Talk:Mohr–Mascheroni theorem

Can the M-M theorem be restricted further?

Suppose your compass has a minimum width or a maximum width. What effect might this have? — Preceding unsigned comment added by 192.182.156.36 (talk) 19:20, 22 November 2020 (UTC)

Number of operations

I'm puzzled. How can be the number of steps in the construction using compass "almost squared", as the article says? Each step in the ruler-and-compass construction is by itself a single geometric construction (e.g., finding the intersection points of a circle and straight line) which can be simulated by a fixed (thus using constantly many steps) construction using compass. Therefore, the total number of steps needed should be linear in the number of steps in the original ruler-and-compass construction. Or am I missing something? — Emil J. 14:54, 7 October 2008 (UTC)

I looked at the proof linked from the article. At one point it uses the idiom "Let K be an arbitrary (but large enough) circle through Q₁ and C". If that is allowed as a single action, then the number of steps in the compass construction is linear in the number of steps in the original construction, as I wrote above. On the other hand, assume that taking "large enough" circles is not allowed in a single step, i.e., one can only use the distance of two already constructed points as a radius of a new circle. Then the diameter of the set of constructed points can at most double at each step. However, one can arrange four points in a pattern of diameter 1 so that the intersection point of the two lines defined by the points is arbitrarily far away, thus simulation of a single step in a ruler and compass construction may require arbitrarily many steps in the compass construction. Hence the number of steps in the compass construction cannot be bounded by any function in the number of steps of the original construction, let alone a quadratic one. Either way, the claim in the article is wrong, and I will remove it. — Emil J. 12:44, 8 October 2008 (UTC)

Did you ever get your 2008 post resolved?  When I came across this article a few days ago, though it was horribly written, it did not mention anything about "almost squared".  Someone must have edited the article since your post.  I'm curious if your inquiry still stands or if its been resolved by the current article.  I have made quite a few edits of my own over the past couple of days. 134.204.1.226 (talk) 16:35, 24 May 2024 (UTC)

Incomplete: descriptions of constructions needed

This article is incomplete: it needs descriptions of the non-trivial constructions, with diagrams. -- The Anome (talk) 09:00, 27 May 2018 (UTC)

I have made quite a few edits over the past few days. I hope its cleared some things up for you. 50.34.41.50 (talk) 14:12, 29 May 2021 (UTC)

The Theorem

Id point out that the actual theorem in its full glory, properly worded and expressed mathematically, was never even given. Most sources state that one can only start a construction with a finite set of discrete points and finish a construction with a finite set of discrete points. As stated in the first line:

"any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone."

It is incorrect. Too broad and generic, written too layman, and allows for circles and lines to start a construction.

Im not saying its necessarily wrong to word it like this; it might be equivalent. I just know that most legitimate mathematical sources that take care to word things properly are more restrictive in their statement of the theorem.

50.35.103.217 (talk) 20:00, 20 July 2018 (UTC)

I see no prohibition on having lines in the page already. The MMT is about what you can do with compass alone. Prohibiting a straightedge is not the same thing as prohibiting the preexistence of lines. And considering the fact that you can intersect circles with lines with compass alone, even if those lines are defined only by two points, then there is no loss in generality in allowing lines to be in the plane already. I think that stipulating no lines is unnecessarily restrictive language. As you point out the theorem is frequently worded such that you start and end with a finite set of points, but oddly doesnt allow for circles to be drawn in the finished product, or at the start, which I find inexplicable.134.204.1.226 (talk) 18:46, 27 May 2021 (UTC)

Generalization of the theorem

The Mohr-Mascheroni theorem states you need to start out with a finite set of discrete points and the desired construction has to contain a finite number of discrete points. This implies to me that one cannot start out nor finish with circles, but I think it would be an equivalent result.

Given the use of a compass, and the ability to construct discrete points (say a center and an arbitrary point) one can easily enough use a compass to produce circles in the final product. Thus the theorem should read if one starts with discrete points, one can finish with discrete points or circles, using just a compass.

