Talk:Klein bottle

message: Mathematicians try hard to floor us
 With a non-orientable torus
    The bottle of Klein
    They say is divine
 But it is so exceedingly porous.
:Frederick Winsor, The Space Child's Mother Goose The Space Child's Mother Goose
I had that book, but do not remember that limerick!  There is a quatrain of which I remember only alternate lines: "they went to sea in a bottle by Klein" and "for the sea was entirely inside the hull." —Tamfang (talk) 20:53, 15 March 2023 (UTC)
thread_title: Inside vs. Outside

message: I'm trying to bring order to the image layout in this article. It also means I'm throwing out images we don't need -- for the moment. 
❝Sverdrup❞ 23:36, 12 Aug 2004 (UTC)
replies: text: I removed some more images today.  dbenbenn | talk 14:57, 3 Mar 2005 (UTC)
replies: text: Considering these are objects that disappear into themselves, I suppose a certain irony is inescapable. – AndyFielding (talk) 09:49, 23 February 2022 (UTC)
thread_title:  Unused image

message: Should we mention that the Klein bottle arises as the connected sum of three copies of \mathbb{R}P^2?
:It's the sum of two real projective planes.  And that was already mentioned in the version you saw when you asked your question.  Taking the connected sum of real projective planes is the same as gluing together two Moebius bands along their boundaries.  --C S (Talk) 13:41, 21 February 2006 (UTC)
thread_title:  Connected sum construction 

message: I took out the following text
Topologically, the Klein bottle can be defined as the square [0,1] × [0,1] with sides identified by the relations (0,y) ~ (1,y) for 0 ≤ y ≤ 1 and  (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram:
: Image has been deleted. CiaPan 18:24, 26 January 2006 (UTC)
replies: text: You're wrong about the text — it is correct. So I've restored it (and replaced the wrong :Image:Klein.jpg with a proper one).CiaPan 07:20, 15 November 2005 (UTC)
thread_title:  Wrong Image 

message: Like the poetry, guys!  It's a nice touch to what can sometimes be a dry topic (I'm a math major, so I'm allowed to say that :) ).DonaNobisPacem 22:49, 23 December 2005 (UTC)
thread_title:  Poetry 

message: What happens if you pour water (or some other liquid) into the "opening" of the bottle? --Jfruh 21:30, 22 February 2006 (UTC)
replies: text: If you're talking about a three dimensional immersion, then yes, it would hold water. A four dimensional embedding would not, largely due to the fact that it's a two dimensional manifold, and would hold four dimensional water about as well as a 1 dimensional manifold (i.e. a piece of string) holds three dimensional water.James pic 14:40, 13 October 2006 (UTC)
replies: text: I don't understand, after studying it carefully, how the bottle could "hold water". From where I stand, whatever you pour into "the mouth" of the water, assuming it traveled on it's way without respect to gravity, would exit the bottle at the place where the curve enters, and then go back up out of the opening and meet where it entered, just like JIP said. In this sense it doesn't "hold water" at all - at the most it is containing it, like if you put a drop of water on top of a plastic sheet.
I think there are some pictures of them containing water in the external links, or you can try google. What is you definition of hold? --Cronholm144 22:37, 17 July 2007 (UTC)
thread_title:  Use as an actual bottle 

message: Any sources on this? (other than circular wiki-page references) K is the connected sum of two projective planes, so in the world of non-orientable closed manifolds K is considered genus 2, as far as I know. MotherFunctor 04:00, 15 May 2006 (UTC)
replies: text: In my experience, the word "genus" is reserved for orientable surfaces.  Though if the Klein bottle were going to have a genus, it ought to be 2 as you say.  We do have a formula like χ = 2 – k, and that number k might be called the genus by some authors.  For now, I'm deleting the mention of genus, but if someone finds a reference which says that k is called the genus, then they can put it back (with k =2 instead of –1 obviously).  -lethe talk [ +] 10:08, 15 May 2006 (UTC)
replies: text: I looked around.  The springer online encyclopedia says the genus of the Klein bottle is 1, while the topology textbook says it is 2.  I prefer 2. -lethe talk [ +] 11:59, 15 May 2006 (UTC)

