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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.
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)
From the main page (see my comments below):
That's very interesting, but as a mathematician my response is that it's ill-formed: it merely shows that the concepts of "inside" and "outside" were not properly defined. A circle has an inside and an outside if it is embedded in a 2-dimensional plane, but a loop of string in the real world does not. However, there is certainly "string" and "not-string"... Similarly with the Kelin bottle. It may be interesting to see symbolism in it -- but that symbolism does not rest upon its mathematical properties, only on popular conceptions --
Tarquin 06:48 Aug 9, 2002 (PDT)
Dear Tarquin, thanks for your remarks.
I am using the Klein bottle and the Moebius loop as analogies; but they are not arbitrary analogies. My intuition is that the structural analogies go quite deep, and I'm very interested in formulating the analogies coherently.
Your point about an 'inside/outside' distinction as relevant when a higher dimensional form intersects a lower dimensional space is clear (as a sphere intersecting a plane in the form of a circle). Your point about what 'is' and what 'is not' an element of the given form is more relevant to what I am trying to say.
I am conceptually (logically) projecting the spatiotemporal-causal field (what I call the phenomenal field-event) into the form of such a 'surface' as that of the KB or the ML. This may be a conceptual device, but it isn't a merely arbitrary projection. I can justify it ontologically, and with reference to sciences such as physics. E.g., physics would not be possible at all, would have no foundation, if its logic did not correspond with its ontology – the set and field of events (including putative or theoretically useful posits or 'entities') with which it is concerned and with which it interacts. This field is logically a continuum – even when it exhibits the characteristics of discreteness. The discreteness itself is accountable for by the continuity of the logic in which physics is based: in the main this means mathematics, but it is not only mathematics. It is also logic in a more general (philosophical) sense, and it is (therefore) also ontology. That is why there can be experimental verifications of mathematical physical theories: because there is a logical translatability between and applicability of the mathematics with reference to the events of an experiment. This translatability I think of as a continuity: i.e., as a logical continuity. It is also (thereby) a spatiotemporal-causal (or phenomenal) continuity – just because thinking is spatiotemporal feature of the field itself. It doesn't stand or exist 'outside' of that field – which, in essence, is my point.
So, you can say that the KB and ML are 'symbols'; but they are something more than that. They are 'analogies' or 'analogues', in a rather deep sense. There is something about the 'logic' of their definition which seems to correspond (co-respond) very neatly and nicely with the structure of experience that I am far too briefly indicating here. If you can see my point, that the spatiotemporal-causal-logical continuity (I won't say 'continuum': that's a different concept; I mean here continuity, logical continuity, which supports exactly that translatability that I mentioned above) can be conceptually projected or thought as 'like' the 'surface' of a KB or ML, then we can get to the next point: namely, how do we define what is NOT a point on that 'surface'? If that 'surface' represents the logical continuity of the spatiotemporal-causal field, then what could possibly be defined as NOT on or part of that surface? From the perspective of metaphysics, the answer is: what is NOT on such a 'surface', i.e., what is NOT qualifiable in terms of spatiotemporality, is what is technically names 'transcendent'.
In terms of the topological analogy, I take this as corresponding to what I think of as the 'space of possibility' of a geometrical form (of any number of dimensions). What is such a 'space'? Is it itself already dimensional and even metrical? Or is it not so at all? Is it simply, and primordially, and quite literally, the possibility of dimensionality and metricality; of geometricality? Is it 'transcendent' with respect to all possible articulations of 'form'? (Clearly, this perspective does not conform to that notion of General Relativity that takes logical (or mathematical) 4-dimensionality as representing an actual 4-D 'substrate in which 'mass' and 'events' are somehow embedded, or against which they appear as against some kind of inhrently metrical backdrop! To the contrary, such a 4-dimensionality is simply itself a logical feature of the field of eventfulness. The 'space of possibility' that I am referring to is metaphysically and logically 'prior' to this.)
This is what I'm getting at with the argument that the KB is a very interesting and neat analogy for the structure of consciousness. Let me use, first, your point of the intersection of a sphere and a plane. Suppose that the spatiotemporal field-event is a continuity without an 'outside' (this shouldn't be an unfamiliar concept: isn't that the way that the 4D continuum is defined?). And suppose that an individual's embodied experience is just like a 'slice' through this continuum - except that the 'continuum' does not, on this view, 'exist' like a 4D 'entity'; rather, the 4D-ness of the field is a logical feauture of it which can be represented topologically, but which does not 'exist' topologically, if you see what I mean. That individual's experience, then, would exhibit (to the individual) the characteristics of a field that was divided between 'inside' and 'outside' at some apparent, putative 'boundary'. But if the individual sought to determine just where that boundary 'is' - whether 'conceptually' and/or 'empirically', it doesn't finally matter, as the two are logically continuous procedures, as should be evident from the nature of the schema and the analogy - they would simply be unable to do so. All that they would find is a continuity.
