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Author Topic:   Mathematics and Nature
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 48 of 90 (269582)
12-15-2005 5:43 AM
Reply to: Message 47 by Dr Jack
12-15-2005 4:43 AM


Confusions of Topology
I think there is a confusion of terms here. There is no idealised Mobius Band as such, despite layman artistic or computer-genberated depictions of such. An MB is a loop with certain global topological properties. It doesn't exclude other topological properties, local or global.
My pieces of paper exhibited the required global topological behaviour, amongst other behaviours. You can imagine an idealised surface wrapped into a Mobius Band, but this surface also has topological properties that are irrelevant to its Mobius topology such as the nature of its width - is it open or closed, or is there a distance function defined.
If you want to do this properly, you need to strip it all down to an S1 base space with a line bundle (tangent bundle being the obvious) and then choose suitable transition functions. But this is not how we generally use the "concept" of a MB.

This message is a reply to:
 Message 47 by Dr Jack, posted 12-15-2005 4:43 AM Dr Jack has replied

Replies to this message:
 Message 49 by Dr Jack, posted 12-15-2005 5:49 AM cavediver has replied
 Message 56 by RAZD, posted 12-15-2005 3:41 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 50 of 90 (269587)
12-15-2005 6:14 AM
Reply to: Message 49 by Dr Jack
12-15-2005 5:49 AM


Re: Confusions of Topology
There is most certainly an idealised mobius strip:
As I mentioned, this is an idealised "surface" that is then given the topology of a Mobius Band (Loop, Strip, etc). It is just one way of depicting something with the properties of a MB. Of course, you can define an MB to be this idealised surface as depicted in Wolfram, but you are somewhat missing the point. The Wolfram entry is appallingly 3d geometrical, which is a shame, and certainly not how I would write the entry. It is the topology that gives the MB its properties, not its geometry. All that geometry is very nice but utterly restricted to this particular representation and 3d embedding. Great for engineers using real MBs in applications, but wholly insufficient for the mathematician.
This message has been edited by cavediver, 12-15-2005 06:17 AM

This message is a reply to:
 Message 49 by Dr Jack, posted 12-15-2005 5:49 AM Dr Jack has replied

Replies to this message:
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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 51 of 90 (269589)
12-15-2005 6:16 AM
Reply to: Message 49 by Dr Jack
12-15-2005 5:49 AM


Re: Confusions of Topology
I'm not sure, but I think Mobius Band may be more engineer speak (shame on me). That said, my biography of Mobius is called Mobius and his Band (pun intended on referring to his contemporaries)

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 53 of 90 (269596)
12-15-2005 7:34 AM
Reply to: Message 52 by Dr Jack
12-15-2005 6:39 AM


Re: Confusions of Topology
I'm too tied up to do this justice at the moment, but would like to continue this later.
For now: thickness is irrelevant. Topology is about properties invariant under deformation, and thickness can be deformed to flatness. The join does break rotational symmetry and could cause problems, depending upon what you are defining as the MB: is it the paper itself, arrows upon the paper, etc. I don't have to view it as a problem. The topological properties are still there. An apple is a perfectly good realisation of SO(3) rotations, unhampered by its nobblyness and stalk!

This message is a reply to:
 Message 52 by Dr Jack, posted 12-15-2005 6:39 AM Dr Jack has replied

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 55 of 90 (269608)
12-15-2005 9:19 AM
Reply to: Message 54 by Dr Jack
12-15-2005 7:49 AM


Re: Confusions of Topology
Even a sheet of paper is not topologically equivalent to a bounded surface.
Of course. We are talking across each other here and I think it's my fault for not having time to explain myself coherently. Think of arrows on the paper loop revealing the coarse-grained topological properties. The fine-structure of the paper and the ink marks of the arrows does not have to be important.

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 58 of 90 (269791)
12-15-2005 8:27 PM
Reply to: Message 57 by RAZD
12-15-2005 3:47 PM


Re: Why mathematics is so useful in the sciences
Or am I blindingly obtuse and invariably wrong on this topic too?
Well, perhaps I wouldn't go quite that far But...
quantum behavior of subatomic particles to change from one to another or to more than one or disappear altogether for brief moments in time
is incompatible with
What the particle is has to be described as a probability cloud, never certain.
The latter is QM, which is inadequate to describe the former, which is the world of QFT. QM and its probability cloud failed to describe the real world, where particle creation and annihilation were observed. It required the addition of SR (which amongst other things introduced the concept of antiparticles) which led to the first formulations of QFT. We call this "second quantisation".
But the mathematics of pair creation/annihilation and virtual particles is very well understood AND inextricably linked with the "indistinguishability" of those particles. This is not a random world where anything can happen. There are strict rules that are observed to be obeyed every day at every particle lab around the world.

