Poincare conjecture
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I am sure the fact that this Russian mathematician has won this award[^] is a re-post today but when I was reading this article I came across this sentence explaining what this problem entailed: "The riddle Perelman tackled is called the Poincare conjecture[^], which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere." Now trust me when I say that I am by no means a math whiz, and I in no way want to belittle his efforts or the efforts of other mathematicians at all. My quandary here probably just points out a shortcoming in my understanding of both higher level math constructs and the effort it takes to solve these problems, but... Please first tell me why this was so hard? I think, given the problem statement, any reasonable person that has ever seen and held some basic solids in their hands would agree that you cannot form a doughnut shaped object into a sphere without ripping it and that any other solid shape can be. I mean it really goes without saying I thought. That is to say that any dimensional solid with a genus >=1 cannot be transformed into a dimensional solid with a genus of 0 without breaking some surfaces. Second please explain to me why we needed a mathematical proof for this? Is it not enough of a proof to simply see the shapes, even a basic genus 1 solid, and just know by looking at it that unless you tear a surface there is no way you are going to get it into a genus 0 form? Maybe I am tired and not seeing this clearly right now but this has been bugging me all day.
I am not that good at maths but i'd like to share my thoughts on this. The base of mathimatics are a few simple axioms. Like "1+1=2" and "there is only one line that joins 2 points". Those axioms are accepted as true only based on human experience and cannot be proved". All axioms were set somewhat 2500 years ago. Taking those axioms we can prove other things, like "5+6=11" and "the volume of a box is n^3". Those are theorems which can later be used along with axioms to prove (or disprove) more complex problems. Seeings those axioms as a grammar, there is a limit to the number of statements you can constract, and some of those statements can not be proved based on this set of axioms, even if they are indeed true! (see incompleteness_theorem[^]) The question with the "Poincare conjecture" was if it can be proved or not using maths. (as you have stated, everyday experience tell as that it has to be true). Do we really need a proof since 100 out of 100 people can see using common sence that it's true? Taking it as graded (without a proof), whould simply mean that we enter it as an axiom in the system of mathimatics. Now, adding a new axiom is something that has not be done before as far as i know. It's like euclid holds a nonexpiring patent to it. I don't see any mathematician beeing happy with the idea of adding such a complex problem side by side with the rest of very simple axioms. Supposed we did added a problem X that can't be proved as an axiom, would had the following efect. 1)We can now prove those problems that required that X problem to be true. 2)Again, Seeings those axioms as a grammar, there would be an expansion to the number of statements that we can construct. There whould be a number of new statements that can be constructed, statements that neither be proved nor desproved. (see incompleteness_theorem[^])
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I am sure the fact that this Russian mathematician has won this award[^] is a re-post today but when I was reading this article I came across this sentence explaining what this problem entailed: "The riddle Perelman tackled is called the Poincare conjecture[^], which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere." Now trust me when I say that I am by no means a math whiz, and I in no way want to belittle his efforts or the efforts of other mathematicians at all. My quandary here probably just points out a shortcoming in my understanding of both higher level math constructs and the effort it takes to solve these problems, but... Please first tell me why this was so hard? I think, given the problem statement, any reasonable person that has ever seen and held some basic solids in their hands would agree that you cannot form a doughnut shaped object into a sphere without ripping it and that any other solid shape can be. I mean it really goes without saying I thought. That is to say that any dimensional solid with a genus >=1 cannot be transformed into a dimensional solid with a genus of 0 without breaking some surfaces. Second please explain to me why we needed a mathematical proof for this? Is it not enough of a proof to simply see the shapes, even a basic genus 1 solid, and just know by looking at it that unless you tear a surface there is no way you are going to get it into a genus 0 form? Maybe I am tired and not seeing this clearly right now but this has been bugging me all day.