But more general than that...

Similarly at the start of a construction, if one has a circle, one can find its center and select an arbitrary set of discrete points on it, thus making starting with a circle no weaker than not. And if one starts with points that would define the circle, one can construct the appropriate circle, thus from points a circle is formed. From points a circle is formed, and from circles points are formed. There is no difference in my way of thinking. The theorem should then read that one can start with circles or points to arrive at circles or points, using just a compass. 50.35.103.217 (talk) 19:50, 20 July 2018 (UTC)

Removal of Lengths and a^2 = bc nonsense

An earlier version of this article, prior to my edits, was horribly and circularly written. For one it revolved around line segments whose lengths solved the equation a^2 = bc. In the context of Euclidean geometry it sickened me to see that, as lengths are explicitly rejected. We can talk about ratios of lengths, but not lengths themselves - a unitary segment must be defined to make sense of such things. That is the point of a straightedge as opposed to a ruler, after all. We arent dealing with rulers or working in $\mathbb {R} ^{2}$, we are working in the Euclidean plane like Euclid did, with straightedges and compasses. This reality wasnt respected by the earlier author.

Furthermore, the various proofs and constructions throughout the article relied on this a^2=bc notion, so I revamped everything. One such construction was impossible. The original author gave a hand-wavy set of instructions that, when attempted, proved impossible, nonsensical, and circular in reasoning. Suggesting to me that they never actually attempted the construction despite feeling comfortable enough to teach others. Naturally, my frustrations compelled me to totally rewrite this article. 134.204.1.226 (talk) 17:12, 21 February 2024 (UTC)

There is nothing wrong with a^2 = bc. In Euclid's time they would be referred to as areas and one doesn't need a unit length to deal with them. How do you think Pythagoras' theorem was dealt with except using areas just like this? Dmcq (talk) 08:58, 24 July 2018 (UTC)

Are you defending the use of a graduated ruler in Euclidean constructions? Are you saying we can talk about area in the Euclidean plane but not length? Im going to call that an absurdity. How did Pythagoras deal with the theorem? Presumably with a unit length and dealt with ratios. Maybe not. I didnt say Euclid didnt have a notion of length or area; I said they disallowed themselves from using such notions in Euclidean constructions. You are explicitly prohibited from using rules and area-o-meters. Which is why basing a proof off of it is suspicious, its weak, it disrespects the fundamental principles. 134.204.1.226 (talk) 16:34, 24 May 2024 (UTC)

There is a reason that a^2 is called the square of a. No graduated ruler is required. No unit length is required. As I said a^2 would be represented by an area - the area of the square with side a. bc would be the area of a rectangle ofwith one side of length b and the other of length c. We don't have to convert the lengths to any scale. They are of the same dimesnsion as each other and can be compared directly. Saying a^2 = bc in Greek terms is saying a square of side a has the same area as a rectangle with sides b ad c. Dmcq (talk) 14:55, 24 July 2018 (UTC)

Regarding the "manual or guidebook" citation

I will admit that this does have the feel of a step-by-step instructional. This was not my intention in writing it. I stand by it, however (though it could be reformulated perhaps), because the purpose of providing these instructions is to demonstrate to the reader how the basic constructions are done. In doing so, this constitutes a proof of the theorem. This was not my idea or my original mentality when coming across this page; I merely replaced what was already here and the format that already existed with better and more detailed constructions. This is meant to build confidence in the reader that indeed these constructions are possible. A true instructional would explain far more compass-only constructions than just what is necessary to prove the claim, such as tangents and the like, which is beyond the scope of the proof. My recommendation is not to remove these instructions, but to instead provide a more rigorous proof of why they work. Undoubtedly any proof of the theorem, which is perfectly suitable for the article, would inevitably have to show these constructions anyway. If someone manages to prove the theorem without showing these constructions, many readers would be greatly dissatisfied. I'm not sure what the person who flagged this as an instructional would suggest we replace it with. The original content prior to my edits failed to satisfactorily prove the theorem and lacked any instruction or rigor entirely so for whatever it is worth, the content now is a huge improvement over the previous. 50.34.41.50 (talk) 14:12, 29 May 2021 (UTC)

