Talk:Dehn plane

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Hyperreals, surreals, or any nonarchimedean field?

The article seems muddled. It's clear that the Dehn plane is defined in terms of cartesian coordinates (x,y), where x and y are finite members of a nonarchimedean system. However, the article says in various places that the system could be any nonarchimedean field, or that it has to be the hyperreals, or that you can get a model from the surreals. If it's defined using the hyperreals, as stated in the first sentence, then there's no reason to construct a model using the surreals, because the definition is in terms of an explicit construction on the hyperreals. If the definition had been given axiomatically, then it might have made sense to talk about two different models, one using the hyperreals and one using the surreals. Since Dehn died in 1952, he can't have defined it in terms of either the hyperreals or the surreals.-- (talk) 17:01, 22 March 2009 (UTC)


Google scholar, mathscinet, and the arxiv all come up blank on "Dehn plane". Does this object actually exist outside of Wikipedia? Chenxlee (talk) 15:50, 12 October 2010 (UTC)

Bold page move

Hi all

I have moved the page to a non-named version and put "Dehn plane" as "sometimes known as"

I suspect that best gets around the neologisms and lack of sources for Dehn plane as the page title. Chaosdruid (talk) 00:29, 11 February 2011 (UTC)

I agree that the name "Dehn plane" is not ideal, but it seems better than any obvious alternatives. As the deletion discussion mentioned, there is at least one source for it. The new name you apparently invented does not make sense and has no sources. r.e.b. (talk) 12:56, 11 February 2011 (UTC)
I did not invent it lol. It is not a name, it is a descriptive title. Dehn's theorem is of a non-parallel planar geometery is it not?
The problem is that the page title refers to something which only has one reference. That is unfortunately not going to pass muster on notability. The descriptions of a Dehn non parallel planar geometry is sourced to at least four refs and so would pass muster. Indeed Hilbert refers to it on page 81 as This new geometry may be called a semi-euclidean geometry. PDF of the translated book I added this because Rucker says "...described in David Hilbert..." - although I am assuming that is the maths behind it rahter than the name :¬)
I cannot see what the alternatives are apart from merging it with one of the other candidates in the AfD discussion. Chaosdruid (talk) 17:32, 11 February 2011 (UTC)
I think "Dehn plane" is the lesser of two evils. The term "non parallel planar geometry" seems to be a worse neologism. Sławomir Biały (talk) 18:03, 11 February 2011 (UTC)
I take it that you have read the deletion discussion? If so then you must realise that the weight of opinion was against retaining this name for the page.Chaosdruid (talk) 18:22, 11 February 2011 (UTC)
There was some discussion, yes, but hardly a solid consensus developed there, and certainly nothing about the particulars of what title to use. The AfD closure stipulated that such details should be decided here. Sławomir Biały (talk) 18:28, 11 February 2011 (UTC)
(ec) Rucker appears to be starting the neologism. He first describes it in a problem set, "9. Points in the Cartesian plane are given as pairs (x,y) of real numbers. Let the points in the Dehn plane be given as pairs...Show that Euclids Fifth Postulate fails in the Dehn plane..." and "This construct is due to Max Dehn, and is described in David Hilbert, The foundations of Geometry (Chicago:Open Court, 1902), p.129. The curious thing about the Dehn plane is that, although he Fifth Postulate fails, the ..." That, alone, is not enough to establish that it is neither a neologism nor that it exists other than as a mere descriptor to separate Cartesian from Dehn's work, especially as Hilbert nor anyone else use the term Dehn plane. Chaosdruid (talk) 18:22, 11 February 2011 (UTC)
(ec again) P.S. There was a fair amount of discussion as to what titles to use - "Nonarchimedean geometry", "How Dehn constructed a non-Legendrean geometry", "pick a descriptive title" as well as the ones I mention at the top. Chaosdruid (talk) 18:46, 11 February 2011 (UTC)
I think uneasiness about using the term "Dehn plane" is overstated. The title is both natural and descriptive. It has been used by at least one other published source, so we are clear of WP:NEO. The present title should be kept unless a better one is proposed. Sławomir Biały (talk) 18:42, 11 February 2011 (UTC)
(ec) We cannot use it as a title as it is not sourced as Dehn plane, but descriptively it should be "Dehn's (something or other)" such as the one I had or "Dehn's planar geometry" Chaosdruid (talk) 18:48, 11 February 2011 (UTC)
Re "cannot use": why not? Rucker calls it this, and besides it is descriptive. "Dehn's plane" would also work, but overall follows a less common naming convention, at least in mathematics. Also, the other titles that have been proposed are all clearly worse. Sławomir Biały (talk) 18:56, 11 February 2011 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Simply because it cannot be sourced to that as its name. "Dehn's Theorem" is quite clearly called such in numerous textbooks (The Foundations of Euclidean Geometry, Combinatorics of train tracks etc.) and so would pass muster on notability, although Invitation to mathematics [1] uses "Dehn-like invariants". These books, nor any others I can find, do not mention the "Dehn plane". Chaosdruid (talk) 20:32, 11 February 2011 (UTC)