I think the point is to develop a tool or method against a known case so you can apply it to unkown cases. Elaine :rose:
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Yes, both Bassam Abdul-Baki and I posted it yesterday. :-D This is a very historic and important problem and does not involve just "shapes" but 3-dimensional manifolds which are an extremely important class of mathematical objects. Much like Fermat's last theorem (though arguably more important) it seems simple, yet the general mathematical proof is extremely difficult. Perelman solved it using Ricci flow. If you think of collapsing a geometric object down to a point, you can assign a "flow", similar to heat diffusion, to what is called the Reimann metric. The Reimann metric is basically a way of describing how space is shaped and so the Ricci flow describes how the Reimann metric deforms. In other unsuccessful approaches, many mathematicians encountered a difficulty in studying Ricci flow to solve the conjecture. Singularities or infinite points arose which rendered a proof beyond their reach. The idea that Perelman had was to remove those singularities - a method called Ricci flow surgery. By being able to remove these singularities each time they arose Perelman was able to establish a proof of the conjecture. It's an extremely important problem because it's proof tells us much about how space behaves and this has application in many sciences, not the least of which is general relativity. I think it is a very, very fundamental mathematical result.
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I am sure the fact that this Russian mathematician has won this award[^] is a re-post today but when I was reading this article I came across this sentence explaining what this problem entailed: "The riddle Perelman tackled is called the Poincare conjecture[^], which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere." Now trust me when I say that I am by no means a math whiz, and I in no way want to belittle his efforts or the efforts of other mathematicians at all. My quandary here probably just points out a shortcoming in my understanding of both higher level math constructs and the effort it takes to solve these problems, but... Please first tell me why this was so hard? I think, given the problem statement, any reasonable person that has ever seen and held some basic solids in their hands would agree that you cannot form a doughnut shaped object into a sphere without ripping it and that any other solid shape can be. I mean it really goes without saying I thought. That is to say that any dimensional solid with a genus >=1 cannot be transformed into a dimensional solid with a genus of 0 without breaking some surfaces. Second please explain to me why we needed a mathematical proof for this? Is it not enough of a proof to simply see the shapes, even a basic genus 1 solid, and just know by looking at it that unless you tear a surface there is no way you are going to get it into a genus 0 form? Maybe I am tired and not seeing this clearly right now but this has been bugging me all day.
Some of the most obvious problems have turne out to be wrong. some of the easiest mathematical problems can be solved using mathematical induction. However, to the layman, if s/he tries the first few examples and sees a pattern, they will automatically believe it to be true. However, some problems work for the first few dozen numbers and then start to fail way down the line. That is why a proof is needed to say if it's absolutely true without a doubt.
"I know which side I want to win regardless of how many wrongs they have to commit to achieve it." - Stan Shannon Web - Blog - RSS - Math - LinkedIn
Last modified: Wednesday, August 23, 2006 7:18:01 AM --
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I am not that good at maths but i'd like to share my thoughts on this. The base of mathimatics are a few simple axioms. Like "1+1=2" and "there is only one line that joins 2 points". Those axioms are accepted as true only based on human experience and cannot be proved". All axioms were set somewhat 2500 years ago. Taking those axioms we can prove other things, like "5+6=11" and "the volume of a box is n^3". Those are theorems which can later be used along with axioms to prove (or disprove) more complex problems. Seeings those axioms as a grammar, there is a limit to the number of statements you can constract, and some of those statements can not be proved based on this set of axioms, even if they are indeed true! (see incompleteness_theorem[^]) The question with the "Poincare conjecture" was if it can be proved or not using maths. (as you have stated, everyday experience tell as that it has to be true). Do we really need a proof since 100 out of 100 people can see using common sence that it's true? Taking it as graded (without a proof), whould simply mean that we enter it as an axiom in the system of mathimatics. Now, adding a new axiom is something that has not be done before as far as i know. It's like euclid holds a nonexpiring patent to it. I don't see any mathematician beeing happy with the idea of adding such a complex problem side by side with the rest of very simple axioms. Supposed we did added a problem X that can't be proved as an axiom, would had the following efect. 1)We can now prove those problems that required that X problem to be true. 2)Again, Seeings those axioms as a grammar, there would be an expansion to the number of statements that we can construct. There whould be a number of new statements that can be constructed, statements that neither be proved nor desproved. (see incompleteness_theorem[^])
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Ray Cassick wrote:
"The riddle Perelman tackled is called the Poincare conjecture[^], which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere."