From here, we can get to your other point, the more interesting and important one, concerning what is 'part' of the 'surface' and what is not. This has one meaning (solution), if we presuppose a metric or co-ordinate space, for example, according to which we can define (presumably by some formula) what co-ordinates belong to the 'surface' and what co-ordinates do not. But what if we take the mathematical analogy as an analogy (or as a logical-conceptual model), and state that all possible co-ordinates, of any number of dimensions, are generated by principles that are only effective within the differential spacetime-causal field itself: that is to say, where there is logic and mathematics, there must be (primordial logical) difference; without such difference, there could be no logic and no mathematics, and no definition of topology, let alone of 'space' or 'time' (of spatiotemporal differentiation). What this means, in sum, is that any 'point-moment' that can in any respect and according to any number of dimensions (greater than zero) be spatiotemporally 'located' ('co-ordinated') is thereby immediately implicated in the spatiotemporal field; hence, is already thereby a point-moment of the 'surface' in question.
In other words, to NOT be on this 'surface' (the 'surface' that here 'represents' the logical continuity of 'spacetime' itself) entails to NOT be in any sense or respect qualifiable spatiotemporally: to be, technically speaking, transcendent to spatiotemporality (to the 'surface' that 'represents' the the logical continuity of spatiotemporal-causal field). That 'transcendent' is equivalent, here, to what I called the 'space of possibility' of any spatiotemporal dimensionality whatsoever. In that it is transcendent in this absolute sense, it is also obviously transcendent in the sense that it is absolutely non-geometrical and non-topological; and, yes, even 'non-logical'; but please don't confuse this with any popular notions of 'illogical' or the like; the transcendent is just transcendent per se. It is the metaphysical possibility of 'logic', 'spatiotemporality', 'phenomena'.
The point of the argument, and its recourse to the analogy of the KB and the ML, therefore, is that our conscious experience is in fact structured just in this way. The phenomenal (spatiotemporal) field, which is logically continuous (as we know from detailed experience) is 'just like' a 'single-surface' topological form (such as the KB or ML), but, from the spatiotemporally localised-limited perspective of an 'embodied being', it appears (for reasons I won't go into here) to be inherently demarcated into two divided domains: the 'internal' and the 'external', concepts which often are superimposed upon the 'mental' and the 'physical', the 'private' and the 'public', and so on. However, under a thorough-going phenomenological analysis, this turns out to be quite erroneous. And the analogy of the KB and ML are a neat device for indicating the nature of such an analysis. But I think that's enough for now. I'd like to hear your comments; especially if you can see a way for clarifying - or else dismissing - the functionality of the analogy.
On the other hand, you could take this as an article in the encyclopedia, if you could find a useful title for it. From my point of view, this is a 'theory' that has a good deal of experimental (phenomenological) proof, already. Monk 0
I'm not sure exactly how to write up something about the other form of the klein bottle, but here is a link to a website that describes both types. — siro χ o 01:20, Jul 31, 2004 (UTC)
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.
[[User:Sverdrup| ❝Sverdrup❞]] 23:36, 12 Aug 2004 (UTC)
Anyone here own an Acme Klein bottle and a camera? This article could use a good photograph.
Also, could we please take out the gigantic parametric equations? I seriously doubt that anyone ever actually uses them, and even if someone somewhere has needed them, they don't seem necessary to an encyclopedia article. "Encyclopedias synthesize and highlight" ( Indrian). dbenbenn | talk 05:15, 29 Jan 2005 (UTC)
Perhaps someone skilled in Mathematica could add the figure-8 immersion? See the MathWorld reference for a picture to work from. dbenbenn | talk 14:56, 3 Mar 2005 (UTC)
Hoping not to sound rude, I must say that I disagree with this claimed immersion of the Klein bottle. Could anybody supply an exact link to it in MathWorld? I claim that it is orientable and topologically equivalent to the toroidal surface. The 1/2 twist in it does not affect the topology; it is a metric feature for a coordinate atlas choice. Besides, it is graphically manifest that it distinguishes inside and outside spaces (which indeed the twist could not change). I'm amazed that this has survive unchallenged since March 2005 on Wiki (and since when on MathWorld).
Just my 2 cents, 37.180.43.216 ( talk) 12:17, 11 December 2014 (UTC) Chris
I have uploaded some new images to Wikimedia commons. The first is a slight different immersion of the Klein bottle into R3 and the second is the figure-eight version requested above (cut-aways added for clarity). I have included the parameterizations of these immersions on the image description page on the commons. These parameterizations are much simpler than those used in this article (IMHO).
