Talk:Klein bottle

Mathematicians try hard to floor us
With a non-orientable torus
   The bottle of Klein
   They say is divine
But it is so exceedingly porus.

Frederick Winsor, The Space Child's Mother Goose (http://www.purplehousepress.com/space.htm) The Space Child's Mother Goose

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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)

Contents

1 A reply in dialogue 2 Figure 8 Klein bottle 3 Unused image 4 Acme Klein bottle? 5 Figure 8 immersion 6 New immersions, pictures, and parameterizations 7 Connected sum construction

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 (http://www.geom.uiuc.edu/zoo/toptype/klein/) 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.

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

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

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)

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

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

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

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)

Connected sum construction

Should we mention that the Klein bottle arises as the connected sum of three copies of <math>\mathbb{R}P^2<math>?