Talk:Klein bottle

Inside vs. Outside

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)

2002

From the main page (see my comments below):

The Klein bottle is important – if you need an image to help you solve a 'philosophical problem' - because it gives an actual example of a surface that is continuous and unitary and yet appears to display the features of 'inside/outside'. The age-old 'problem' of the relationship between 'one and many' requires that difference is possible. How do you get One to self-differentiate? An apparent distinction between 'inside/outside' would be one way. In other words, in terms of your actual experience, this amounts to the distinction between yourself (as the good old classical 'subject') and the 'outside world' (as the good old classical 'object').. Like the Moebis Loop, the Klein Bottle indicates, in concrete (and mathematical – that's important, because we need both kinds of discourse) terms, how you can have the logical appearance of 'twoness' (duality) where you actually have only a unity and continuity. Of course, Moebius Loops and Klein Bottles are still 'objects' of logic, of logical intellect:and that means (paradoxically, as it would seem) that they are 'objects' constructed by dualistic thinking that indicate what must ultimately be transcendent (and this means: transcendent to dualistic thinking itself).. So, in effect, they are excellent meditative devices, a bit like Zen koans, but their paradoxical value for meditation is obviously not mathematical (mathematically, there is no great problem about them, no paradox) but comes into play when you begin to ask: if the experience of my 'self' is like one apparent surface of a Moebius Loop or a Klein Bottle, and if the experience of my 'world' is like the other apparent surface of one of these topological figures, then what does this tell me about what my experience of self/world is really all about? One more thing: a 'shape' in space means that there must be a 'space' for the 'shape': what, in experience, is this 'space', if it is not the 'shape' itself, but prior to it?

The Sphere for a long time was a symbolic figure of Totality; but a Sphere actually presupposes two incommensurable surfaces. The Moebius Loop and the Klein Bottle are far more itneresting and useful symbols for Totality (or Unity, Oneness).. Make of this what you will! Happy journeys on the one surface of being.

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)

A reply in dialogue

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.

However Wikipedia is an encylopedia, not an experiment in progress. --rmhermen

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

Figure 8 Klein bottle

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. —siroχo 01:20, Jul 31, 2004 (UTC)

Unused image

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.

Sketch of a Klein bottle

[[User:Sverdrup|❝Sverdrup❞]] 23:36, 12 Aug 2004 (UTC)

I removed some more images today. dbenbenn | talk 14:57, 3 Mar 2005 (UTC)

Considering these are objects that disappear into themselves, I suppose a certain irony is inescapable. – AndyFielding (talk) 09:49, 23 February 2022 (UTC)

Acme Klein bottle?

Anyone here own an Acme Klein bottle and a camera? This article could use a good photograph.

I own one! But no camera. Maybe I borrow one? -Lethe | Talk 08:53, Mar 3, 2005 (UTC)

Sure, you can borrow mine. It's in Vail, Colorado... :) dbenbenn | talk 14:02, 3 Mar 2005 (UTC)

I got a camera, and now you have a picture, though I think the article is too cluttered with pictures, at this stage. -lethe ^talk 09:45, 8 January 2006 (UTC)

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)

One serious problem with that parametrization is that it describes not a Klein bottle, but rather an immersion of a Klein bottle in R^3. This immersion is not really a Klein bottle. In fact, I believe a parametric description of a Klein bottle could be useful, and I think I've seen such descriptions that are far more succinct (they are, of course, embeddings in R^4, rather than immersions in R^3). So. I agree that this parametrization should be removed. But let's replace it with something nicer instead of just deleting it. -Lethe | Talk 09:09, Mar 3, 2005 (UTC)

Yes, good idea. Can you dig up your succinct parametrization? dbenbenn | talk 14:02, 3 Mar 2005 (UTC)

Figure 8 immersion

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

The 1/2 twist is exactly the same half twist as seen in the Mobius strip. The "figure-8" is simply a means to bring together (identify) the two side edges of the Mobius strip without any tears or discontinuities, hence turning it into a Klein bottle. To see that it is one sided, start on the inside of the outer lobe of the figure-8 and walk longitudinally around the "torus". Once you come back around, due to the half twist, you'll be in the inside of the inner lobe of the figure-8. Now walk along the figure-8 and, after you pass through the self intersection, you end up on the outside of the outer lobe of the figure-8. QED.