I put that tag in, not because of the constructions, but because of the comments that were sprinkled throughout these constructions. As I have been going through the sections to improve format and tone, I have been removing the objectionable bits. When I am finished, I'll remove the tags. However, the bigger problem here is that none of the construction material is referenced and unless reliable secondary references are obtained all of your work (and mine) will be for naught, as any editor can come by and remove it as unsourced. I have not, as yet, determined if this is really an improvement and I am not swayed by the argument you have given on this talk page. For the moment I am suspending judgment as I put your contribution into a more acceptable form.--Bill Cherowitzo (talk) 04:32, 30 July 2018 (UTC)

Regarding the Intersection of Circles

Why is it necessary to have centers or to have known radii? Any two circles that cross through one another have intersections that are apparent. These intersections are trivial. I imagine the only reason you'd want radii or centers is for an analytical approach to geometry, which I think steps outside the realm of discussion a touch. Personally I think a purer, Euclidean approach is more appropriate. 166.184.168.82 (talk) 22:33, 5 August 2018 (UTC)

This was put in because all of the references I have access to have used this and incorporated it into their proofs. Since we are supposed to be reporting on what is in the literature, this is very appropriate.--Bill Cherowitzo (talk) 17:56, 6 August 2018 (UTC)

Find new literature then. Are you using Euclid's Elements? This is unnecessary in strict Euclidean constructions. Clearly since centers can be found using a compass alone, this should be obvious one does not need a center. Intersecting two circles are a fundamental operation in Euclid's work and dont require anything about centers or radii being known beforehand. Euclid intersected two circles directly, without any reference to centers or radii.

That's not the way it works here. I have found the citations, if you want to remove the references to centers, it is up to you to find the citations that support that point of view. Geometry has come a long way since Euclid, are you using anything published after The Elements? What is a strict Euclidean construction?—a figment of your imagination?--Bill Cherowitzo (talk) 02:19, 10 August 2018 (UTC)

I am going to agree with the original poster here. Especially with regard to the "known radii". The implication of a statement like that is that distances are recognized and measured. This, however, distinguishes the constructions from traditional Euclidean constructions, where length (i.e. marked straightedge) was not a notion to be considered. With regard to the claim that any circles center can be constructed, this is a claim that I am hesitant to use. It is true from one perspective but not another. In order to construct the center of a circle, using compass only, it becomes necessary to place points arbitrarily in space. When discussing constructability, such as with the constructable numbers, there is a real prohibition on doing this. In fact, the circle center cannot be constructed with compass only if you arent also allowed to place points arbitrarily. Whether doing this is allowed becomes a necessary statement premising any construction. 134.204.1.226 (talk) 21:55, 20 May 2021 (UTC)

Grammar

Someone has modified the content from good grammar TO bad grammar, for some inexplicable reason. I know my writing can be a bit antiquated in form, but it ought not be replaced with a third graders text messaging style. 166.184.168.82 (talk) 22:43, 5 August 2018 (UTC)

Your grammar improvements have mostly been incorporations of personal pronouns and that is considered un-encyclopedic according to Wikipedia:MOSMATH. Also, this way of writing is very consistent with what is found in the construction literature when only the steps involved are listed.--Bill Cherowitzo (talk) 17:56, 6 August 2018 (UTC)

Personal pronouns? No, I dispute that accusation. I do not use personal pronounces, unless youre referring to the ubiquitous "we" that appear EVERYWHERE in mathematical documentation. If you want to remove that phrasing, fine, but dont replace it with childish babble. It is much worse. Please do get basic English grammar down sufficiently well it isnt a pain to read. Correcting grammar and making it worse in the process is just ridiculous.