text: So I checked up on the encyclopedia, am confused why they say genus of K is 1, but I don't really recognize the sources either. Massey's Algebraic topology mentions genus for non-orientable surfaces, as does John Lee's book on topological manifolds. Massey's gives the \chi = 2 - g. formula for nonorientable surfaces too. I think the definition in terms of maximum number of non-intersecting Jordan curves, such that their complement is path connected is nice, it's concrete anyway, without being 2 seperate definitions, which is weak. MotherFunctor 02:03, 23 May 2006 (UTC)
thread_title:  genus-1?? 

message: The initial name given was "Klein Fla-e-che" (Fläche = Surface); however, this was wrongfully interpreted as Fla-s-che, which ultimately, due to the dominance of the English language in science, led to the adoption of this term in the German language, too.
Any reference for that? --Trigamma 10:22, 9 December 2006 (UTC)
replies: text: If memory serves, this is mentioned as a plausible speculation in Game, Set and Math by Ian Stewart. Algebraist 10:19, 29 May 2007 (UTC)

text: The "dominance of the English language" reason is wrong. The english mistranslation has been adopted cause it looks in fact more like a bottle than a surface. I'm german and especially in math the english language is very rare. --87.172.145.241 09:21, 22 July 2007 (UTC)

text: Although it looks like a bottle isn't its original intent to convey that it represents a surface?  Was being hollow just to convey this sense of surface?  Was also one of the original purposes to try to convey a 4th dimensional equivalent of a Möbius strip (which is a surface in our 3D world just as the equivalent object would be in a 4D world)?  Or are none of those things so? Gonegahgah (talk) 04:57, 2 August 2015 (UTC)

text: * Any non-pathological surface with no edges will look "hollow". — I imagine the name "bottle" stuck partly because it's more distinctive than "surface".  ("Klein surface? Which one is that again?" "The one that looks like a bottle.") — It's not the hyperspatial equivalent of the M-strip, it's two ordinary M-strips stitched together. —Tamfang (talk) 05:32, 2 August 2015 (UTC)
thread_title:  Klein Fla-e-che 

message: Could the author possibly mean "three dimensions" in the following? After all, it is (as suggested in the second sentence here) a four dimensional object so if a visualization is sufficient for heruistic use but not quite correct then it must be a three-d visualization because if it was a four-d visualization it would be completely correct. I won't change it because it's possible I'm missing a subtlety, but the author might have a look.
Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions.
:The Klein bottle is two dimensional, as it is glued together from a two-dimensional sheet.  It can be embedded into four dimensions, but you can do that in different ways.  The visualization given is a particular way of doing that.  The next paragraph (unquoted) needs to be improved, but I'll do that.  --C S (Talk) 10:31, 8 April 2007 (UTC)
replies: text: I see that I was mistaken. Thanks--Gtg207u 20:43, 8 April 2007 (UTC)
thread_title: Couple of Questions

message: The text and figures refer to Klein bottles as "2d."  The text also refers to a sphere as "2d."  Shouldn't these all be classified as "3d" objects?  --algocu 16:51, 27 August 2007 (UTC)
replies: text: They are 2-dimensional manifolds, so the text is correct. (I can understand why you might think of a sphere as 3-dimensional, as it embeds in R3 but not in R2. But by that reasoning, a Klein bottle would be 4-dimensional.) --Zundark 20:52, 27 August 2007 (UTC)
thread_title:  3d vs 2d 

message: It may be useful to have an explicit description of the fundamental group of the Klein bottle, as well as the presentation as connected sum of two copies of the real projective plane. Katzmik (talk) 09:24, 24 October 2008 (UTC)
thread_title:  fundamental group 