Hoping not to sound rude, but I am amazed that someone would challenge such a long standing, patently obvious immersion. Just my two cents. Cloudswrest (talk) 15:18, 11 December 2014 (UTC)

One-sidedness is subtly demonstrated in another way in my model [1], made of a single strand which cuts the (u,v) rectangle obliquely. If the manifold were orientable, this oblique path would not cross itself. —Tamfang (talk) 05:26, 2 January 2015 (UTC)

New immersions, pictures, and parameterizations

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).

A parameterization for an embedding of the Klein bottle into R⁴ = C² is given by

$z_{1}=(a+b\cos v)e^{iu}\,$

$z_{2}=(b\sin v)e^{iu/2}\,$

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)

Thanks, Fropuff! I've put your new diagrams in the article. dbenbenn | talk 23:09, 3 Mar 2005 (UTC)

Hello, i have made some parametrization myself, rather simple parametrization that has klein bottle topology and in 4D is most like moebius strip. It's simply torus that does 4D flip. there's viewer and equations http://dmytry.pandromeda.com/klein_bottle.html (i think somebody there might be interested in just looking at that, and can tell me how this specific surface is named if it is named somehow.) -Dmytry.

If a=0 you get Lawson's surface in S3. —Tamfang (talk) 20:17, 1 January 2015 (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)

Done. I used a lemniscate figure-8 curve. IMHO a lemniscate cross section gives a more pleasantly appearing Klein bagel than one with a lissajous figure-8 cross section. Cloudswrest (talk) 22:44, 21 October 2010 (UTC)

MHO too. —Tamfang (talk) 08:05, 3 April 2015 (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)

The R⁴ parameterization above is basically just the basic torus parameterization as shown here with z₁ = x + iy, and a real z height replaced by a complex or two dimensional z₂ = z + iw. The minor cross section is a circle of radius b that "flips over" in the z-w space and presents its "backside" as it comes around and re-connects. In the orthogonal R³ x-y-z projection the torus pinches or flattens to a line/crosscap at u = π. A possible figure could show the flattening torus with a periodic circular cross section that becomes horizontal at the cross over, and is then upside down after the cross over. In order to portray the "upside downess" perhaps the circle could have different colors for the top and bottom. Cloudswrest (talk) 17:55, 26 June 2014 (UTC)

FYI: I recently added a 3D projection figure of the above to the main article. Cloudswrest (talk) 22:11, 1 January 2015 (UTC)

Connected sum construction

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)

Wrong Image

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:Klein.jpg]] Image has been deleted. CiaPan 18:24, 26 January 2006 (UTC)

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

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)

Poetry

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)

Use as an actual bottle

What happens if you pour water (or some other liquid) into the "opening" of the bottle? --Jfruh 21:30, 22 February 2006 (UTC)

• The liquid will go inside the bottle, if it's held in a proper way. With practice, it should be possible to have the liquid go all the way to the "bottom" of the bottle, which is the same place as the "opening", only on the other side of the "opening's" surface. JIP | Talk 11:57, 12 May 2006 (UTC)

  Acme klein bottles sells Klein mugs, which are appropriately shaped for drinking beer out of. But note that these are immersions of Klein bottles in three dimensions, not actual Klein bottles. -lethe ^{talk +} 12:24, 12 May 2006 (UTC)

  Can someone explain the difference to me? Is a klein bottle actually four-dimensional? Dansiman 04:40, 7 September 2006 (UTC)

  It can be embedded in 4-dimensions, whereas it can only be immersed in 3 (the map is not one-to-one). In layman's terms, yes, it is 4-dimensional. --King Bee 13:41, 4 October 2006 (UTC)

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)

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)

genus-1??

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)

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)

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)

Possibly someone confused the orientable and non-orientable genus, and used the wrong formula $\chi =2-2g.$ For $\chi =0$ this gives 'genus' 1. --CiaPan 17:39, 22 May 2006 (UTC)

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)

Klein Fla-e-che

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)

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)

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)

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)

• 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)

Parametric equations

Thanks Tamfang. Here are what I have worked out to be the parametric equations for a Klein Strip (a 4th dimensional Möbius strip equivalent) - or at least a 3D slice of them at 90° & 270° rotation - if anyone is interested?

4th dimensional Klein Strip

x(u,v) = (R + r * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v / 2))) * sin(v)

y(u,v) = (R + r * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v / 2))) * cos(v)

z(u,v) = r * (sin(v / 2) * sin(u) * sin(v / 2 + π / 2) + cos(u) * sin(v / 2))

Where R is the radius of the donut and r is the radius of the cross section at the back, and u and v both step from 0 to 2π.