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A parameterization for an embedding of the Klein bottle into R4 = C2 is given by
where a > b > 0 are constants and u,v run from 0 to 2π. Obviously I can't draw this one.
I don't have time to edit this article right now. So someone should feel free to incorporate these images and their parameterizations into the article. -- Fropuff 18:23, 2005 Mar 3 (UTC)
Can we put the cutout image of the figure 8 immersion in the article beneath the current one? I didn't understand the figure 8 immersion until I saw the cutout diagram on this talk page, it makes it much clearer. I'd put it in the article myself but the image syntax gives me nightmares. Maelin ( Talk | Contribs) 08:22, 29 May 2007 (UTC)
How would you express a torus in a form most closely analogous to Fropuff's parametrization? — Tamfang ( talk) 21:41, 16 March 2012 (UTC)
Should we mention that the Klein bottle arises as the connected sum of three copies of ?
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:
Because that describes and depicts a regular torus, not a Klein bottle. http://mathworld.wolfram.com/KleinBottle.html for more. 209.6.124.246 16:31, 13 September 2005 (UTC)EricN
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)
Being in or out of love is somewhat easier to take if one remembers to search for the beloved along two dimensional manifolds. Or try to escape through the crawl space. After all, since space-time has no intrinsic distinction between inside and out, all such reliable distinctions must be made of substances or solid object. In the case of living beings, that means molecules, membranes, shells, skin or clothing. Mathematics is a living, vital field! SyntheticET ( talk) 22:44, 8 November 2009 (UTC)
What happens if you pour water (or some other liquid) into the "opening" of the bottle? -- Jfruh 21:30, 22 February 2006 (UTC)
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? -- Cronholm 144 22:37, 17 July 2007 (UTC)
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)
Possibly someone confused the orientable and non-orientable genus, and used the wrong formula For this gives 'genus' 1. -- CiaPan 17:39, 22 May 2006 (UTC)
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)
Question, how is this different from the Mobius Tube parameterization in the main article? Cloudswrest ( talk) 14:01, 29 July 2016 (UTC)
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.
--Fourth dimension ?-- I'm pretty sure the first dimension is width, second is length, third is width, and fourth is time.
See the fourth dimension page, especially the "The fourth spatial dimension and orthogonality" section, to see what is meant by 4D coordinates. DMacks 00:59, 2 May 2007 (UTC)
If we take a Klein bottle (see the picture) and cut a round hole in the "wall" of the bottle in the place where the "handle" intersects the wall, we obtain a non-orientable surface with one boundary component and without self-interections. What is it? It is not Mobius strip: according to Mobius strip article, gluing a disk to a Mobius strip produces the real projective plane. So, what is it? `' Míkka 23:25, 17 July 2007 (UTC)
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)
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)
here a presentation:
What would happen if you poured water into a klien bottle? It boggles my mind. Twinkie Ding Dong ( talk) 03:02, 22 January 2009 (UTC)
First off, I found reference to the Klein Bottle in 'The Number of the Beast' by Robert Heinlein. My question is this, what is the purpose of a Klein Bottle? 1:50am 03/11/09 Arizona, USA —Preceding unsigned comment added by 65.103.204.18 ( talk) 08:53, 11 March 2009 (UTC)
Acme sells a Klein Stein beer mug. —Preceding unsigned comment added by SyntheticET ( talk • contribs) 22:36, 8 November 2009 (UTC)
o---------------o / |\ / o---------o | \ / / \ | | \ / / \ o | o / / \ / o | o o o---/ / \ | | |\ / o / \ | | | \ / | / \| | | \ / o--+--o o | | o / |. \ / | | | / o . \ / | o----o / \ . \ / | / \ .--o o-----------o \| \ \ o \ \ / \ \ / o-----------o
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:
where
for 0 ≤ u < 2π and 0 ≤ v < 2π.
The problem is g(0). I found:
According to the article, 0≤u. Now, we cannot compute and with , because g(u) is dividing some terms in their formulae.