Mohr's Alternative Proof

Under Alternative Proof, Mohr's is the only one whose methods are unspecified. The text only states that "it was different", whatever that means. I think his method ought to be specified for consistency or the language changed. 166.184.168.82 (talk) 22:43, 5 August 2018 (UTC)

I put in what was in the source. If you have additional (referenced) material please add it. --Bill Cherowitzo (talk) 17:56, 6 August 2018 (UTC)

out of curiosity, it may have been a verbatim quote from the source, but did you accidentally neglect to include a greater context that would disambiguate the statement? 134.204.1.226 (talk) 21:49, 20 May 2021 (UTC)

Figure Numbers

There is no need for the midpoint construction image to be Figure 1. This is like the sixth image in the entire document, so why Figure 1? Additionally, there is no standard across all of the images in the article. 166.184.168.82 (talk) 23:21, 5 August 2018 (UTC)

Reverted the format. Removed the only "figure " number.

Unfortunately I have been working on this article piecemeal and have a grander vision of its final form. I had planned to label all the diagrams and this was only the first one I had gotten around to. --Bill Cherowitzo (talk) 17:56, 6 August 2018 (UTC)

I dont know if you grasp English too well, my friend, but Im fairly certain English is not your strong suit. Probably a second language? No offense. Between article contributions and the like, its revealed. Please do humble yourself a bit with regard to the English. The original article before my edits were chalked full of problems and I see them emerge constantly since youve been interacting.

I dont care if you use figure numbers. And I dont care what your "grander vision" is. Have at it. But you clearly missed the point of my criticism. One does not start the count at 1 in the middle of the article after a dozen figures have already come before. Please use 1 for the FIRST image, and 2 for the SECOND image. You might also use 3 for the THIRD image. There is a pattern to this. If you need help with the counting still, let me know.

Offense is taken. However, coming from someone with a clear weakness in spelling and a lack of understanding of the use of the apostrophe (for example, your last response in the section Grammar above) I can not take it too seriously. Snarky responses are not going to get you very far. If there is a problem with any edit that I have made, please be explicit and discuss it on these pages.--Bill Cherowitzo (talk) 02:29, 10 August 2018 (UTC)

Midpoint Split-up Suggestion

I broke the midpoint construction up into two subsections. The first doubles the line segment AB, and the second finds the inverse of a point across the circle. All I did was made it a bit more bite-sized and conceptual. I however suggest splitting this up into two legitimate sections and constructions with their own images. I feel as though doubling a line segment and inverting a point across a circle are two very valuable constructions in their own right. Inverting applies more generally than just halving. And since the doubling aspect is referenced elsewhere in the document, it is fairly standalone anyway. The midpoint construction can be made simply into three steps: draw circle, double the segment external to circle, invert the point across the circle. 166.184.168.82 (talk) 23:29, 5 August 2018 (UTC)

I reformatted. Created a Point Inverse construction, a Segment Doubling construction, and reworked the midpoint to reference them. 166.184.168.82 (talk) 00:24, 6 August 2018 (UTC)

As mentioned above, I was far from finished working on this article. I was going to include a section on the proof using circle inversion. I find this approach to be much cleaner and quicker, but it is slightly more sophisticated than the technique using reflection in a line, so I was delaying putting it in until the rest of the article was cleaned up. I will expand on the material you have added, with references.--Bill Cherowitzo (talk) 17:56, 6 August 2018 (UTC)

I see. You were the original composer of this article? Or the bulk of the content anyway? Read some of the commentary and criticisms I had. No offense but it was HORRIBLY written. Im not knocking the approach, per se, but the utter lack of rigor and in some cases circular reasoning that was involved. In fact, one construction that was provided in this article prior to my coming across it, was literally impossible. No construction was explained, and attempts to do it failed, because it relied on information not provided. And thats not even counting the borderline English illiteracy baked into the convoluted mathematics. I dont know if that was you or not because your writing seems more coherent now. Basically it was a horrible mess which is why I totally revamped this whole article. Id appreciate your help getting it up to par. 50.35.103.217 (talk) 21:59, 9 August 2018 (UTC)