I can't find that anyone else has worked these out before unless someone can correct me on that. Gonegahgah (talk) 14:14, 6 August 2015 (UTC)

The improvements suggested by Tamfang (below) would make these parametric equations now:

$x(u,v)=(R+r*(\sin({\frac {v}{2}})*\sin(u)*-\sin({\frac {v}{2}})+\cos(u)*\cos({\frac {v}{2}})))*\sin(v)$

$y(u,v)=(R+r*(\sin({\frac {v}{2}})*\sin(u)*-\sin({\frac {v}{2}})+\cos(u)*\cos({\frac {v}{2}})))*\cos(v)$

$z(u,v)=r*(\sin({\frac {v}{2}})*\sin(u)*\cos({\frac {v}{2}})+\cos(u)*\sin({\frac {v}{2}}))$Gonegahgah (talk) 10:18, 29 July 2016 (UTC)

• Simpler to write $\cos({\frac {v}{2}}+{\frac {\pi }{2}})$ as $-\sin({\frac {v}{2}})$ ...
• I've made models of stereographic projections of this:

  w = cos(u) cos(2v)

  x = cos(u) sin(2v)

  y = sin(u) cos(v)

  z = sin(u) sin(v)

Of course there are several equivalent ways to write this. —Tamfang (talk) 23:35, 27 August 2015 (UTC)

Thanks for the suggested improvements Tamfang and also for using an example of math formatting.

I've come to understand 4D Klein Strips a little better now I believe...

I hope this is of use to others interested in 4D Space wanting to explore Klein Bottles in 4D...

The type of Klein Strip (or bottle) that I am concentrating on are what I might term 'simple' Klein Strips.

That is they (like simple Mobius Strips) follow a ring path. Maybe I should call them 'simple ring' Klein Strips?

It seems in 4D that the 'twist' can occur in 360° of sideways directions whereas a Mobius Strip can only twist left or right in 3D.

The interesting thing from this would be that you can have a whole 360° of varieties of 'simple ring' Klein Strips in 4D.

I also need to clarify that it now appears that my first picture above only depicts a middle 3D slice and of only one variety of a Klein Strip.

I've written the following parametric equations to cycle through a middle 3D slice of all the 'simple ring' Klein Strip varieties:

4D Klein Bottle animation of all varieties, 3D slice of middle only

$x(u,v,t)=(R+r*(\sin({\frac {v}{2}})*\sin(u+t)*\cos(t)*-\sin({\frac {v}{2}})+\cos(u+t)*\cos({\frac {v}{2}})))*\sin(v)$

$y(u,v,t)=(R+r*(\sin({\frac {v}{2}})*\sin(u+t)*\cos(t)*-\sin({\frac {v}{2}})+\cos(u+t)*\cos({\frac {v}{2}})))*\cos(v)$

$z(u,v,t)=r*(\sin({\frac {v}{2}})*\sin(u+t)*\cos(t)*\cos({\frac {v}{2}})+\cos(u+t)*\sin({\frac {v}{2}}))$

where t is the variety of Klein Strip.

Thanks CiaPan for the fix. I've removed the π altogether using cos() instead of sin(); my apologies.

I'm hoping to expand this to further related formulas in the future...Gonegahgah (talk) 09:56, 29 July 2016 (UTC)

Question, how is this different from the Mobius Tube parameterization in the main article? Cloudswrest (talk) 14:01, 29 July 2016 (UTC)

Hi Cloudswrest. The main difference is that my form adds a twist throughout the length of the Klein shape. The pinched torus doesn't have that twist.

I add this is because the Klein shape in 4D, like a Mobius Strip in 3D, rotates from being a path, to being on its side (to being underneath the path, to the backside, and back to the path...)

This means that it is flat at one point and a wall at the opposite point. This has implications for how a 4D Klein shape would be seen our 3D space.

Rather than moving directly sideways (pinched torus like) it will tend to have a rotational change that has one side higher than the other like a Mobius Strip.

This will bleed through to our 3D space and will be seen as a rotation throughout the length of the pinched torus.

As a contrast a 3D sphere that rolls into 4D will change to a bump down to a circle but will remain centered because no part goes any higher when it rotates.

A Klein Ring on the other hand, when we twist the back into 4D space to show the varieties, changes its vertical orientation to us; just as twisting a Mobius does.