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 59 of 90 (269801)
12-15-2005 8:51 PM
Reply to: Message 56 by RAZD
12-15-2005 3:41 PM


Re: Confusions of Topology
You are equivocating. The model displays the topology to a crude degree sufficient to convey the concept in model form, but it is still not the mathematical concept
I am not equivocating. I am disagreeing with your understanding of the mathematical concept. Once again, this does come down to a matter of philosophy... and perhaps of more relevance, mathematical training. I am a topologist, and I thus have different working definitions to someone not trained in topology, or for example someone emphasising geometry. I am also a mathematical physicist so I am bringing in the technology of representations and realisations.
The other properties you say are "irrelevant to it's mobius topology" show that you recognize that it is not the mathematical concept, as they do not exist for the concept
No, it shows that I do not agree with your understanding of the concept. Adding extra structure does not destroy the original property. The concept is a property, not an object.

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 Message 56 by RAZD, posted 12-15-2005 3:41 PM RAZD has replied

Replies to this message:
 Message 60 by RAZD, posted 12-15-2005 9:20 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 61 of 90 (269816)
12-15-2005 9:32 PM
Reply to: Message 60 by RAZD
12-15-2005 9:20 PM


Re: Confusions of Topology = 1 math for another
Perhaps you should describe for me my understanding of the concept?
you are not comparing the concept with the model that is the reality of the paper strip.
As I said, the concept is a property. I am not trying to say that the paper strip is the same as that idealised concept described in the Wolfram article.

This message is a reply to:
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Replies to this message:
 Message 62 by RAZD, posted 12-15-2005 9:44 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 63 of 90 (269903)
12-16-2005 4:43 AM
Reply to: Message 62 by RAZD
12-15-2005 9:44 PM


Re: Confusions of Topology = 1 math for another
the mathematical concept is a single sided surface.
This is how the concept was first realised but is now just an example of some idealised 3d object having the Mobius topology. I appreciate that this is how it is always depicted in popular mathematics, and constitutes the general understanding even amongst mathematicians who are perhaps not topologically trained outside of basic 3d concepts.
The Mobius topology is a 2d property. What does "single sided" mean in the context of a 2d surface? Singled sided is just an artifact of the embedding in 3d. Furthermore, what is a "surface"? Do we need such a vague concept to understand the topology? Not at all. That is why we have algebraic topology: to remove all of the vagueness of the pictures and extract the pertinent properties.
This is the problem here. For you to have any hope of appreciating what I am saying, you have to substantially broaden your understanding.
But in the context of your definition of a Mobius Strip (and as you brought up the term, I have been at fault in immediately substituting my understanding for your understanding without adequately making this clear), you are quite correct. But given that your definition is essentially just an idealisation of a real object: Mobius' first piece of paper "cellotaped" into a loop with a twist, it is not too surprising that you say that my loop cannot possibly satisfy your definition, because it is not idealised. I am happy to concede this rather obvious point
This message has been edited by cavediver, 12-16-2005 05:50 AM

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 65 of 90 (269918)
12-16-2005 6:47 AM
Reply to: Message 64 by Dr Jack
12-16-2005 6:01 AM


Re: Confusions of Topology = 1 math for another
The mathematical concept is of a bounded 2d surface with an equivalence map making points on the top and bottom edges equivalent to one another, in reversed order.
Quite. Though I would go further and re-state this in the algebraic language of a bundle, as you can then dispense with the idea of a surface altogether leaving just the topological data.

This message is a reply to:
 Message 64 by Dr Jack, posted 12-16-2005 6:01 AM Dr Jack has replied

Replies to this message:
 Message 66 by Dr Jack, posted 12-16-2005 8:42 AM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 67 of 90 (269952)
12-16-2005 9:13 AM
Reply to: Message 66 by Dr Jack
12-16-2005 8:42 AM


Re: Confusions of Topology = 1 math for another
but I would argue that the definition I gave is the primary definition
Well, you can I'm not going to spend time on fighting "primary" vs "more fundemental"...

This message is a reply to:
 Message 66 by Dr Jack, posted 12-16-2005 8:42 AM Dr Jack has replied

Replies to this message:
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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 71 of 90 (270365)
12-17-2005 4:37 PM
Reply to: Message 69 by RAZD
12-17-2005 7:41 AM


Re: Confusions of Topology = 1 math for another
While you guys are busy patting yourselves on your backs about the various esoteric definitions you are using ... 2 things:
Esoteric??? Go get a degree in mathematics, or preferably a PhD before you start commenting on our usage please.
Wikipedia? Give me a f'ing break.