I dont know what you mean by sophistication. Sounds a bit pretentious? The point of this article wasnt to show every possible compass-only construction for every possible figure there is. Thus the new section on "extending a line segment" seems unnecessary and a more complicated contribution to this article than what is required. The point of this article, in my assessment, was to merely prove the theorem by demonstrating how the basic constructions of geometry are possible with compass alone. Not to make it "easy", or "sophisticated" or to add advanced tools to the geometers toolbelt. Just to prove it. Nothing more. Anything else goes beyond the scope of the article, in my opinion. Ive tried to break up complicated constructions into more bite-sized parts, like I did with the midpoint - breaking it up into a doubling and a point inversion. Dont really need to make arbitrary integer multiples of a line-segment, do we? Or like I did with line-line intersections - we have point reflections across a line now as a construction, and the line-line and line-circle constructions reference them, making it all a bit more streamlined. These are useful tools in their own right and well worth explaining (not unlike arbitrary segment multiples), but the difference is they werent beyond the scope since they were essential in proving line-line and line-circle constructions. I dont know, if you want to turn this article into a users manual for all compass-only constructions then that's on you.50.35.103.217 (talk) 21:59, 9 August 2018 (UTC)

Just to set the record straight, I did not do any editing in this article until after you made your changes. I have only glanced at what the article was before your input, just enough to realize that there certainly were problems. I appreciate your attempt to correct some of them. Unfortunately, your technique seems to be to provide your own constructions and while your work is fine mathematically, it is unacceptable by wikipedia policy WP:NOR. I have been going through the article, reformatting to bring it into compliance with wikipedia style WP:MOSMATH and providing references for the more standard constructions. I have now reached a point where I will have to replace some of your original work by referenced constructions. Up until now I have tried to keep your diagrams and their labels, but this may not be possible any longer. My aim is exactly the same as the one you stated, to provide the constructions needed to prove the theorem, but do realize that an encyclopedic article is about the theorem and not a particular way to prove it. Since there are two major approaches (with or without circle inversion) both need to be discussed.

As to the section on extending a line... a more careful reading would show that this form was needed to give a complete proof of inversion in a circle, and not just the version where the point is outside of the disk. The construction itself is extremely simple and far more transparent than your doubling construction and much easier to extend. Your version was of course used in your proofs that come later, but these will be replaced by simpler constructions that don't use this doubling. Of course, if I am wrong about this being OR, simple citations will set me straight. --Bill Cherowitzo (talk) 23:31, 9 August 2018 (UTC)

Im really not interested in wikipedias policies, in general. By all means, bring my contribution up to par if you wish. Im not disputing a need for standardization but content is what Im after - considering the article was so lacking in the first place. Lets get the content before we worry about font style, shall we? Priorities, and all.

I dont dispute that the article should be streamlined. But in order to prove the theorem, one has to show how to prove it. You cant prove it without these constructions so again, Im at a loss for how to do it any differently.

I dont see that multiple different approaches NEED to be discussed simply because they exist. You said it yourself, a proof is all we need. Only one approach will suffice.

If as you say the math Ive contributed is fine, then why would you want to delete it and replace it? Seems unreasonable unless you genuinely have something better to replace it with.

And in fact, the inversion in a circle can be made complete through repeated doubling. Segment extending by an integer is not strictly required. It is excessive for the needs of the article. The Doubling I had was a simpler construction that accomplishes the same goal, completes the construction and does it in fewer operations. In fact I mentioned doubling for points too far inside of the circle (less than 1/2 radius from center). Id already confronted that issue. Is that the "more careful read" you were referring to?