This is why the rotation, in the representation, is necessary because of the Ring's change in vertical orientation, to us, throughout its length.

At heart it is still a Mobius Strip. Does that ring true from my explanation?Gonegahgah (talk) 11:23, 7 August 2016 (UTC)

Couple of Questions

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)

I see that I was mistaken. Thanks--Gtg207u 20:43, 8 April 2007 (UTC)

--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)

Cutting and making Klein bottles

The Klein bottle immersed in three-dimensional space.

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)

Take a cross — a planar concave figure like a Greek cross. Bend horizontal arms backwards, twist one of them by 180° and glue their edges; that will make a Möbius strip shape. Bend vertical arms forwards and glue their edges; that will make a shape of a rubber band. The result is your Klein bottle with a hole.

Does it have any name? (That was my actual question, sorry for bad phrasing) IMO it is a quite distinguished shape. Now that we are to it, is there any topological classification of 2D manifolds with a single edge? `'Míkka 15:47, 28 August 2007 (UTC)

Sorry, I can not help you with this — I am not a mathematician, and know not more than epsilon about manifolds taxonomy. Possibly you could ask your question on maths ref. desk. --CiaPan 09:36, 29 August 2007 (UTC)

If you take a Klein bottle immersion model like this on the picture and cut it in halves with its only symmetry plane, you would get two Möbius strips with opposite chirality (that is, one left-twisted and the other one right-twisted).

--CiaPan 05:39, 28 August 2007 (UTC)

3d vs 2d

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)

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)

fundamental group

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: $\pi _{1}(K)=\langle a,b|aba^{-1}=b^{-1}\rangle$

Question

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)

You should be aware that there is no such thing as 'into' a Klien Bottle - that's the point. —Preceding unsigned comment added by 207.189.106.4 (talk) 18:25, 20 October 2010 (UTC)

Heh, heh. Oh yes there is. Klein's demonstration that it lacks an interior is based on morphing the neck in the fourth dimension to disentangle it from the body. But in four dimensions no 2-manifold, not even a 2-sphere, has an interior - here there is indeed no "into". Back in familiar 3D, a Klein bottle always has a crossing place to provide a "stopper" to the bottle: temporarily pull a stopper and you can indeed pour water into it, then reseal the "bottle". Here is an ASCII art drawing of a morph of the Klein bottle which helps to demonstrate the point:

        o---------------o
       /                |\
      /   o---------o   | \
     /   / \        |   |  \
    /   /   \       o   |   o
   /   /     \     /    o   |
  o   o       o---/    / \  |
  |   |\     /   o    /   \ |
  |   | \   /    |   /     \|
  |   |  \ /  o--+--o       o
  |   |   o  /   |.  \     /
  |   |   | /    o .  \   /
  |   o----o    / \ .  \ /
  |            /   \ .--o
  o-----------o     \|
   \           \     o
    \           \   /
     \           \ /
      o-----------o

Think of it as a "c" shaped prism joined to a cross-quadrilateral prism to close the loop. Cut a hole for the stopper anywhere you like. — Cheers, Steelpillow (Talk) 20:17, 20 October 2010 (UTC)

When you pour water into the bottom of a physical model of a Klein bottle, like the ones shown on the Acme web site, it usually splashes back at you because there is air trapped on the inside. The easiest way to fill one with water is to turn it sideways and immerse it. A13ean (talk) 20:30, 20 October 2010 (UTC)

Parametrization

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:

${\begin{aligned}x&={\frac {{\sqrt {2}}f(u)\cos u\cos v(3\cos ^{2}u-1)-2\cos 2u}{80\pi ^{3}g(u)}}-{\frac {3\cos u-3}{4}}\\y&=-{\frac {f(u)\sin v}{60\pi ^{3}}}\\z&=-{\frac {{\sqrt {2}}f(u)\sin u\,\cos v}{15\pi ^{3}g(u)}}+{\frac {\sin u\cos ^{2}u+\sin u}{4}}-{\frac {\sin u\,\cos u}{2}}\end{aligned}}$

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)

How did someone even find this parametrization? 98.113.222.37 (talk) 17:45, 20 June 2010 (UTC)

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)

Consider removing trivia / popular culture sections?

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)

A common sense tells me to always apply two basic wikipedia rules:

• There must be references cited
• WP:NOTABILITY: if the klein bottle is mentioned in the text of another wikipedia as a notable component of the description, plot, etc., then it may be reasonable to list the case here as well.