message: I have tried to draw the Klein's bottle immersion using the parametrization given
in the main article.
Right me if I'm wrong but I think there is a mistake:
:\begin{align}
x & = \frac{ \sqrt{2} f(u) \cos u \cos v (3\cos^{2}u - 1) - 2\cos 2u}{80\pi^{3}g(u)}-\frac{3\cos u -3}{4}\\
y & = -\frac{f(u)\sin v}{60\pi^{3}}\\
z & = -\frac{\sqrt{2}f(u)\sin u \,\cos v}{15\pi^{3}g(u)}+\frac{\sin u \cos^{2} u + \sin u}{4}-\frac{\sin u\,\cos u}{2}
\end{align}
where
:f(u) = 20u^{3}-65\pi u^{2}+50\pi^{2}u-16\pi^{3}\,
:g(u) = \sqrt{8\cos^{2}2u-\cos 2u (24\cos^{3}u-8\cos u + 15) + 6\cos^{4}u (1 - 3\sin^{2}u)+17}
for 0 ≤ u < 2π and 0 ≤ v < 2π.
The problem is g(0).  I found:
:g(0) = \sqrt{8\cos^{2}(2\cdot 0)-\cos(2\cdot 0) (24\cos^{3}(0)-8\cos(0) + 15) 
                                     + 6\cos^{4}(0) (1 - 3\sin^{2}(0))+17}
:g(0) = \sqrt{8\cdot 1 - 1 (24\cdot 1 -8\cdot 1 + 15) 
                                      + 6\cdot 1 (1 - 3\cdot 0)+17}
:g(0) = \sqrt{8 -  (24 -8 + 15) + 6 +17} = \sqrt{31-31} = 0
According to the article, 0≤u.  Now, we cannot compute x(u,v) 
and z(u,v) with u=0,  because g(u) is dividing some
terms in their formulae. 
Eviruena (talk) 21:03, 28 January 2009 (UTC)
replies: text: How did someone even find this parametrization? 98.113.222.37 (talk) 17:45, 20 June 2010 (UTC)

text: I also couldn't get this to plot correct, and I am very dubious as to it's correctness. I have updated the parameterisation availble here with a different one found by Robert Israel. Wridgers (talk) 12:12, 30 November 2012 (UTC)
thread_title:  Parametrization 

message: I suggest a separate section on physical models of Klein bottles.  This should collect pointers to various makers (such as Mitsugi Ohno, Alan Bennett, Acme, etc).  Also discuss approximations and compromises that happen when an R3 immersion of the Klein bottle is made of glass, fabric, paper, etc.  Also The section would follow the Construction section, just before the Properties section.
If there are no objections and I find time in the next few weeks, I'll try to do this.
NoahVail (talk) 23:15, 4 November 2009 (UTC)
thread_title:  Suggest section on Physical Models of Klein Bottles 

message: "Clean" or "Kline"? Richard W.M. Jones (talk) 09:58, 7 July 2010 (UTC)
replies: text: German spelling is almost completely regular. Klein was German, and the regular German pronunciation is , roughly rhyming with English 'incline'.  If it were pronounced  (similar to English 'clean'), it would be spelled 'Klien'. I have added the IPA to the article. --Macrakis (talk) 14:01, 7 July 2010 (UTC)
thread_title:  How do you pronounce "Klein"? 

message: The bundle projection was incorrect. One should map to the parallel edges in order that it is well-defined on equivalence classes of the total space. See for example Steenrod, The Topology of Fibre Bundles, section 1.4.  —Preceding unsigned comment added by DrTroublemaker (talk • contribs) 06:15, 2 August 2010 (UTC)
thread_title:  Klein Bottle as circle bundle over the circle. 