Eviruena ( talk) 21:03, 28 January 2009 (UTC)
It seems that the references to trivia (see WP:TRIVIA) are a distraction to the actual subject of this article, which is a mathematical concept. Should these references remain or be removed? Spectre9 ( talk) 01:57, 4 February 2009 (UTC)
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Designers of high reliability closed systems such as submarines, spacecraft, underwater tunnels, ferries, and fuel tanks for gasoline, liquid hydrogen, fluorine or other gases, etc., must take special account of the problem Klein's bottle presents. No mission-critical vessel that must be absolutely sealed can be inspected merely by examining its surfaces for the edges of holes. One can depict a scenario where a Klein bottle type of accident might occur. A ship, craft or tunnel containing interior tubing (submarines have a great deal of that) with legitimate openings to the outside or the inside of the ship or tunnel must be carefully planned. A tube may have a valve that opens or closes that tube to fluid transport and is closed during construction and testing. If one end of the tube is to the interior, and the other to the exterior, it could open the valve during operation (combat, flight or occupancy) and then be flooded with water or drained of air. A rule, not to construct tubes with only one control valve in the interior of a sealed volume to the exterior environment, is of course a much, much too simple minded rule to handle the vast number of problems that can exist in modern complex systems. No fundamental distinction exists between the interior and exterior of a volume. A vortex at the center of a galaxy does not distinguish between north and south directions until spin differentiation occurs in charges interior to each star drawn into the whirling vortex. When charges start to move, positive charges move one way, negative the other and the star eventually explodes. A similar condition appears to exist in the photon, which is a quantum h of action moving along at the speed of light c, and gradually losing energy to wave-time and momentum to wavelength.
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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)
"Clean" or "Kline"? Richard W.M. Jones ( talk) 09:58, 7 July 2010 (UTC)
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)
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. citation needed
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)
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)
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)
Would it be appropriate to add the Lawson Klein bottle to this article, or would it merit an article of it's own ? (I think it needs a mathematician to judge !)
There are some examples at e.g.: http://vimeo.com/2495945
Darkman101 ( talk) 18:55, 11 September 2011 (UTC)
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)
Presumably there's a continuous deformation between them? An animation would be nice. — Tamfang ( talk) 18:33, 6 June 2012 (UTC)
In the video that I linked yesterday, Carlo H. Séquin mentions that the Lawson surface is homotopy-equivalent to the ‘bagel’. I would love to see a movie … — Tamfang ( talk) 20:46, 5 February 2022 (UTC)
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)
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)
I've made a ‘bottle’ rather simpler (and prettier imho) than Robert Israel's:
— Tamfang ( talk) 08:54, 6 January 2015 (UTC)
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)
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)
I may just be a dummy, but this section makes absolutely zero sense to me. Pariah24 ┃ ☏ 23:06, 3 June 2017 (UTC)
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)
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
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)
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)
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)
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:
Not ideal -- maybe some others can take a stab at this? -- Macrakis ( talk) 20:33, 15 May 2022 (UTC)
There needs to be something more said about Pinch point (mathematics).
I would like to flip the introduction, to make it more useful for the casual reader, who currently has to get through a bit of (to them) intimidating gobbledygook to get to the Klein bottle's salient feature, that it is one-sided. In other words, change the lead-in from sounding like a mathematics article to being a general-encyclopedia article, hopefully making it more useful for the many people who come here after a google search.
Basically I would move / rewrite the layperson's description to the first sentence, creating a short paragraph that includes the Mobius strip mention (which is very helpful for lay understanding). I'd move with the mathematical definition and details to the second and subsequent paragraphs. I wouldn't change anything other than the introduction.
Any thoughts? - DavidWBrooks ( talk) 18:41, 7 March 2023 (UTC)
"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."including a couple examples of other nonorientable surfaces, and one which is orientable. That seems reasonably approachable for someone without a mathematical background. If you're worried about the casual reader's inability to skip over a technical sentence to get to the rest, ehh. 35.139.154.158 ( talk) 14:14, 16 March 2023 (UTC)
Anybody else want to come into this - help me respond to an editor who thinks that an actual physical Klein bottle (thanks, Cliff Stoll!), thousands of which exist around the world, somehow aren't actually Klein bottles because ... um, not sure why. Particularly since there has long been a photo of a similar glass construction, although without demonstrating the ability to hold liquid. - DavidWBrooks ( talk) 14:44, 18 March 2023 (UTC)
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)
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)
A Klein bottle is a 2D manifold (sheet) that is a 4D shape. Just like a cup is a 2D manifold that is a 3D shape. In fact, Cliff Stoll's website explicitly states this. His "Klein bottles" are MODELS of Klein bottles, not actual Klein bottles. In mathematical language, glass "Klein bottles" that you can purchase are 3D "immersions" of a 4D object. D.Lazard keeps undoing my edit that makes this clear in the article, even though it is well-understood that Klein bottles are 4D shapes. (As an aside, I'm quite confused by his understanding here. I said that I don't think he understands the concept of the difference between something made of a 2D manifold and being a 3D object, and he accused me of WP:PA. So bizarre, especially considering that his original undo of my edit accused me of being pedantic, which is clearly an actual WP:PA. But whatevs.) Nandor72 ( talk) 00:58, 7 November 2023 (UTC) This post was misplaced in the middle of an older discussion than the edit war that motivated it. So, I move it in a new section. D.Lazard ( talk) 10:22, 7 November 2023 (UTC)
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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.