They usually allow me to quickly eliminate lots of trivia, and mostly without trouble. - 7-bubёn >t 04:03, 4 February 2009 (UTC)

Lacan uses the klien bottle to explain the self and Levi-Strauss to explain the structure of mythology. Would these topics be trivia? The klien bottle is not only a mathematical concept. In Ellipse there are sections on Ellipses in physic and Ellipses in statistics and finance for instance. --Timtak (talk) 23:24, 2 April 2015 (UTC)

Practical cautions required by Klein's nuisance

Extended content

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. —Preceding unsigned comment added by SyntheticET (talk • contribs) 20:43, 21 October 2009 (UTC) The BP oil rig could have used a few extra valves. 98.118.62.140 (talk) 16:50, 24 May 2010 (UTC) Yes, but no idea how that's relevant to Klein bottles. A well+platform isn't a closed system and there's no evidence of a detectible problem in an "unseen/uninspectable" area is at fault. DMacks (talk) 18:35, 24 May 2010 (UTC) There's obviously no containing it, though. SomeAvailableName 09:59, 03 June 2010 (PST)

Suggest section on Physical Models of Klein Bottles

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)

How do you pronounce "Klein"?

"Clean" or "Kline"? Richard W.M. Jones (talk) 09:58, 7 July 2010 (UTC)

German spelling is almost completely regular. Klein was German, and the regular German pronunciation is [kla͡ɪn], roughly rhyming with English 'incline'. If it were pronounced [kliːn] (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)

Klein Bottle as circle bundle over the circle.

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)

Fläche/Flasche

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)

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)

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)

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)

The claim in this wikipedia article dates from 2005 [2]. 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, [3] ) on this wikipedia article. --Trigamma (talk) 17:49, 21 October 2010 (UTC)

Quotient space construction

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 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)

Yes, of course. I was misreading. Thank you. 24.220.188.43 (talk) 16:20, 10 September 2011 (UTC)

Construction from a Mobius strip

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)

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)

Lawson Klein bottle

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)

How does the Lawson Klein bottle differ from the usual Klein bottle? Jowa fan (talk) 03:23, 12 September 2011 (UTC)

Lawson's figure is an immersion in 4space of the Klein surface, which apparently has some interesting properties (that I'm not advanced enough to appreciate). —Tamfang (talk) 21:43, 16 March 2012 (UTC)

I found a page that says it's a minimal surface in the 3sphere, so that's pretty nifty. —Tamfang (talk) 22:32, 26 November 2014 (UTC)

It looks like there's a good array of sources for this, it should probably be included whenever someone has a chance. a13ean (talk) 17:12, 27 November 2014 (UTC)

It can be parametrized as

• cos(u) cos(v),
• cos(u) sin(v),
• sin(u) cos(2v),
• sin(u) sin(2v).

I have made some models of this in stereographic projection. —Tamfang (talk) 19:15, 5 February 2022 (UTC)

equivalence under projection?

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)

I'm pretty sure, but someone else can probably confirm this. a13ean (talk) 22:21, 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)

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)

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)

Perhaps 3Blue1Brown would take it on. —Tamfang (talk) 02:12, 6 February 2023 (UTC)

Figure 8 immersion equations.

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)

proposition

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 ᗰOᕼᗩᗰEᗪŽ╦ 13:15, 26 August 2014 (UTC)

We can't, because it's WP:OR. --CiaPan (talk) 11:15, 27 August 2014 (UTC)

okay, but it stilles a good idea. --ᔕGᕼᗩIEᖇ ᗰOᕼᗩᗰEᗪ (talk) 13:11, 27 August 2014 (UTC)

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)

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)

OR

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)

The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. It need not be! —Tamfang (talk) 02:09, 6 February 2023 (UTC)

Klein group?

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)

Bottle shape formula wrong?

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)

Construction from rectangle

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

P,C1,C2,D

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)

Nonsense

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

1. The "traditional" Klein bottle
2. Left handed figure-8 Klein bottle
3. 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)

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)

Should the word embeddings be replaced with immersions? —Tamfang (talk) 17:09, 5 February 2022 (UTC)

Certainly. D.Lazard (talk) 17:33, 5 February 2022 (UTC)

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)

The illustration "Time evolution of a Klein figure in xyzt-space" is quite wrong

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)

What am I missing?