Reading that quote, a question that comes to my mind is why? What is the purpose of this exercise? :~
I'd love to help, but unfortunatley I have prior commitments monitoring the length of my grass. :Andrew Bleakley:
Understanding the very nature of space time and the forces that hold this universe together is one big study in topology (the study of shapes and surfaces and how they change from one form to another). Einstein proved that gravity is merely spacetime being bent by mass, so understanding this requires topology. The more we understand the harder the questions become and the more powerful the tools we need. Answering questions such as this force mathematicians to develop new tools and methadologies to solve the problem. These, in turn, often give rise to solutions in other areas that had previously been unsolvable. Working on questions like these are very, very important.
cheers, Chris Maunder
CodeProject.com : C++ MVP
FIX: A MFC program created in Visual Studio .NET 2003 unexpectedly quits when you try to close it[^]
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I am sure the fact that this Russian mathematician has won this award[^] is a re-post today but when I was reading this article I came across this sentence explaining what this problem entailed: "The riddle Perelman tackled is called the Poincare conjecture[^], which essentially says that in three dimensions, a doughnut shape cannot be transformed into a sphere without ripping it, although any shape without a hole can be stretched or shrunk into a sphere." Now trust me when I say that I am by no means a math whiz, and I in no way want to belittle his efforts or the efforts of other mathematicians at all. My quandary here probably just points out a shortcoming in my understanding of both higher level math constructs and the effort it takes to solve these problems, but... Please first tell me why this was so hard? I think, given the problem statement, any reasonable person that has ever seen and held some basic solids in their hands would agree that you cannot form a doughnut shaped object into a sphere without ripping it and that any other solid shape can be. I mean it really goes without saying I thought. That is to say that any dimensional solid with a genus >=1 cannot be transformed into a dimensional solid with a genus of 0 without breaking some surfaces. Second please explain to me why we needed a mathematical proof for this? Is it not enough of a proof to simply see the shapes, even a basic genus 1 solid, and just know by looking at it that unless you tear a surface there is no way you are going to get it into a genus 0 form? Maybe I am tired and not seeing this clearly right now but this has been bugging me all day.
Ray Cassick wrote:
think, given the problem statement, any reasonable person that ...
Ah - but therein lies the beauty of Mathematics. Often you may say this about things that seem totally reasonable to a given person but then when you try and prove it - meaning show that for every single case no matter what - you find a subtlety, a teeny, tiny hole in your logic. Statements such as "time progresses at the same rate for everyone" or "the more you push something the faster it goes" get a little bent out of shape when you start rigourously trying to prove them, and out of that attempt to prove those sensible "anyone knows this" we have General Relativity.
cheers, Chris Maunder
CodeProject.com : C++ MVP
FIX: A MFC program created in Visual Studio .NET 2003 unexpectedly quits when you try to close it[^]
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Whatever. I was trying to make a point there. I don't remember what are the actual axioms.
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Whatever. I was trying to make a point there. I don't remember what are the actual axioms.
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Ray Cassick wrote:
think, given the problem statement, any reasonable person that ...
Ah - but therein lies the beauty of Mathematics. Often you may say this about things that seem totally reasonable to a given person but then when you try and prove it - meaning show that for every single case no matter what - you find a subtlety, a teeny, tiny hole in your logic. Statements such as "time progresses at the same rate for everyone" or "the more you push something the faster it goes" get a little bent out of shape when you start rigourously trying to prove them, and out of that attempt to prove those sensible "anyone knows this" we have General Relativity.
cheers, Chris Maunder
CodeProject.com : C++ MVP
FIX: A MFC program created in Visual Studio .NET 2003 unexpectedly quits when you try to close it[^]
Chris Maunder wrote:
time progresses at the same rate for everyone
Any guy out shopping with his wife/girlfriend knows this one isn't true.
Software Zen:
delete this;