message: I have deleted the sentence
It was originally named the Kleinsche Fläche "Klein surface"; however, this was incorrectly interpreted as Kleinsche Flasche "Klein bottle," which ultimately led to the adoption of this term in the German language as well.
because I have searched Google Books and found no backup for it; German texts that would be expected to mention such a change do not, simply saying it is called Kleinsche Flasche after its inventor. Please do not restore the claim unless you have better evidence than the German Wikipedia article (equally unrefererced). Languagehat (talk) 19:45, 31 August 2010 (UTC)
replies: text: I added it back in with two refs, but even so the claim does seem somewhat shaky.  Should this be phrased as "may have originally been called..." etc?  Let me know what you think.  A13ean (talk) 00:10, 1 September 2010 (UTC)
replies: text: I'm glad you agree that it's shaky!  I softened the wording and deleted your second reference, which is simply a copy of the German Wikipedia article; I don't have access to your first reference, so I'll take your word for it that it supports the claim.  But I am reasonably certain the claim is false, because otherwise it would be part of the standard history of the word (having edited dictionaries, I am quite familiar with this stuff).  Thanks for being so reasonable! Languagehat (talk) 21:31, 1 September 2010 (UTC)
replies: text: Good catch on the second ref, I was in a rush and didn't take the time to look at it closely.  I like your current phrasing.  Thanks, A13ean (talk) 00:57, 2 September 2010 (UTC)
replies: text: The claim in this wikipedia article dates from 2005 . The book in the reference was printed in 2009. So maybe the author of the book has based his statement ("Another interpretation (unverified, and not incompatible with the previous one) claims that it comes from a bad pun, or a bad translation from the German, in which the Kleinsche Fläche (Klein surface) became the Kleinsche Flasche (Klein bottle).", p. 95,  ) on this wikipedia article. --Trigamma (talk) 17:49, 21 October 2010 (UTC)
thread_title:  Fläche/Flasche 

message: Which equivalence classes are the corners in? (As written, they're in both of the supposedly disjoint edge classes, so the construction doesn't quite work.) Does it matter, so long as they're put in one or the other? 24.220.188.43 (talk) 21:18, 22 June 2011 (UTC)
replies: text: The edges aren't disjoint, they meet at the corners!  Each edge point of the square not at a corner belongs to an equivalence class of size 2.  The four corner points belong to a single equivalence class of size 4.  The word "disjoint" doesn't appear in the description: there's no problem with (0,0)~(1,0) and (0,0)~(1,1) both being simultaneously true.  The construction looks OK as far as I can tell. Jowa fan (talk) 00:23, 23 June 2011 (UTC)
replies: text: Yes, of course. I was misreading. Thank you. 24.220.188.43 (talk) 16:20, 10 September 2011 (UTC)
thread_title:  Quotient space construction 

message: The article currently claims that a Klein bottle can be constructed froma single Mobius strip, and vice versa. Is this true? I was under the impression that this is not possible. — Cheers, Steelpillow (Talk) 12:39, 23 June 2011 (UTC)
replies: text: What was meant by this edit is unclear, but it's misleading in any case since the Klein bottle has two orientation-reversing loops.  I have removed the line.    A13ean (talk) 15:17, 24 June 2011 (UTC)
thread_title:  Construction from a Mobius strip 

message: Is there a 4space form whose 3space projections include both the '8' form and the familiar 'bottle' form? —Tamfang (talk) 21:47, 16 March 2012 (UTC)
replies: text: I'm pretty sure, but someone else can probably confirm this.  a13ean (talk) 22:21, 16 March 2012 (UTC)

text: Hmm, I thought there might have been one of these at some point but I may be mistaken.  A quick google search didn't turn up anything, and as far as I know finding a parametrization of a homeomorphism between two parametrized immersions is non-trivial, unless you get lucky (for example, there's an easy way to go from one parametrization of Boy's surface to Steiners, although the latter is not an immersion).  If no one else can find one it might be a fun summer project to try to find one.  a13ean (talk) 18:58, 6 June 2012 (UTC)

text: Perhaps 3Blue1Brown would take it on. —Tamfang (talk) 02:12, 6 February 2023 (UTC)
thread_title:  equivalence under projection? 

message: In edit "05:11, 20 February 2013‎" I updated the equations and mentioned in the comment that the previous equations were "incorrect".  On further analysis I see both the updated and previous equations produce the exact same rendering.  The old equation starts going around the sideways figure-8 at <0,0> and starts off clock wise.  The updated equations go around the same figure-8, starting at the more traditional <1,0> and going (initially) counter-clockwise. Cloudswrest (talk) 18:07, 20 February 2013 (UTC)
thread_title:  Figure 8 immersion equations. 

