approximate constructions exist for every single point in the plane, since the constructible points are dense in the plane (I have added a statement to this effect). Hence adding this note would be like adding a note in the article Rational number to say that Pi could be approximated as accurately as one wished with a rational number. This is true, but also true of all other real numbers and that wouldn't be the right place for it. Elroch 21:42, 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).

## "Extended constructions" section

Shouldn't this be moved to the "Constructible Number" article, since they have little to do with compass and straightedge directly?

Whether moved or not, the second sentence in the section on marked rulers is completely obtuse. Someone who understands it in detail should just yank this particular sentence and rewrite it wherever it belongs. "This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects both lines, and such that the distance between the points of intersection equals the given segment." Between which points of intersection -- all three? How does a distance "equal" a segment? Can we stop jamming "for example" appositives everywhere they are grammatically legal? (140.232.0.70 (talk) 21:47, 14 January 2011 (UTC))

## "Constructible Angles" section

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)

## Constructing with only ruler or only compass

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)

## Origami

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)

## Newton

Obsessed as he was with a belief that gravity was known to the ancients, Isaac Newton used a compass and straightedge construction to illustrate his theory of gravitation, rather than the fluxions he had invented (differential equations), which would have been more natural. He chose this more awkward method so as to demonstrate that the theory would have been accessible to the ancient pythagorans, unaware that Archimedes had infact known of greatly more advanced analytical geometric methods.

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)

In my long-lost undergraduate days, I came across a sourcebook with an interesting letter from Newton in which he defined what we now know as the derivative with *exactly* the "modern" notion of a limit (given verbally, but precisely). It's bizarre that this took centuries to be rediscovered.--OinkOink 21:22, 20 February 2007 (UTC)

## Featured Article

Do you think that the article is ready to become a featured article? Tomer T 18:44, 23 March 2007 (UTC)

## Sections on Constructible points and lengths and Compass and straightedge constructions as complex arithmetic

These two section appear to have almost identical content. Should they be merged? --Salix alba (talk) 23:37, 10 May 2007 (UTC)

I absolutely agree that they should be merged. Compass and straightedge constructions as complex arithmetic seems a touch more understandable to me, BTW. Root4(one) 16:13, 15 May 2007 (UTC)

## General Tone

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)

## About the definition of "straightedge"

"Infinite in length and only having one side." Isn't that physically impossible? 207.62.186.233 (talk) 02:59, 3 June 2008 (UTC)

## Information-free example

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)

Do you see how the text "doubling the cube" in that sentence appears in blue, perhaps with an underline? That means that you can click on it, and read in detail about the problem of doubling the cube. In that article there is a section titled "Solutions" that answers your question. —Dominus (talk) 06:36, 28 March 2009 (UTC)

## Reviving name debate

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)

## Incorrect proof

Moved this here from the article:

Suppose an algorithm that gives you a point as a result. The point will always be given by intersection of two lines (or line and circle or two circles. See previous elemental operations), but there are infinite points in a classical euclidean space.
Regardless of the algorithm we use, we can only perform a finite number of steps. Therefore, no matter what algorithm we use to determine them, there will always be points that we cannot mark by crossing of two lines (same holds if we mark the point as intersection of a line and a circle, or of two circles)

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)

## 60 degree angles

Is it mentioned anywhere how to construct 60 degree angles? It is one of the simpler constructions. 04:03, 28 November 2009 (UTC)

## Trisecting a segment with ruler and compass.

thumbnail doesn't seem to animate, tho the gif itself does. --Arkelweis (talk) 07:54, 28 February 2011 (UTC)

## Recent (?) Results

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)

## Three dimensions?

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)

Replacing circles with spheres so you look at points where lines or spheres intersect is the obvious extension. However you still don't get anything beyond doing square roots. Dmcq (talk) 10:51, 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)