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)
From the main page (see my comments below):
That's very interesting, but as a mathematician my response is that it's ill-formed: it merely shows that the concepts of "inside" and "outside" were not properly defined. A circle has an inside and an outside if it is embedded in a 2-dimensional plane, but a loop of string in the real world does not. However, there is certainly "string" and "not-string"... Similarly with the Kelin bottle. It may be interesting to see symbolism in it -- but that symbolism does not rest upon its mathematical properties, only on popular conceptions --
Tarquin 06:48 Aug 9, 2002 (PDT)
Dear Tarquin, thanks for your remarks.
I am using the Klein bottle and the Moebius loop as analogies; but they are not arbitrary analogies. My intuition is that the structural analogies go quite deep, and I'm very interested in formulating the analogies coherently.
Your point about an 'inside/outside' distinction as relevant when a higher dimensional form intersects a lower dimensional space is clear (as a sphere intersecting a plane in the form of a circle). Your point about what 'is' and what 'is not' an element of the given form is more relevant to what I am trying to say.
I am conceptually (logically) projecting the spatiotemporal-causal field (what I call the phenomenal field-event) into the form of such a 'surface' as that of the KB or the ML. This may be a conceptual device, but it isn't a merely arbitrary projection. I can justify it ontologically, and with reference to sciences such as physics. E.g., physics would not be possible at all, would have no foundation, if its logic did not correspond with its ontology – the set and field of events (including putative or theoretically useful posits or 'entities') with which it is concerned and with which it interacts. This field is logically a continuum – even when it exhibits the characteristics of discreteness. The discreteness itself is accountable for by the continuity of the logic in which physics is based: in the main this means mathematics, but it is not only mathematics. It is also logic in a more general (philosophical) sense, and it is (therefore) also ontology. That is why there can be experimental verifications of mathematical physical theories: because there is a logical translatability between and applicability of the mathematics with reference to the events of an experiment. This translatability I think of as a continuity: i.e., as a logical continuity. It is also (thereby) a spatiotemporal-causal (or phenomenal) continuity – just because thinking is spatiotemporal feature of the field itself. It doesn't stand or exist 'outside' of that field – which, in essence, is my point.
So, you can say that the KB and ML are 'symbols'; but they are something more than that. They are 'analogies' or 'analogues', in a rather deep sense. There is something about the 'logic' of their definition which seems to correspond (co-respond) very neatly and nicely with the structure of experience that I am far too briefly indicating here. If you can see my point, that the spatiotemporal-causal-logical continuity (I won't say 'continuum': that's a different concept; I mean here continuity, logical continuity, which supports exactly that translatability that I mentioned above) can be conceptually projected or thought as 'like' the 'surface' of a KB or ML, then we can get to the next point: namely, how do we define what is NOT a point on that 'surface'? If that 'surface' represents the logical continuity of the spatiotemporal-causal field, then what could possibly be defined as NOT on or part of that surface? From the perspective of metaphysics, the answer is: what is NOT on such a 'surface', i.e., what is NOT qualifiable in terms of spatiotemporality, is what is technically names 'transcendent'.
In terms of the topological analogy, I take this as corresponding to what I think of as the 'space of possibility' of a geometrical form (of any number of dimensions). What is such a 'space'? Is it itself already dimensional and even metrical? Or is it not so at all? Is it simply, and primordially, and quite literally, the possibility of dimensionality and metricality; of geometricality? Is it 'transcendent' with respect to all possible articulations of 'form'? (Clearly, this perspective does not conform to that notion of General Relativity that takes logical (or mathematical) 4-dimensionality as representing an actual 4-D 'substrate in which 'mass' and 'events' are somehow embedded, or against which they appear as against some kind of inhrently metrical backdrop! To the contrary, such a 4-dimensionality is simply itself a logical feature of the field of eventfulness. The 'space of possibility' that I am referring to is metaphysically and logically 'prior' to this.)
This is what I'm getting at with the argument that the KB is a very interesting and neat analogy for the structure of consciousness. Let me use, first, your point of the intersection of a sphere and a plane. Suppose that the spatiotemporal field-event is a continuity without an 'outside' (this shouldn't be an unfamiliar concept: isn't that the way that the 4D continuum is defined?). And suppose that an individual's embodied experience is just like a 'slice' through this continuum - except that the 'continuum' does not, on this view, 'exist' like a 4D 'entity'; rather, the 4D-ness of the field is a logical feauture of it which can be represented topologically, but which does not 'exist' topologically, if you see what I mean. That individual's experience, then, would exhibit (to the individual) the characteristics of a field that was divided between 'inside' and 'outside' at some apparent, putative 'boundary'. But if the individual sought to determine just where that boundary 'is' - whether 'conceptually' and/or 'empirically', it doesn't finally matter, as the two are logically continuous procedures, as should be evident from the nature of the schema and the analogy - they would simply be unable to do so. All that they would find is a continuity.
