Hm turning a sphere inside out what's the point?

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Discussion Overview

The discussion revolves around the mathematical concept of turning a sphere inside out, exploring its implications within topology and the perceived purpose of such mathematical endeavors. Participants question the practical applications of this concept and its relevance to other scientific fields.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the mathematical proof of turning a sphere inside out and its purpose.
  • Others argue that this concept illustrates the complexities of three-dimensional problems compared to two-dimensional ones.
  • A participant questions the realm of mathematics involved, suggesting it is topology.
  • There are claims that not all mathematics needs to have a practical application, with some emphasizing the intrinsic value of pure mathematics.
  • Some participants suggest that concepts in pure mathematics can lead to applications in other fields, although this is not always immediately apparent.
  • A later reply discusses the potential relevance of this concept to string theory and other scientific applications, such as fluid dynamics and electromagnetics.
  • Several participants express skepticism about the necessity of practical applications for mathematical concepts, suggesting that the pursuit of knowledge for its own sake is valid.
  • One participant highlights that while the specific proof may not have immediate consequences, the techniques used could be beneficial for proving other results.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of practical applications for pure mathematics. While some argue for the intrinsic value of mathematical exploration, others seek tangible benefits from such concepts.

Contextual Notes

The discussion reflects a range of opinions on the relationship between pure mathematics and its applications, with some participants emphasizing the historical context of mathematical discoveries and their eventual relevance.

Who May Find This Useful

This discussion may be of interest to those exploring the philosophy of mathematics, the field of topology, or the connections between pure mathematics and applied sciences.

Mathematics news on Phys.org
To you it is of no importance at all. To mathematicians it shows that things are not quite as simple as we might expect and that problems in 3 dimensions can be much different from problems in 2 dimensions (where turning a circle inside out in exactly the same way is impossible).
 
Here's the full video:

http://video.google.com/videoplay?docid=-6626464599825291409
 
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Sorry for the novice-ness, but what realm of mathematics is that? It is topology?
 
horrors, does pure math have a point?
 
mathwonk said:
horrors, does pure math have a point?

Before taking an advanced mathematical logic course, I would have said that pure math should be studied for its own sake. Now I'm not so sure.
 
ok


MY QUESTION IS... ok...
they figure out how to turn that sphere inside out...

WHAT IS THE PRACTICAL USE OF THIS...or there isn't any and matematicians are just doing their research for the sake of nothing...

Does that help other sciences to create some assumptions or models for something that can be practical or bring improvements to something not so developed...


i don't get it...

and furthermore...they set conditions to whatever they want to do...

what if that matter can be bended SHARPLY or be creased - and the surface can't go thru itself... what would then happen... idk..

they prove something that it is impossible in practice and even it is impossible in its abstract version if they do not set the needed conditions and rules in this abstract world :)... why do they do that then?
 
Not everything needs a purpose applicable to the real world. If you can not appreciate this, you will never appreciate pure mathematics.
 
  • #10
Gib Z said:
Sorry for the novice-ness, but what realm of mathematics is that? It is topology?

Yes. Modern topology was founded by Poincare and Brouwer in their study of the behavior of solutions of ordinary differential equations. Theorems of topology have unbelievably powerful applications in many areas of pure and applied mathematics. Ie., the Brouwer fixed point theorem, the Poincare index theorem, the generalized Jordan curve theorem/Mazur's theorem, the Hopf fibration, knot theory, the study of cohomology and the discovery of category theory. As in many areas of pure mathematics, the results far outweigh the initial seed of studying ordinary differential equations.
 
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  • #11
we have all asked them from time to time, but a general question of "what is that good for?" risks being a bit like a parent wondering why his kids listen to rock and roll, it misses the point and shows a rigidity and lack of imagination or intellectual curiosity.

obviously, this theorem is interesting because it reveals a surprizing aspect of geometry, one that contradicts ordinary intuition. the point is not that one can imagine a sphere turning inside out, but rather that it can be done without ever introducing any wrinkle in the sphere as it passes through itself.

the intuitive operation is to push inward on the sphere at each pole, and continue pushing the sphere through itself and inside out. but this leaves a circular wrinkle at the last stage at the equator. amazingly this can be avoided.

any fact about the geometry of manifolds may or may not have applications in engineering and physics or cosmology.

topologically it was a question of the connectedness of a certain set of deformations of a smooth manifold, and was proved by Smale.

It is also an interesting story of mathematical imagination and communication, since the original expository article in scientific american reports that the chain of explanation from the man who envisaged the transformations to the one who drew the pictures passed through one link of explanation by a blind person, I believe it was bernard morin.
 
  • #12
does anyone ask what is the point of a picasso? nevermind the quite cliche comparison of art to math but why do people necessarily assume math needs a purpose? damn our primary education system.
 
  • #13
mathwonk said:
horrors, does pure math have a point?

A concept in pure maths can have a use that is not apparent until several decades after it's been discovered.
 
  • #14
that's why maybe people call that math is the QUEEN/king of all SCIENCES.

Everything is based on math but math is based and depends purely on one's imagination and intellectual capacity.

The world is so weird... I think I am going crazy when I am trying to understand it or think about the meaning of life ...etc...


Maybe we will never find out why or how the world was created.