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)

@AndyFielding: 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)

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)

Mention three-dimensional representations in lead

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)

Pinch point (mathematics)

There needs to be something more said about Pinch point (mathematics).

rewrite intro for casual reader

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)

I gave it a shot. - DavidWBrooks (talk) 16:54, 12 March 2023 (UTC)

I sympathize (heaven knows I have often enough been bewildered by WP math articles), and it's a good start, but I am uneasy about the phrase "a solid structure"; that describes a representation, not the essence. How about leading with

A Klein bottle is a one-sided surface with no edges, which can be made by gluing the edges of two Möbius strips. …

—Tamfang (talk) 20:50, 15 March 2023 (UTC)

But it's a bottle and a bottle is a structure as lay folks understand the term - an object, something solid. "One-sided surface" sounds like a weird sheet of paper to a non-topologist, not like something that can be filled with water which I can hold in my hand (as I am doing right now, thanks to Cliff Stoll). Since we can almost immediately go into the mathematical explanation, I don't think using terms in mathematically suspect ways for the first sentence or two is a problem. - DavidWBrooks (talk) 22:06, 15 March 2023 (UTC)

It's not a bottle, that's just its name (which the article even talks about how the misnomer got applied in the first place). Nor is it a solid. A solid is a three-dimensional region (with a two-dimensional surface as its boundary). The Klein bottle can't be embedded into R^3 (i.e. it's non-orientable)...it doesn't have an inside or outside, so it's not the boundary of a solid, either. This wasn't just mathematically suspect, it was just plain wrong. It also gave the article a very awkward introduction with the standard blurb a paragraph in. If you want a friendlier opening sentence (not a bad idea really), looking at sources is probably a better idea 35.139.154.158 (talk) 22:29, 15 March 2023 (UTC)

It's a glass object that holds liquid - that's what a bottle is! (Or a jar, but the opening width is wrong.) You are insisting that the introduction be mathematically exact with mathematical wording, which makes it irrelevant to the casual reader. That's the problem I'm trying to fix. - DavidWBrooks (talk) 12:33, 16 March 2023 (UTC)

No, it is most definitely not a glass object that holds liquid. Should we also start our article on a torus telling the reader it's a fried, edible doughy object 🤦‍♂️? I'd be okay with putting the formal description as a manifold a few sentences later, at the end of the first paragraph, after the informal one that's already in place. But the casual reader is already getting "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)

Casual readers read “mathematics” before “Klein bottle”, so they will know immediately that, if they are not interested in mathematics, this article is not for them. D.Lazard (talk) 14:57, 16 March 2023 (UTC)

Cliff Stoll's website emphasizes that the bottle has zero volume … —Tamfang (talk) 18:53, 16 March 2023 (UTC)

The bottle has no volume at all, as it has no interior. The interior can be defined only for oriented surfaces. D.Lazard (talk) 19:08, 16 March 2023 (UTC)

I am staring at a glass object in my hand that holds liquid after being told that it is "most definitely not a glass object that hold liquid." I don't think this is going to go anywhere. (This discussion is, incidentally, a good example of the difficulty of explaining mathematical concepts in ways approachable to non-mathematicians. There's a reason not many science writers do articles about higher mathematics.) - DavidWBrooks (talk) 21:21, 16 March 2023 (UTC)

I've added a photo of my glass Klein bottle holding blue-colored liquid, which I think is useful for readers. (Earlier comments are correct, I've come to realize, that "bottle" is a poor name even in lay terms because it's very difficult to get the liquid out, which is a basic function of a bottle. "Container" is a better term.) - DavidWBrooks (talk) 12:58, 17 March 2023 (UTC)

I have again reverted your addition (reason in edit summary). Please, again, understand that just because one can make a piece of glass that represents one of the regular homotopy classes of the immersion of the Klein bottle into R^3, that doesn't mean that's what a Klein bottle actually is. The figure-8 immersion can't reasonably work like a bottle, yet it's got no less claim to being a Klein bottle as the more traditional one. Please, this continued insistence on thinking that this thing is fundamentally a bottle (in the everyday use of the word) isn't helpful. 35.139.154.158 (talk) 19:14, 17 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)

I just realized that this article doesn't even mention Stoll's Klein bottles - easily the best-known instance outside topology classrooms - presumably because they're not "really" Klein bottles in the mathematical sense. That's ridiculous, a painful example of reducing the value of a wikipedia article out of misplaced exactitude. - DavidWBrooks (talk) 14:55, 18 March 2023 (UTC)