From here, we can get to your other point, the more interesting and important one, concerning what is 'part' of the 'surface' and what is not. This has one meaning (solution), if we presuppose a metric or co-ordinate space, for example, according to which we can define (presumably by some formula) what co-ordinates belong to the 'surface' and what co-ordinates do not. But what if we take the mathematical analogy as an analogy (or as a logical-conceptual model), and state that all possible co-ordinates, of any number of dimensions, are generated by principles that are only effective within the differential spacetime-causal field itself: that is to say, where there is logic and mathematics, there must be (primordial logical) difference; without such difference, there could be no logic and no mathematics, and no definition of topology, let alone of 'space' or 'time' (of spatiotemporal differentiation). What this means, in sum, is that any 'point-moment' that can in any respect and according to any number of dimensions (greater than zero) be spatiotemporally 'located' ('co-ordinated') is thereby immediately implicated in the spatiotemporal field; hence, is already thereby a point-moment of the 'surface' in question.
In other words, to NOT be on this 'surface' (the 'surface' that here 'represents' the logical continuity of 'spacetime' itself) entails to NOT be in any sense or respect qualifiable spatiotemporally: to be, technically speaking, transcendent to spatiotemporality (to the 'surface' that 'represents' the the logical continuity of spatiotemporal-causal field). That 'transcendent' is equivalent, here, to what I called the 'space of possibility' of any spatiotemporal dimensionality whatsoever. In that it is transcendent in this absolute sense, it is also obviously transcendent in the sense that it is absolutely non-geometrical and non-topological; and, yes, even 'non-logical'; but please don't confuse this with any popular notions of 'illogical' or the like; the transcendent is just transcendent per se. It is the metaphysical possibility of 'logic', 'spatiotemporality', 'phenomena'.
The point of the argument, and its recourse to the analogy of the KB and the ML, therefore, is that our conscious experience is in fact structured just in this way. The phenomenal (spatiotemporal) field, which is logically continuous (as we know from detailed experience) is 'just like' a 'single-surface' topological form (such as the KB or ML), but, from the spatiotemporally localised-limited perspective of an 'embodied being', it appears (for reasons I won't go into here) to be inherently demarcated into two divided domains: the 'internal' and the 'external', concepts which often are superimposed upon the 'mental' and the 'physical', the 'private' and the 'public', and so on. However, under a thorough-going phenomenological analysis, this turns out to be quite erroneous. And the analogy of the KB and ML are a neat device for indicating the nature of such an analysis. But I think that's enough for now. I'd like to hear your comments; especially if you can see a way for clarifying - or else dismissing - the functionality of the analogy.
On the other hand, you could take this as an article in the encyclopedia, if you could find a useful title for it. From my point of view, this is a 'theory' that has a good deal of experimental (phenomenological) proof, already. Monk 0
I'm not sure exactly how to write up something about the other form of the klein bottle, but here is a link to a website that describes both types. — siro χ o 01:20, Jul 31, 2004 (UTC)
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.
[[User:Sverdrup| ❝Sverdrup❞]] 23:36, 12 Aug 2004 (UTC)
Anyone here own an Acme Klein bottle and a camera? This article could use a good photograph.
Also, could we please take out the gigantic parametric equations? I seriously doubt that anyone ever actually uses them, and even if someone somewhere has needed them, they don't seem necessary to an encyclopedia article. "Encyclopedias synthesize and highlight" ( Indrian). dbenbenn | talk 05:15, 29 Jan 2005 (UTC)
Perhaps someone skilled in Mathematica could add the figure-8 immersion? See the MathWorld reference for a picture to work from. dbenbenn | talk 14:56, 3 Mar 2005 (UTC)
Hoping not to sound rude, I must say that I disagree with this claimed immersion of the Klein bottle. Could anybody supply an exact link to it in MathWorld? I claim that it is orientable and topologically equivalent to the toroidal surface. The 1/2 twist in it does not affect the topology; it is a metric feature for a coordinate atlas choice. Besides, it is graphically manifest that it distinguishes inside and outside spaces (which indeed the twist could not change). I'm amazed that this has survive unchallenged since March 2005 on Wiki (and since when on MathWorld).
Just my 2 cents, 37.180.43.216 ( talk) 12:17, 11 December 2014 (UTC) Chris
I have uploaded some new images to Wikimedia commons. The first is a slight different immersion of the Klein bottle into R3 and the second is the figure-eight version requested above (cut-aways added for clarity). I have included the parameterizations of these immersions on the image description page on the commons. These parameterizations are much simpler than those used in this article (IMHO).