You should be concerned with Wikipedia policy, editors who regularly flaunt them will quickly find themselves banned from editing here. I don't think that you have come to appreciate how seriously we take the NOR policy. It doesn't matter that the math is correct, it does matter that the correct math is cited in a reliable secondary source. These are the rules that the Wikipedia community has set for itself and in reverting your original research (under Wikipedia's definition) I am only acting as an agent of that community. Other editors, if they caught wind of this, would do the same. As to the inversion in a circle; I didn't claim that doubling couldn't be used to complete the argument, it certainly could–but I find that integer multiple form makes the application of Archimede's axiom more direct and clearer. Also, your argument about being too close to the center is in a construction that will be replaced. --Bill Cherowitzo (talk) 03:05, 10 August 2018 (UTC)

theorem wording

I feel as though the phrasing

 "The theorem can be stated more precisely as:
    Any Euclidean construction, insofar as the given and required elements are points, may be completed with the compass alone if it can be completed with both the compass and the straightedge together.

is in fact less precise. I appreciate why this line was included. But does the wording not imply that:

1. You cannot start with any circles already constructed
2. Circles are not a part of the finished product, and
3. that the set of constructible objects are in fact a subset of those with straightedge included.

I agree with you. Your first two points, anyway, Im having trouble understanding your third. I originally wrote the line, or a variation of it, and it had been edited. I have tried several times to rewrite it but my edits keep getting undone. It also implies that you cannot have a line already drawn on the plane. Im actually the one that began serious contributions to this page; it was a snub when I came across it. But as soon as I did build it up, someone else has hijacked the page, destroyed quality work, refuses to allow my edits, and most baffling of all, they remade content more complicated than they needed to be (e.g. the line-line intersection could be done in 9 applications of the compass (three reflections and one inversion) instead of the 30+ my work has been replaced with). 192.182.156.36 (talk) 19:17, 22 November 2020 (UTC)

One of the things you don't seem to appreciate is that your work is considered WP:OR unless you can provide some reliable sources. OR material has no place on Wikipedia and will be consistently removed once found. Also, please sign your comments with --~~~~ in the future, rather than having a bot do it.--Bill Cherowitzo (talk) 21:27, 22 November 2020 (UTC)

Bill, MY work was valid. There is plenty of unsourced content on wikipedia; I dont have to do all the work, do I? And why would edits in grammar and syntax be undone? Do I need to cite that as well? In fact Ive provided sources in the past, but that too gets deleted. By definition the wikipedia is a community constructed reference; it does not require the utmost rigor. Why does valid, true content need to be deleted from any reference just because someone doesnt like the citations? Wouldnt the proper course of action be to request citations from the community rather than delete the contribution entirely? Is the wikipedia a source of content or just an enumerated list of external resources? I think maybe you fail to grasp the original purpose of this site. I think you fail to grasp that anyone can contribute sources where they are lacking, rather than deleting the content itself. I cannot fathom why anyone who claimed to be in support of mathematical truths be so belligerent to those who would contribute meaningfully. What gives any one individual the right to steamroll all the other contributors to any page? Also, I do sign off... I just dont sign in. Perhaps you cant take a moment to figure out how that works. Just take a look at yourself, man. You have to put your two cents into every single comment made on the talk page; criticizing every question or suggestion made. If that isnt someone trying to take control, I dont know what is. Do you OWN this page? Or do you just think you know better than everyone else? 192.182.156.36 (talk) 23:21, 29 November 2020 (UTC)

History citation

The bit about the rusty compass in the second paragraph of the history requires a citation. I havent got an external citation but these statements were also made in the history section of the poncelet-steiner theorem. Perhaps these can be verified and copied?

Reflections and Inversions

It has been stated to me from several sources that every mohr-mascheroni construction can be done with "either reflections or with inversions". I dont know what this means but Id like to see some mention of it, if its true, somewhere in the article. And a reference to how one might construct, say, a reflection using only inversions, or an inversion using only reflections, etc., as that seems to be the implication. Id point out also that the compass-only construction of a point inversion into a circle comprises of a reflection.

Other restricted constructions

The Poncelet-Steiner theorem has a section on "other restricted constructions". It seems reasonable for the M-M theorem to have a similar section, perhaps echoing what the P-S theorem says. This article makes no real mention other than a casual reference to P-S that alternative restrictions do exist. 192.182.156.36 (talk) 23:37, 29 November 2020 (UTC)