I have added mention of Stoll as well as Japanese glassblower Mitsugi Ohno (whose article discusses his creation of an early Klein bottle) next to the long-existing mention of the physical collection at the Science Museum in London. I also re-added the photo of a Stoll version holding liquid, right under a long-standing photo of a Stoll version that is not holding liquid! I think the entire paragraph about physical construction should be moved to its own section - it's stuck in the middle of the construction section, hard for the casual reader to find. - DavidWBrooks (talk) 14:44, 19 March 2023 (UTC)

Okay, this is getting utterly ridiculous. I don't even know what your goal is here anymore. I've reverted you yet again since a) the text about Ohno was completely unsourced, and b) the source you did add was just primary to some Geocities-esque business page offering these for sale. A quick look on Etsy finds dozens of people selling these, not only in glass, but other materials as well..should we link these too? We also don't need yet another image of a glass model here (especially since there's no way to verify who actually made it, not that it should even matter). The fact that we already have multiple images of glass models should be sufficient for anyone to realize that yes, they can be made out of glass. We have no business promoting a specific seller of these. The bit about the history viz. Ohno would be perfectly fine......if you can source it properly. I'm just...flabbergasted and dumbfounded at this point about what you're even trying to do here. 35.139.154.158 (talk) 17:40, 19 March 2023 (UTC)

I have been on wikipedia long enough to recognize an editor who thinks they own an article and I have better things to do that beat my head against them. Enjoy your day. - DavidWBrooks (talk) 21:09, 19 March 2023 (UTC)

Well, Stoll bottles have a hole in the wall, which allows you to put liquid in. Should we say a flaw in the model (an ideal Klein bottle has no such hole) is a defining feature? —Tamfang (talk) 05:14, 20 March 2023 (UTC)

That would be an excellent point to mention in the article as part of a discussion about physical models of the Klein bottle - using wikipedia to inform the public. - DavidWBrooks (talk) 11:56, 20 March 2023 (UTC)

Given your little tantrums above and here, I don't think you should be participating until you retract. I.e., if you decide to take your toys and go home because you didn't get your way, you don't get to keep playing with everyone until you can play nice. And if you've really been on Wikipedia long enough to ____, then you should understand that proper sourcing is a pretty core requirement of adding material to an article. 35.139.154.158 (talk) 00:19, 21 March 2023 (UTC)

I attempted to make this edit, but another user is insistent upon undoing the edit. See my edit https://en.wikipedia.org/w/index.php?title=Klein_bottle&oldid=1183811213 Nandor72 (talk) 01:01, 7 November 2023 (UTC)

Tamfang, I appreciate your desire to mediate this, but c'mon, do you really expect our article here to talk about personal observations of one particular guy's products? Do we have any reliable, independent, secondary sources discussing the history of these things as novelty items and/or what goes into constructing one? If so, there might actually be a little blurb to add (this is certainly not the focus, nor should we be promoting one particular seller...as I said, lots of people sell these, and made of different materials), but otherwise, we still have things like WP:NOR, etc to keep in mind. 35.139.154.158 (talk) 00:19, 21 March 2023 (UTC)

if you glue the edges of two

by joining the edges of two (mirrored) Möbius strips

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)

@Tamfang: '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)

The spherinder Klein bottle is not a surface

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)

Edit war about the dimension of Klein bottles

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)

This post of Nandor72 follows an WP:edit war about the caption of an image of a physical realization of a Klein bottle. Nandor72 changed the previous caption from "A hand-blown Klein Bottle" to "A hand-blown 3-D immersion of the 4-D Klein Bottle" ([4]). This has been reverted by an IP user with the edit summary "if you want to get pedantic, a Klein bottle is two dimensional, not four....but I don't think we really need to get pedantic here". Since then, Nandor72 did twice this change and I reverted them twice, with edit summaries "Misplaced (and wrong) pedantry (nobody is saying that he trinks in a 3D coffee mug)" and "Do not edit war. If somebody disagree with your edit, you must discuss on the talk page of the article in view of a consensus. Please do not comment on the supposed competence of other editors. This is WP:persnal attacks, that are forbidden on Wikipedia." (sorry for the typo). The mentioned WP:personal attack appears in the last (to date) edit summary by Nandor72, which begins with "I don't think you understand the concept of 3-dimensional", a clear accusation of incompetence.

