Talk:Klein bottle

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--Cronholm¹⁴⁴ 03:57, 16 June 2007 (UTC)

1 Inside vs. Outside
2 A reply in dialogue
3 Figure 8 Klein bottle
4 Unused image
5 Acme Klein bottle?
6 Figure 8 immersion
7 New immersions, pictures, and parameterizations
8 Connected sum construction
9 Wrong Image
10 Poetry
11 Use as an actual bottle
12 genus-1??
13 Klein Fla-e-che
14 Couple of Questions
15 Cutting and making Klein bottles
16 3d vs 2d

[edit] 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

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)

[edit] 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

[edit] 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. —siro χ o 01:20, Jul 31, 2004 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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

$z_1 = (a + b\cos v)e^{i u}\,$

$z_2 = (b\sin v)e^{i u/2}\,$

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.

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 $χ = 2 - 2 g .$ For $χ = 0$ 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

χ = 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)