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Author | Topic: Mathematics and Nature | |||||||||||||||||||||||||||
cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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.
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RAZD Member (Idle past 1404 days) Posts: 20714 From: the other end of the sidewalk Joined: |
the mathematical concept is a single sided surface.
the paper model does not have a single surface, it has thickness dimensions with an inside as well as an outside as well as edges, and we aren't even getting to the rather difficult discontinuity at the joint formed by cut ends roughly lined up or lapped and glossed over with tape. if you extract the mathematical topological concept from the model you are comparing two mathematical concepts and not the model to the original concept. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
the mathematical concept is a single sided surface. No, it isn't. 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. So (x, 0) = (1-x, 1) for all x in [0, 1] (taking the surface as [0,1]x[0,1] for simplicity of notation). This is then expanded to all objects topologically equivalent to this.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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.
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
You could indeed, but I would argue that the definition I gave is the primary definition since the algebraic topological definition was defined to match the formal definition I gave.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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"...
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Dr Jack Member Posts: 3514 From: Immigrant in the land of Deutsch Joined: Member Rating: 8.7 |
Touche
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RAZD Member (Idle past 1404 days) Posts: 20714 From: the other end of the sidewalk Joined: |
While you guys are busy patting yourselves on your backs about the various esoteric definitions you are using ... 2 things:
(1) from Wikipedia:M·bi·us strip n. The Mbius strip or Mbius band is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It was co-discovered independently by the German mathematicians August Ferdinand Mbius and Johann Benedict Listing in 1858. Common usage still uses the definition I gave, and it still serves to describe the fundamental mathematical concept involved, and (2) you are still ignoring that the model created by the strip does NOT in fact have these properties itself, that you are mentally extracting those properties from the model to compare it to the concept. For the model to have the properties even your esoteric definitions use, the points on one face would have to project through the paper -- they don't. Enjoy. {corrected typo} This message has been edited by RAZD, 12*17*2005 02:24 PM we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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Son Goku Inactive Member |
The way I understand it, the Mobius strip is a topology, just like a Klein bottle.
Topology is used to examine connectness in mathematical entities independant of their geometry or other specifics, just like differential geometry dicusses geometry without the need for embedding. So something can have the property of a mobius strip topology even if it is not a surface, simply by having that connectivity. Examples would be mobius energy in knots and junction ladders in conformal field theory have boundaries with mobius topology. So even if we cannot embed a 2D figure in 3D space which has the mobius topology, that does not mean that the mobius strip does not exist, as it is a topology not a shape. I'm not the best on topology so there might be some mistakes. This message has been edited by Son Goku, 12-17-2005 01:55 PM
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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.
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RAZD Member (Idle past 1404 days) Posts: 20714 From: the other end of the sidewalk Joined: |
ah yes. attack the messenger and not deal with the message.
cool.
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cavediver Member (Idle past 3643 days) Posts: 4129 From: UK Joined: |
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...
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RAZD Member (Idle past 1404 days) Posts: 20714 From: the other end of the sidewalk Joined: |
Wikipedia? Give me a f'ing break. Nature study shows errors in wikipedia similar to enc. britannica. dictionary.com
es·o·ter·ic adj. 1.a. Intended for or understood by only a particular group: an esoteric cult. See Synonyms at mysterious. 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. 2.b. Not publicly disclosed; confidential. If you find either 1b or 2a insulting, then have at it. 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. Comparing the (mathematical) topological characteristics that you extract from the model with the original mathematical concept does not get you to the point of having realized the concept in actual fact. Personally I don't see what the problem is with just admitting this basic fact. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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Son Goku Inactive Member |
To quote John Listing himself, the inventor/discoverer of the mobius strip:
By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity. In this manner the Mobius strip was a connectivity he discovered in oriented three-dimensional polyhedra, as it came up again and again when discussing connectivity it became a hot topic in topology. August Mbius then invented the one sided representation of this topology to aid him in thinking about it, as it is often easier to understand connectivity when it is related to shapes. The Mobius strip is real, in that many things posses that connectivity. However no sequence of atoms can arrange themselves into the shape of August Mobius.
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