The reasons of my reverts are the following:

• Discussing the dimension of a physical object in the caption of an image of this object is blatant pedantry (this is not a personal attack, it is an opinion on an edit)
• The discussion on the dimension of the (mathematical) Klein bottle belongs to the article body, not to the image of a physical realization.
• It is definitively wrong to say that a Klein bottle is a 4D object: It is a 2D manifold that can be embedded in a 4D Euclidean space and immersed in a 3D Euclidean space. It is true that the image represents an immersion of a Klein bottle in the usual 3D space, but there is no 4D here.

D.Lazard (talk) 11:28, 7 November 2023 (UTC)

For what little it's worth, I agree with Lazard's last bullet. —Tamfang (talk) 03:54, 9 November 2023 (UTC)

First, I apologize for re-editing your undo. I was aware of neither the etiquette nor the use of Talk, as I have never needed to do this before.

Second, saying that I think you don't understand something is not an attack. I'm sorry you think it is. But when you claim that a Klein bottle is a 2D object, it causes me to think you made a typo or there is some other issue.

Third, I am not wrong about the dimensionality.

• A circle is a 1-manifold that can only be represented fully in a 2-space (or higher). It is a 2D object.

• A cup is a 2-manifold that can only be represented fully in a 3-space (or higher). We can draw its immersion in 2D (a picture of it), but I think we can all agree that a cup is a 3D object.

• A sphere is a 3-manifold that can only be represented in 3-space (or higher). It is a 3D object.

• A spherical shell is a 2-manifold that can only be represented in 3-space. It is a 3D object.

• A wire frame model of a sphere is a 1-manifold that can only be represented in 3-space (or higher). It is a 3D object.

• A surface hyper-cube (tesseract) is a 2-manifold that can only be represented in 4-space (or higher). It is a 4D object.

• Continuing on to a Klein bottle, it is a 2-manifold that can only be full represented in 4-space (or higher). A Klein bottle is a 4D object. We can have its immersion in a 3-space.

Yes, any small section is 2-dimentional, so a topologist would call it a 2-manifold. Part of the issue here is the imprecision of language. In every-day usage we call something that can only be represented in 2-space or higher a "2D object." But can we agree that Cliff's Klein bottle models are 3-dimensional? Yes, they are made from a 2-manifold (ideally), and a topologist might informally call it "a 2D object," in conversation, but "2D object" is not really a topological term, and formally a topologist would just call it a 2-manifold. I looked in my three topology books and my differential geometry book, and none of the pictures label things as nD objects; they just label them n-manifolds.

By the way, please read Cliff Stoll's page at https://www.kleinbottle.com/whats_a_klein_bottle.htm . He HIMSELF says that Klein bottles are 4D objects AND says that his glass bottles are only 3D immersions, and the picture in question comes from his website.

Yes, no one would label a picture of someone drinking from a mug and label it as "woman drinking from a 3D mug." However, dimensionality is an important (and confusing) aspect of Klein bottles since most people can't picture 4-space (I myself can't picture 5-space at all), and labeling Cliff's models just as "Klein bottles" will confuse the readers into thinking that they are Klein bottles... which they are not. I all-too-often hear students (and others) who see my Cliff Stoll Klein bottles actually claim that they are indeed Klein bottles. People who should know better, too. In this way, re-labeling the picture is neither pedantic nor unnecessary. At the very least the picture should be labeled "3D model of a Klein bottle" or "3D immersion of a Klein bottle." Nandor72 (talk) 21:21, 10 November 2023 (UTC)

You do not define what are 2D, 3D and 4D objects. I did not find any definition in Wikipedia, and it is not clear whether "object" refers to mathematical objects or to physical objects. Even for mathematical objects, there are several possible definitions. In particular, for a manifold, nD may refer to:

• The dimension of the manifold (2 for a Klein bottle)
• The smallest dimension of a Euclidean space in which it can be immersed (3 for a Klein bottle)
• The smallest dimension of a Euclidean space in which it can be embedded (4 for a Klein bottle)

The ambiguity of the term "nD"-object implies that it cannot be used in Wikipedia, whithout a detailed and sourced discussion that is out of the scope of this article.

Also, Cliff Stoll is apparently not a mathematician, and his personal web site cannot be used as a WP:reliable source (even if he were a reputed mathematician). D.Lazard (talk) 12:10, 11 November 2023 (UTC)