This message is a reply to:
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Replies to this message:
 Message 72 by RAZD, posted 12-17-2005 4:57 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 73 of 90 (270371)
12-17-2005 5:06 PM
Reply to: Message 72 by RAZD
12-17-2005 4:57 PM


Re: Confusions of Topology = 1 math for another
When the message is written in such insulting terms I will happily attack the messenger and ignore the message. I'm not paid to do this you know...

This message is a reply to:
 Message 72 by RAZD, posted 12-17-2005 4:57 PM RAZD has replied

Replies to this message:
 Message 74 by RAZD, posted 12-17-2005 6:39 PM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 78 of 90 (270444)
12-18-2005 4:10 AM
Reply to: Message 74 by RAZD
12-17-2005 6:39 PM


Re: Confusions of Topology = 1 math for another
1.b. Of or relating to that which is known by a restricted number of people.
2.a. Confined to a small group: esoteric interests.
We are already discussing mathematics, so I can only assume you a referring to some subset of the mathematically literate as your "small group" or "restricted number of people". What subset is this? Any that excludes you?
While you guys are busy patting yourselves on your backs about the various...
This is what I find insulting. MrJack and I are two mathematicians discussing some exceptionally basic concepts. We are not discussing personal choice or philosophy here. You seem to have a problem with being corrected over a definition, and then bring up Wikipedia to defend your position.
As I have mentioned before, you seem to value your own amateur "expertise" over a consensus of professionals, and seem to take umbridge when you are presented with new information. I have not once detected an ounce of interest in any new concepts presented (and I am talking about accepted facts about reality, not my own esoteric ideas), other than to attempt to deride or ridicule them from your position of skeptism and, more importantly, ignorance. I find this immensely disheartening and is why I am losing patience discussing anything with you... You sound more and more like Randman as time goes by.
You still do not have a real mobius strip, but just a poor approximation that is good for demonstrating the concept. The model is not the concept.
Back to the philosophy, but I really have lost interest now. I will finish by re-stating my point that it is not the paper loop itself that is my mobious strip; in the same way that I point to not the apple but rotations of the apple as the realisation of the group SO(3).
This should continue with SGs comments in his last post becasue that is far more pertient: is Mobius behaviour (or any other mathematical concepts) exactly realised by fundemental particles?
This message has been edited by cavediver, 12-18-2005 05:11 AM

This message is a reply to:
 Message 74 by RAZD, posted 12-17-2005 6:39 PM RAZD has replied

Replies to this message:
 Message 81 by RAZD, posted 12-18-2005 8:24 AM cavediver has replied

  
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 85 of 90 (270466)
12-18-2005 8:53 AM
Reply to: Message 81 by RAZD
12-18-2005 8:24 AM


Re: Confusions of Topology = 1 math for another
Which does show a common definition, does it not? There are other sources of the same definition, but why bother if you so readily dismiss anything that doesn't match your "high" level of understanding eh?
Mathematics is fortunately not defined by popular consent of internet references. Dictionaries and encycolpedias of mathematics and physics are full of inaccuracies, and they need pointing out. I do this regularly with one of the UKs best selling dictionaries of science, as an assistant editor.
Funny, I don't remember you saying this particular point in just this way before.
quote:
But in the context of your definition of a Mobius Strip (and as you brought up the term, I have been at fault in immediately substituting my understanding for your understanding without adequately making this clear), you are quite correct. But given that your definition is essentially just an idealisation of a real object: Mobius' first piece of paper "cellotaped" into a loop with a twist, it is not too surprising that you say that my loop cannot possibly satisfy your definition, because it is not idealised. I am happy to concede this rather obvious point
What you are really saying here is that the model is not the concept of the mobius strip.
I have been saying that the paper loop on its own is not the mobius strip.
And deal with a mathematical abstaction of the apple instead.
Describe how the rotations of the apple are an abstraction.
This has been my point all along - you keep looking at mathematical abstractions instead of the real object.
I am looking at something different to what you think of as the "real object". But what I am looking at appears to be equally real to me.
It's not philosophy but reality intruding on your philosophy.
I'm glad you have reality so well sown up to be able to claim this. I wish I had your insight.
I deal everyday with making my involved, technical, esoteric mathematical approximations of reality work in the real world and take into account the fact that the math is not the reality to make it work.
Let me know when you start working in the real world of sub-nuclear to Planck scale physics and we will see how long your hard and fast ideas of reality last.
This message has been edited by cavediver, 12-18-2005 08:57 AM

This message is a reply to:
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Replies to this message:
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