I just did a check using "straightedge-and-compass construction" on Google Books and that did bring up the hyphens somewhat. There were six without hyphens and four with on the first page, the first five did not have a hyphen. It was 9 hyphens and 20 without and one I couldn't tell on the first three pages. Dmcq (talk) 16:14, 24 May 2013 (UTC)
Just noticed I used "straightedge and compass construction" instead of "compass and straightedge construction". In fact straightedge and compass construction came up 1560 times on Google books but compass and straightedge construction only came up 900 times.
I think on this evidence the article should be titled "Straightedge and compass construction" rather than "Compass-and-straightedge construction". Dmcq (talk) 16:22, 24 May 2013 (UTC)
I was also jarred by this move and did a little investigation myself. I have certainly seen the hypenated version but only rarely and a quick pass through my bookshelf did not turn up any instances of it. I also looked for the ruler and compass version with or without hyphens. The variations may be due to some grammatical rules (and I am no expert on those). The hyphens make straightedge-and-compass a compound noun, denoting a class of constructions, but one can also refer to the tools used in such a construction, in which case the hyphens should not appear. In using the compound noun, one can say that construction X is a straightedge-and-compass construction, but this article is concerned with straightedge-and-compass constructions. When the move was made, there were two changes, the hyphens were added and constructions was singularized. I think that the title should either be hyphenated with constructions, or no hyphens with construction. Bill Cherowitzo (talk) 18:11, 24 May 2013 (UTC)
Well I think we should just follow WP:COMMONNAME unless it comes up wit something really stupid. Dmcq (talk) 19:33, 24 May 2013 (UTC)
The usual rule for hyphens is to include them between words that together form a single adjective for something else in the sentence, and to leave them out when the words form a noun phrase that's not functioning as an adjective. For example, "a high-school opera" ("high-school" functions as an adjective for "opera") but "an opera performed in high school" ("high school" is a noun, the object of "in"). Similarly, "compass-and-straightedge constructions" but "constructions made with compass and straightedge". —Ben Kovitz (talk) 14:48, 25 May 2013 (UTC)
Maybe but it doesn't follow WP:COMMONNAME that I can see and that is part of a Wikipedia policy. Is there a good reason to prefer grammar to common name in this instance? Also straightedge and compass construction is more common in books than compass and straighterdge construction. Dmcq (talk) 16:09, 25 May 2013 (UTC)
I think including the hyphens follows the ordinary rules for punctuation, and forms the common name. Or am I misunderstanding you? I don't have an opinion (yet) on whether "compass" or "straightedge" customarily comes first. I'll do a little googling and see if my results agree with yours. —Ben Kovitz (talk) 16:21, 25 May 2013 (UTC)
Googling books is what I did so please go ahead and check for yourself. I understand common name as meaning the name that is more common as written down in the major reliable sources rather than the name that follows grammar or is otherwise official in some way. Dmcq (talk) 16:27, 25 May 2013 (UTC)
"compass and straightedge constructions" came up with 2250 books, and "straightedge and compass constructions" came up with 2,660. The books in the first couple pages of the latter seem more authoritative than the former: Springer-Verlag, etc. In the first 19 books in the straightedge-and-compass group, about a third had hyphens and two-thirds didn't. —Ben Kovitz (talk) 19:03, 25 May 2013 (UTC)
More results from Google books: "compass and straightedge construction": 832 books, 4 out of the first 20 with hyphens. "straightedge and compass construction": 1,560 books, earliest hits from more-authoritative publishers, 6 out of the first 19 with hyphens.
Here's what I think is going on regarding the hyphens: While the hyphenated version clearly follows standard English punctuation for compound attributive nouns, sometimes copyeditors omit the hyphens when they seem unwieldy and there's no real ambiguity. This happens especially often in titles.
Personally, I find the version without hyphens jarring because it leads to a "garden-path" interpretation after "compass and". I'd prefer to stay with the hyphens. —Ben Kovitz (talk) 05:44, 29 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)

I think "construction" is definitely necessary in the title. It's part of the standard terminology of geometry, which distinguishes constructions from proofs. —Ben Kovitz (talk) 05:21, 29 May 2013 (UTC)

## polygons

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)