message: i figured out a simple way to make a "sweater" look like a klein bottle by inverting a sleeve and linking it with the other normal one. how can we mention this in the article ? 
--╦ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ╦ 13:15, 26 August 2014 (UTC)
replies: text: We can't, because it's WP:OR. --CiaPan (talk) 11:15, 27 August 2014 (UTC)

text: okay, but it stilles a good idea. --ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ (talk) 13:11, 27 August 2014 (UTC)
replies: text:  BTW, see http://www.google.com/#q=mobius+scarf&tbm=isch and http://www.google.com/#q=klein+bottle+hat&tbm=isch :) CiaPan (talk) 11:40, 28 August 2014 (UTC)

text: I'd buy one—and despite its being impossible to wear in this dimension, my fashion sense is so terrible I don't think anyone would notice. – AndyFielding (talk) 09:45, 23 February 2022 (UTC)
thread_title:  proposition 

message: I've made a ‘bottle’ rather simpler (and prettier imho) than Robert Israel's:
: r = k (2 + sin(2u) + sin(4u)/2)
: x = sin(2u)/3 - sin(4u)/5 + r cos(v) cos(u-sin(2u))
: y = cos(2u) - r cos(v) sin(u-sin(2u))
: z = r sin(v)
—Tamfang (talk) 08:54, 6 January 2015 (UTC)
replies: text: It need not be! —Tamfang (talk) 02:09, 6 February 2023 (UTC)
thread_title:  OR 

message: Is there a relation between the Klein bottle, and the Klein group?  The article gives a presentation of the group a Klein bottle satisfies, and this seems to meet the conditions I recall for the Klein group, except perhaps that the context is that of an infinite manifold.  Do I have this right?
Is there an appropriate sense in which a finite version is valid?  — Preceding unsigned comment added by 70.247.166.192 (talk) 15:39, 6 September 2015 (UTC)
thread_title:  Klein group? 

message: This does show nothing in Mathematica, is the formula wrong?
 ParametricPlot3D[{-2/
    15 cos[u] (3 cos[v] - 30 sin[u] + 90 cos^4[u] sin[u] - 
     60 cos^6[u] sin[u] + 5 cos[u] cos[v] sin[u]), -1/
    15 sin[u] (3 cos[v] - 3 cos^2[u] cos[v] - 48 cos^4[u] cos[v] + 
     48 cos^6[u] cos[v] - 60 sin[u] + 5 cos[u] cos[v] sin[u] - 
     5 cos^3[u] cos[v] sin[u] - 80 cos^5[u] cos[v] sin[u] + 
     80 cos^7[u] cos[v] sin[u]), 2/15 (3 + 5 cos[u] sin[u]) sin[v]}, {u, 0, 
  \[Pi]}, {v, 0, 2 \[Pi]}]
HermannSW  — Preceding unsigned comment added by 2A02:8071:691:6900:922B:34FF:FE4D:56C3 (talk) 12:38, 18 December 2016 (UTC)
thread_title:  Bottle shape formula wrong? 

message: I may just be a dummy, but this section makes absolutely zero sense to me.   Pariah24  ┃ ☏   23:06, 3 June 2017 (UTC)
thread_title:  Construction from rectangle 

message: It would be nice to show these simplices in one of the figures. 
Also, boundary C1=boundary C1 = 0?  I don't feel qualified to edit.
Chris2crawford (talk) 12:08, 6 October 2017 (UTC)
thread_title: P,C1,C2,D