That other stuff you mention isn't relevant. As I've already said, we can at least source it to Rucker, a well-known author on expository mathematics, until such a time as someone proposes a reasonable alternative. There is no rush to blindly follow the rules and substitute a questionable description for a title which is perfectly clear and reasonable. Sławomir Biały (talk) 21:53, 11 February 2011 (UTC)

Proposed change

I think a natural target for the article is semi-Euclidean geometry. This is what Max Dehn himself calls the geometry, as well as Hilbert. Sławomir Biały (talk) 22:03, 11 February 2011 (UTC)

I may have misunderstood Hilbert. It appears that the geometry described here is the non-Legendrian geometry. Sławomir Biały (talk) 22:18, 11 February 2011 (UTC)
Well I am glad that you saw what I was saying in the end :¬) Chaosdruid (talk) 22:52, 11 February 2011 (UTC)


In elliptic geometry, the sum of the angles does exceed 180 degrees, contrary to the claim in the lead. I am not sure what was intended here. Tkuvho (talk) 05:26, 13 February 2011 (UTC)

In the sober light of day, Hilbert's text makes a little more sense, and I have adjusted the text accordingly. However, I still have trouble reconciling his statement that "all of the theorems of Riemann's (elliptic) geometry are valid" with the fact that one of the axioms appears to be contradicted. Ideally, someone whose command of German is better than mine should consult with Dehn's paper to see what he actually did. Sławomir Biały (talk) 21:54, 13 February 2011 (UTC)
My German is nil. As far as the mathematics goes, it seems to me that one already obtains the relevant counterexample by considering an affine plane over the ring of dual numbers. The advantage is that one does not need to "ignore" the points at infinity; they are not there to begin with. Tkuvho (talk) 05:39, 14 February 2011 (UTC)

is this true?

I have to admit I don't understand the claim contained in this page to the effect that the plane over the non-Archimedean field will violate the theorem that the sum of the angles is at most pi. This is certainly false if one starts with the hyperreals. By the transfer principle, every triangle, real, infinitesimal, or hyperreal, will have the property that the sum of the angles is pi on the nose. As far as other non-Archimedean rings are concerned, I am not sure how one would go about defining angles in the first place. Could this be some kind of a joke? Tkuvho (talk) 14:01, 15 February 2011 (UTC)

I see that Chaosdruid initiated the "non-Legendrian" claim on 5 february in the deletion discussion, without providing any sources. The claim is in error. The title should be changed. Tkuvho (talk) 14:11, 15 February 2011 (UTC)

non-Archimedean ordered field is not enough

To define the metric one needs to be able to extract square roots. This is not necessarily possible in a non-Archimedean ordered field, though it is possible in the hyperreals and the surreals. Why does one need the metric anyway? Tkuvho (talk) 14:17, 15 February 2011 (UTC)

suggested redirect

I suggest this page be redirected to Hilbert's axioms, and contents moved to a subsection there. The main significance of Dehn's example is as an illustration of the independence of the various axioms. Tkuvho (talk) 14:42, 15 February 2011 (UTC)

errors in translation of Hilbert

The English translation of Hilbert linked at this page has some obvious problems. For example, examining the table on page 82, hyperbolic geometry appears to correspond to the case when the sum of the angles in a triangle is 2\pi. If anyone has a reading knowledge of German it would be good to have this straightened out. Dehn's example where Legendre's theorem does not hold could not be the same as the one described in this page. Tkuvho (talk) 14:58, 15 February 2011 (UTC)

Here is the google translation of the relevant passage of the review of Dehn's paper from Zentralblatt: "With the help of the given Hilbert not Archimedean geometry can also construct two geometries in which the angle sum greater than two right, rel. rights equal to 2, in which but still possible through each point of each line are infinitely many parallels (the non-Legendre and the semi-Euclidean geometry) is. In contrast curiously follows necessarily from the non-existence of parallels, that the angle sum is greater than two right, even the ones not the Archimedean axiom. The recent views on the relationship between the angle sum and the number of parallels through a point must, therefore, if omitted the Archimedean axiom is essentially modified." At any rate, what emerges is that we are dealing with two separate examples. Tkuvho (talk) 15:09, 15 February 2011 (UTC)

Changed without consensus

This page is getting silly with three name changes taking it bak to a badly owrded version of the same name that consensus was against.

The consensus was to change the name from Dehn plane.

Now we have an editor who has decided it should be called "The Dehn plane", against MoS as "The" is not used in titles.

Kindly put back, observe consensus and discuss changes before taking actions on a whim. As yet there is no new evidence added to support the name and no reason for it to be named like this.