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A parameterization for an embedding of the Klein bottle into R4 = C2 is given by
where a > b > 0 are constants and u,v run from 0 to 2π. Obviously I can't draw this one.
I don't have time to edit this article right now. So someone should feel free to incorporate these images and their parameterizations into the article. -- Fropuff 18:23, 2005 Mar 3 (UTC)
Can we put the cutout image of the figure 8 immersion in the article beneath the current one? I didn't understand the figure 8 immersion until I saw the cutout diagram on this talk page, it makes it much clearer. I'd put it in the article myself but the image syntax gives me nightmares. Maelin ( Talk | Contribs) 08:22, 29 May 2007 (UTC)
How would you express a torus in a form most closely analogous to Fropuff's parametrization? — Tamfang ( talk) 21:41, 16 March 2012 (UTC)
Should we mention that the Klein bottle arises as the connected sum of three copies of ?
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:
Because that describes and depicts a regular torus, not a Klein bottle. http://mathworld.wolfram.com/KleinBottle.html for more. 209.6.124.246 16:31, 13 September 2005 (UTC)EricN
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)
Being in or out of love is somewhat easier to take if one remembers to search for the beloved along two dimensional manifolds. Or try to escape through the crawl space. After all, since space-time has no intrinsic distinction between inside and out, all such reliable distinctions must be made of substances or solid object. In the case of living beings, that means molecules, membranes, shells, skin or clothing. Mathematics is a living, vital field! SyntheticET ( talk) 22:44, 8 November 2009 (UTC)
What happens if you pour water (or some other liquid) into the "opening" of the bottle? -- Jfruh 21:30, 22 February 2006 (UTC)
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? -- Cronholm 144 22:37, 17 July 2007 (UTC)
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)
Possibly someone confused the orientable and non-orientable genus, and used the wrong formula For this gives 'genus' 1. -- CiaPan 17:39, 22 May 2006 (UTC)
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)
Question, how is this different from the Mobius Tube parameterization in the main article? Cloudswrest ( talk) 14:01, 29 July 2016 (UTC)
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.
--Fourth dimension ?-- I'm pretty sure the first dimension is width, second is length, third is width, and fourth is time.
See the fourth dimension page, especially the "The fourth spatial dimension and orthogonality" section, to see what is meant by 4D coordinates. DMacks 00:59, 2 May 2007 (UTC)
If we take a Klein bottle (see the picture) and cut a round hole in the "wall" of the bottle in the place where the "handle" intersects the wall, we obtain a non-orientable surface with one boundary component and without self-interections. What is it? It is not Mobius strip: according to Mobius strip article, gluing a disk to a Mobius strip produces the real projective plane. So, what is it? `' Míkka 23:25, 17 July 2007 (UTC)
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)
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)
here a presentation:
What would happen if you poured water into a klien bottle? It boggles my mind. Twinkie Ding Dong ( talk) 03:02, 22 January 2009 (UTC)
First off, I found reference to the Klein Bottle in 'The Number of the Beast' by Robert Heinlein. My question is this, what is the purpose of a Klein Bottle? 1:50am 03/11/09 Arizona, USA —Preceding unsigned comment added by 65.103.204.18 ( talk) 08:53, 11 March 2009 (UTC)
Acme sells a Klein Stein beer mug. —Preceding unsigned comment added by SyntheticET ( talk • contribs) 22:36, 8 November 2009 (UTC)
o---------------o / |\ / o---------o | \ / / \ | | \ / / \ o | o / / \ / o | o o o---/ / \ | | |\ / o / \ | | | \ / | / \| | | \ / o--+--o o | | o / |. \ / | | | / o . \ / | o----o / \ . \ / | / \ .--o o-----------o \| \ \ o \ \ / \ \ / o-----------o
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:
where
for 0 ≤ u < 2π and 0 ≤ v < 2π.
The problem is g(0). I found:
According to the article, 0≤u. Now, we cannot compute and with , because g(u) is dividing some terms in their formulae.