Maybe our existence is just a state of our mind.

Maybe we can't find the answers we are looking for because we are a part of them. :P

it's getting late :P

ttyl later people
 
  • #15
There are two justifications that I see being offered:

1. Some topics developed by pure math have later been applied to the sciences.

2. Not everything has to be useful.

Of these I would say that (2) is more honest. I think a study should be under taken with respect to (1), since as pure math has flourished in recent times I think (1) is less true than in the past. Never mind anecdotal reports, we need to count how much of the total mathematical content generated in the 19th century is in use today.

As for (2), I often use this line for students and I am tempted to agree with it in general, especially when presented with something like the video in this thread. But getting students to study math is one thing, and paying professionals to dedicate their efforts to it is another. Personally, I don't think (2) is sufficient justification for a lifetime career in pure mathematics. Look at all the scientific efforts that could help the world e.g. better solar panels, recycling methods, fusion power, agriculture, birth control, communications networking, medicine, transportation, etc.
 
  • #16
I don't know about this not being useful. I am just in my first year of engineering, but I read this book on chaos which focused on the qualitative aspects of the subject and it partially introduced string theory and what its trying to accomplish.

As far as I know, string theory is based on topology and if it really is the next big breakthrough, then procedures such as inverting this sphere could have real practical applications. In any case, I think this particular method might even be applicable in fluid dynamics or even in electromagnetics. For example, the Earth's magnetic field is something akin to a sphere, and perhaps this method of inverting the sphere could describe some phenomenon where the magnetic field varies as such. The practical applications of any theoretical concept are only limited by the imagination of the practitioner. Isnt that true?
 
  • #17
Does anything have a point?
 
  • #18
morphism said:
Does anything have a point?
the cartesian plane has points?
 
  • #19
One thing that people often forget when it comes to "useless" proofs is that although the fact itself may not have any immediate consequences, sometimes other results that depend on the proof can be useful. Even more often, although the fact might not be very useful, the techniques used to prove it are often very useful to prove other things.

As for why people care about this one example specifically: eh, it's just something that mathematicians think is cool, unexpected, etc.
 
  • #20
morphism said:
Does anything have a point?

Euclidean geometry has points. Are you calling Euclid a liar?
 
  • #21
Dragonfall said:
Euclidean geometry has points. Are you calling Euclid a liar?

Points are undefined elements in Euclidean geometry. :-p Similar to the direction of this thread.
 
  • #22
zinedine_88 said:
okMY QUESTION IS... ok...
they figure out how to turn that sphere inside out...

WHAT IS THE PRACTICAL USE OF THIS...or there isn't any and matematicians are just doing their research for the sake of nothing...

Does that help other sciences to create some assumptions or models for something that can be practical or bring improvements to something not so developed...i don't get it...

and furthermore...they set conditions to whatever they want to do...

what if that matter can be bended SHARPLY or be creased - and the surface can't go thru itself... what would then happen... idk..

they prove something that it is impossible in practice and even it is impossible in its abstract version if they do not set the needed conditions and rules in this abstract world :)... why do they do that then?

In the spirit of an engineer and physicist, the mathematics has to be created before we can apply it. It might now currently have an application but perhaps in the future it could.

I have a question: How does a sphere pass through itself, if it is supposed to be a solid object?
 
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  • #23
Feldoh said:
I have a question: How does a sphere pass through itself, if it is supposed to be a solid object?
Simple: the sphere we are talking about is not solid
 
  • #24
Dragonfall said:
Euclidean geometry has points. Are you calling Euclid a liar?

are you serious dude? did you seriously try to steal my joke?
 
  • #25
I bring the human element into your joke. You can thank me when you receive your fields medal.
 
  • #26
Wow, I've never been in the presence of people that joke about winning fields medals.
 
  • #27
ice109 said:
are you serious dude? did you seriously try to steal my joke?


Actually you stole the joke from me, I came up with the joke in the late spring of 1976. I first used it over a letter correspondence between myself, and a German Mathematician named Kir Yeltsa.

I soon after patented the the joke with the US Government (Patent http://www.youtube.com/watch?v=eBGIQ7ZuuiU"

Please cease all use of this joke from here on forward.

And we should lock this thread since it got way off topic, and there is very little math.
 
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  • #28
The mathematics developed to prove the theorem will spur more ideas, more theorems, and then more mathematics still. Some results obtained in the process will have applications.
 
  • #29
I am just starting to study/understand topology.

Don't burst my bubble of interest. Please! Invert it ... ok :)
And we should lock this thread since it got way off topic, and there is very little math.

That's so limiting. Where is your sense of humour?
oH YA mATH has NO humour, so it is of no use.

I find it very, very interesting. I want to learn more.

The mathematics developed to prove the theorem will spur more ideas, more theorems, and then more mathematics still. Some results obtained in the process will have applications.

agree - thanks
 
  • #30
apparently "turning a sphere inside out" means this: if f:S-->R^3 is the inclusiion map of the sphere, and g:S-->R^3 equals -f, there is a continuous map H:SxI-->R^3,

such that for every t in I, the restriction of H to Sx{t} has injective derivative, and

when t=0 this restriction equals f, and when t=1 this restriction equals g.

'in particular nobody is actually turning anything solid inside out.
 

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