Chaosdruid (talk) 21:14, 20 February 2011 (UTC)

proposed name change

Since the AfD discussion was based on an erroneous assumption that the plane discussed here is non-Legendrian, I re-suggest moving the page to "Dehn's plane" or "Dehn's counterexample". Tkuvho (talk) 12:25, 23 February 2011 (UTC)

As has already been said, the consensus was that "Dehn plane" would not be used in the title. Please accept this as repeating this is not going to get us anywhere. Please choose a proposed title that does not included Dehn plane in it. Chaosdruid (talk) 21:00, 23 February 2011 (UTC)
There was no such consensus. Tkuvho (talk) 13:57, 24 February 2011 (UTC)
I do not appreciate the suggestion that I am lying.
  • If that plane is (almost) nowhere called Dehn plane, we shouldn't have an article under that name (implicating that it is a common/established term). In such a cse merge or rename might be the appropriate course of action.--Kmhkmh (talk) 01:33, 4 February 2011 (UTC)
  • Keep or merge, either non-archimedean field or projective plane should be reasonable targets for merger. Septentrionalis PMAnderson 19:39, 3 February 2011 (UTC
  • See also this Google Scholar search for 'non-Legendrean.' There are four hits in that search that look like they might be used as references (or further reading) for the Dehn plane article if it is kept. If someone does have time to work on Dehn's idea, they might wind up renaming the article, since 'Dehn plane' gets no hits in Google Scholar and obscure titles are not the best. EdJohnston (talk) 04:36, 5 February 2011 (UTC)
  • I am beginning to think that the page should be renamed then. It seems that the method is notable enough, though not named as the page title currently suggests. Perhaps we could rename to "Dehn geometery", "Dehn planar geometery" or "Dehn non-Legendrean geometery". Chaosdruid (talk) 15:35, 5 February 2011 (UTC)
  • Rename Perhaps a more inclusive subject title such as "Nonarchimedean geometry" can be found, but without the neologism.--RDBury (talk) 13:00, 3 February 2011 (UTC)
I suggest that you apologise, or strike the comment, and explain how you came to the conclusion that there was no consensus. Chaosdruid (talk) 16:34, 24 February 2011 (UTC)
I am sorry if you are offended but I did not mean to imply that you are lying when I said there was no such consensus. If you re-read the AfD you will notice that some editors saw nothing wrong with "Dehn plane", for instance Hardy. Thus were was no consensus. You are fully entitled to your opinions that there was a consensus, but I respectfully disagree. Tkuvho (talk) 17:17, 24 February 2011 (UTC)
Hardy does not express a decision for delete, keep, rename or merge. Without giving an opinion Hardy is not involved in any count on consensus, and his discussion does not state anything which would help establish consensus or non-consensus.
Hardy talks about "If it's not mentioned by a particular name, that might mean only that that name seldom if ever appears" which is a problem as the article title should either be a notable title or a descriptive one. Hardy then goes on to say how he created an article using a descriptive title. His discussion follows the lines of giving a page a descriptive title and mentions that Dehn plane might be also be considered descriptive.
It is apparent to me that the "no consensus reached" is pretty much untenable. You have moved the page back and this feels much more like you defending your decision to go against the other 6 editors in the discussion. I would also like to point to your own statement where you yourself said "To boldly go against the consensus" - if that does not prove what I am saying then why the heck did you say it yourself and then come here to repudiate and argue the point? Chaosdruid (talk) 18:01, 24 February 2011 (UTC)
I did boldly go against the consensus that the example described here is non-Legendrian, and also identified the source of their mistake. I have yet to be thanked by the editors involved in that consensus for correcting their error. Tkuvho (talk) 18:39, 24 February 2011 (UTC)
Good luck on that. My first choice was "Dehn non-parallel planar geometry", that was reverted by an editor not involved at the AfD, then followed another mess of three or four changes and eventually you took it back to Dehn Plane, albeit with the addition of pluralising to "Planes" and adding "The", which I would remind you should not be there and still needs removing. "The" is not used in article titles Wikipedia:Article_titles#Article_title_format. Chaosdruid (talk) 19:31, 24 February 2011 (UTC)


@R.e.b.: Thanks for the clarifications regarding both geometries. Could you please elaborate? What is the "t" in "tx" and "ty"? Also, in the semi-Euclidean example, I don't quite understand the bit about square root of 1 plus omega squared. If you read Dehn's paper, could you give some more details? Tkuvho (talk) 18:05, 26 February 2011 (UTC)

I still don't understand what the "t" is in "tx" and "ty" in the second example. Also, does Dehn's field consist of functions, or rather equivalence classes of functions? It is hard to imagine a total order on the functions themselves. Does Dehn attribute the field he uses to any of the authors active in non-Archimedean fields at the time? Tkuvho (talk) 05:53, 27 February 2011 (UTC)
Thanks for the clarification. Can you comment further on the "certain metric" for the non-Legendrian example? Tkuvho (talk) 16:47, 27 February 2011 (UTC)
Thanks for the further clarifications. The metric on the non-Legendrian plane is still not clear. It could not be the natural projective metric, since the latter would have diameter at most pi/2 (or pi, depending on normalisation), and here we are looking for something of infinite diameter. Tkuvho (talk) 15:44, 28 February 2011 (UTC)
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