Eviruena ( talk) 21:03, 28 January 2009 (UTC)
It seems that the references to trivia (see WP:TRIVIA) are a distraction to the actual subject of this article, which is a mathematical concept. Should these references remain or be removed? Spectre9 ( talk) 01:57, 4 February 2009 (UTC)
Extended content
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Designers of high reliability closed systems such as submarines, spacecraft, underwater tunnels, ferries, and fuel tanks for gasoline, liquid hydrogen, fluorine or other gases, etc., must take special account of the problem Klein's bottle presents. No mission-critical vessel that must be absolutely sealed can be inspected merely by examining its surfaces for the edges of holes. One can depict a scenario where a Klein bottle type of accident might occur. A ship, craft or tunnel containing interior tubing (submarines have a great deal of that) with legitimate openings to the outside or the inside of the ship or tunnel must be carefully planned. A tube may have a valve that opens or closes that tube to fluid transport and is closed during construction and testing. If one end of the tube is to the interior, and the other to the exterior, it could open the valve during operation (combat, flight or occupancy) and then be flooded with water or drained of air. A rule, not to construct tubes with only one control valve in the interior of a sealed volume to the exterior environment, is of course a much, much too simple minded rule to handle the vast number of problems that can exist in modern complex systems. No fundamental distinction exists between the interior and exterior of a volume. A vortex at the center of a galaxy does not distinguish between north and south directions until spin differentiation occurs in charges interior to each star drawn into the whirling vortex. When charges start to move, positive charges move one way, negative the other and the star eventually explodes. A similar condition appears to exist in the photon, which is a quantum h of action moving along at the speed of light c, and gradually losing energy to wave-time and momentum to wavelength.
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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)
"Clean" or "Kline"? Richard W.M. Jones ( talk) 09:58, 7 July 2010 (UTC)
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)
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. citation needed
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)
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)
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)
Would it be appropriate to add the Lawson Klein bottle to this article, or would it merit an article of it's own ? (I think it needs a mathematician to judge !)
There are some examples at e.g.: http://vimeo.com/2495945
Darkman101 ( talk) 18:55, 11 September 2011 (UTC)
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)
Presumably there's a continuous deformation between them? An animation would be nice. — Tamfang ( talk) 18:33, 6 June 2012 (UTC)
In the video that I linked yesterday, Carlo H. Séquin mentions that the Lawson surface is homotopy-equivalent to the ‘bagel’. I would love to see a movie … — Tamfang ( talk) 20:46, 5 February 2022 (UTC)
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)
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)
I've made a ‘bottle’ rather simpler (and prettier imho) than Robert Israel's:
— Tamfang ( talk) 08:54, 6 January 2015 (UTC)
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)
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)
I may just be a dummy, but this section makes absolutely zero sense to me. Pariah24 ┃ ☏ 23:06, 3 June 2017 (UTC)
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)
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
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)
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)
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)
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:
Not ideal -- maybe some others can take a stab at this? -- Macrakis ( talk) 20:33, 15 May 2022 (UTC)
There needs to be something more said about Pinch point (mathematics).
I would like to flip the introduction, to make it more useful for the casual reader, who currently has to get through a bit of (to them) intimidating gobbledygook to get to the Klein bottle's salient feature, that it is one-sided. In other words, change the lead-in from sounding like a mathematics article to being a general-encyclopedia article, hopefully making it more useful for the many people who come here after a google search.
Basically I would move / rewrite the layperson's description to the first sentence, creating a short paragraph that includes the Mobius strip mention (which is very helpful for lay understanding). I'd move with the mathematical definition and details to the second and subsequent paragraphs. I wouldn't change anything other than the introduction.
Any thoughts? - DavidWBrooks ( talk) 18:41, 7 March 2023 (UTC)
"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."including a couple examples of other nonorientable surfaces, and one which is orientable. That seems reasonably approachable for someone without a mathematical background. If you're worried about the casual reader's inability to skip over a technical sentence to get to the rest, ehh. 35.139.154.158 ( talk) 14:14, 16 March 2023 (UTC)
Anybody else want to come into this - help me respond to an editor who thinks that an actual physical Klein bottle (thanks, Cliff Stoll!), thousands of which exist around the world, somehow aren't actually Klein bottles because ... um, not sure why. Particularly since there has long been a photo of a similar glass construction, although without demonstrating the ability to hold liquid. - DavidWBrooks ( talk) 14:44, 18 March 2023 (UTC)
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)
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)
A Klein bottle is a 2D manifold (sheet) that is a 4D shape. Just like a cup is a 2D manifold that is a 3D shape. In fact, Cliff Stoll's website explicitly states this. His "Klein bottles" are MODELS of Klein bottles, not actual Klein bottles. In mathematical language, glass "Klein bottles" that you can purchase are 3D "immersions" of a 4D object. D.Lazard keeps undoing my edit that makes this clear in the article, even though it is well-understood that Klein bottles are 4D shapes. (As an aside, I'm quite confused by his understanding here. I said that I don't think he understands the concept of the difference between something made of a 2D manifold and being a 3D object, and he accused me of WP:PA. So bizarre, especially considering that his original undo of my edit accused me of being pedantic, which is clearly an actual WP:PA. But whatevs.) Nandor72 ( talk) 00:58, 7 November 2023 (UTC) This post was misplaced in the middle of an older discussion than the edit war that motivated it. So, I move it in a new section. D.Lazard ( talk) 10:22, 7 November 2023 (UTC)