message: The section titled Homotopy classes begins as follows:
"Regular 3D embeddings of the Klein bottle fall into three regular homotopy classes (four if one paints them). The three are represented by
# The "traditional" Klein bottle
# Left handed figure-8 Klein bottle
# Right handed figure-8 Klein bottle"
But this is ridiculous, because the Klein bottle — like every compact nonorientable surface without boundary — has no embeddings in 3-dimensional Euclidean space.
It's also entirely unclear what the comment "four if one paints them" means.
173.255.104.66 (talk) 19:29, 26 November 2020 (UTC)
replies: text: I wonder whether the paper cited is the same as this.  Carlo Séquin talks about it here, defining the problem at 13:32 and revealing the answer at 21:35. —Tamfang (talk) 17:01, 5 February 2022 (UTC)

text: Should the word embeddings be replaced with immersions? —Tamfang (talk) 17:09, 5 February 2022 (UTC)
replies: text: Certainly. D.Lazard (talk) 17:33, 5 February 2022 (UTC)

text: It means if you 2-color tile the traditional Klein bottle longitudinally, left and right, it induces chirality on it. I hope this helps. Cloudswrest (talk) 16:27, 27 February 2022 (UTC)
thread_title:  Nonsense 

message: The illustration "Time evolution of a Klein figure in xyzt-space" shows the Klein bottle evolving over time.
But it is at best completely misleading and it is at worst entirely wrong.
The illustration, actually an animation, shows the various phases of the evolution of the Klein bottle as 2-dimensional surfaces. But a 2-dimensional surface over an additional   dimension of time depicts a 3-dimensional manifold and not a surface.173.255.104.66 (talk) 19:43, 26 November 2020 (UTC)
thread_title:  The illustration "Time evolution of a Klein figure in xyzt-space" is quite wrong 

message: :Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
I don't get it. If you entered the bottle at the top and traveled down any surface of the tube, you'd end up inside the bottle, not back where you started (outside the bottle). Is this a limitation of the 3D representation? Should that be clarified? Or am I just not drinking enough coffee? Would coffee served in a Klein bottle make anything clearer? – AndyFielding (talk) 09:48, 23 February 2022 (UTC)
replies: text:   Once you got to the 'bottom' inside the bottle, start climbing up the wall. You will come to the 'ceiling' at the 'top' of the bottle, that is you'll return to your starting point, but on the (locally) 'other side' of the surface. --CiaPan (talk) 10:43, 23 February 2022 (UTC)
replies: text: You are missing nothing. I fail to see the point. ...But then, I'm NOT a mathematician. It's obvious if you filled it with water, it wouldn't leak...but it might be difficult to handle and definitely not particularly useful. 137.119.66.171 (talk) 10:33, 14 May 2022 (UTC)
thread_title:  What am I missing? 

message: I suspect that many people are confusing true Klein bottles with their representations in three-dimensional space, hence the questions about filling it with water, etc. So it would seem worthwhile to clarify this in the lead, something like this:
:Models of Klein bottles in three-dimensional space must have places where part of the surface crosses another part, but true Klein bottles do not exist in three-dimensional space, and do not have self-crossings.
Not ideal -- maybe some others can take a stab at this? --Macrakis (talk) 20:33, 15 May 2022 (UTC)
thread_title: Mention three-dimensional representations in lead

message: :
Per a video by Carlo Séquin, I believe "mirrored" is not necessary.  If you cut the 8-bagel along the "top and bottom" of the '8', you get two M-strips of the same handedness. —Tamfang (talk) 22:58, 19 March 2023 (UTC)
replies: text: 'Mirrored' was probably my WP:OR based on the observation described in #Cutting and making Klein bottles section above. If you think I made a mistake, feel free to fix it, either by reverting or by expanding the description and explaining the ambiguity. --CiaPan (talk) 11:59, 7 November 2023 (UTC)
thread_title:  if you glue the edges of two 

message: In the section « Properties » it is said that it is possible to construct a surface non embeddable in R^4, this is false using the Whitney embedding theorem, a surface being a two dimensionnal manifold, it will always be embeddable in R^4.
The example given, the spherinder Klein Bottle, is a 3-manifold and not a surface. Alexballoon (talk) 13:46, 19 August 2023 (UTC)
thread_title:  The spherinder Klein bottle is not a surface