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Science

Poincaré Conjecture May Be Solved 299

Posted by CmdrTaco
Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."
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Poincaré Conjecture May Be Solved More

Poincaré Conjecture May Be Solved

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  • Y'know (Score:3, Insightful)

    by DarenN ( 411219 ) on Tuesday April 15, 2003 @08:46AM (#5735280) Homepage
    for the first time in ages, I'm looking forward to the discussion on this, in the hope that someone explains it in a manner I can understand

    :)

    • Re:Y'know (Score:2, Insightful)

      by kvn299 ( 472563 )
      I actually thought the article did a great job at explaining the problem. Did you read it?

    • Re:Y'know (Score:5, Funny)

      by LordYUK ( 552359 ) <jeffwright821.gmail@com> on Tuesday April 15, 2003 @08:56AM (#5735362)
      "...in the hope that someone explains it in a manner I can understand"

      You're new here, arent you?
    • please, I hit the link on the conjecture and the first 10 words made my eyes glaze over. i was pretty fair in calculus backin the day, but manohman.. while this is intellectually intresting i have a hell of a time caring since its so seemingly esoteric to be... well over-nerdy.

    • I was very disappointed in the article. I couldn't get past that bit about "2 dimensional spheres". This is either a piece of topological technical jargon that needed an explanation when it was introduced or the article is so screwed up I can't give it any credence.

      Is anyone aware of any better written popular (non-technical) reporting on this?

  • Cool. (Score:3, Funny)

    by Anonymous Coward on Tuesday April 15, 2003 @08:47AM (#5735286)
    Only two years more of eating noodles before he's rich!

  • What about the Dunwoody paper? (Score:5, Interesting)

    by Glyndwr ( 217857 ) on Tuesday April 15, 2003 @08:48AM (#5735296) Homepage Journal

    The link to mathworld.wolfram.com [wolfram.com] from the post says:

    In April 2002, M. J. Dunwoody produced a five-page paper that purports to prove the conjecture. However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected.

    So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?

  • The subject of 3 dimensional objects with holes is quite fascinating... wouldn't it be awesome if it was discovered that toroids are actually some extradimensional manifestation... Or even that Toroids have special properties allowing FTL travel...

  • Explanation (Score:5, Informative)

    by MaestroSartori ( 146297 ) on Tuesday April 15, 2003 @08:52AM (#5735331) Homepage
    Shamelessly stolen from here [claymath.org]:

    If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.


    Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?

    • Re:Explanation (Score:4, Funny)

      by jkramar ( 583118 ) on Tuesday April 15, 2003 @08:57AM (#5735370)
      Silly people... this is TOPOLOGY! It's not meant for people to USE it! It's just for mathematicians to RUMINATE UPON!

      Has Fermat's Last Theorem actually been used in practical applications? I don't think so...

      • Re:Explanation (Score:5, Insightful)

        by Vann_v2 ( 213760 ) on Tuesday April 15, 2003 @09:09AM (#5735455) Homepage
        That's not really fair. There is a lot of mathematics that is useful, especially to scientists, but something like Fermat is just one of those mathematical problems which are interesting because 1) they look very simple, but 2) turn out to be maddeningly difficult to prove. To say Fermat, which is basically a mathematical problem akin to getting Linux running on your toaster, is indicative of the field of mathematics is unfair.

      • sigh (Score:5, Insightful)

        by danro ( 544913 ) on Tuesday April 15, 2003 @09:13AM (#5735482) Homepage
        Silly people... this is TOPOLOGY! It's not meant for people to USE it! It's just for mathematicians to RUMINATE UPON!
        Has Fermat's Last Theorem actually been used in practical applications? I don't think so...

        If everyone thought like you we'd still be living in caves.
        Just because practical applications aren't totally obvious for a layman (or even a matematician) doesn't mean this will never be of practical use.
        Even if no practical applications are ever found, this proof (if it survives peer review) may well pave the way for something else that is immensly useful.
        There's just no way to tell right now.

        • Re:sigh (Score:3, Funny)

          by jensend ( 71114 )
          Remember the mathematicians' toast, though: "Here's to pure mathematics! May it never be of any use to anybody!" I think it's attributed to GH Hardy.

        • It's like kids in school saying they shouldn't have to learn algebra since it is never going to do them any good in real life... like balancing their checkbook, or knowing how to make change.

          You're right, we would still be living in caves if people sat around waiting for great, really pratical ideas to show up.

      • Re:Explanation and George Boole (Score:5, Insightful)

        by SystematicPsycho ( 456042 ) on Tuesday April 15, 2003 @09:59AM (#5735859)
        Are you on a computer right now? Ever heard of a guy called George Boole? Does a "boolean" sound familiar? Well you see, this guy called George Boole he hated mathematicians so much he decided to invent this thing called Boolean Logic. You know the, 1 & 1 == 1, 1 || 0 == 0 stuff? As it turns out it was totally useless and that's what he intended, to invent something mathematically correct that is totally useless. So thanks to George Boole for accidentally inventing the foundation of computer architecture, logic gates and boolean logic - and he has something to do with you being on the computer right now. Indeed he is pissed off as he intended it to be useless.

        Give maths time and it will applicable to your everyday life. What has been going on for the past 3,000 years?
      • Just leave mathematicians alone long enough, and they'll occasionally produce eye-poppers like the atom bomb, or the computer.

        I live in fear that one day advanced mathematicians will discover proof we really don't exist, and we all die in an instant.

      • Re:Explanation (Score:4, Informative)

        by Gleef ( 86 ) on Tuesday April 15, 2003 @10:38AM (#5736220) Homepage
        Some uses for topology:
        http://www22.pair.com/csdc/car/carhomep.htm [pair.com]

        Granted, none of this is stuff I would expect a gas station attendant to be playing with, but it's apparently increasingly important for researchers and engineers.

      • "Useless" mathematics that we use (Score:5, Insightful)

        by Len ( 89493 ) on Tuesday April 15, 2003 @11:07AM (#5736497)
        100 years ago a proof of the difficulty of factoring large numbers might only have been interesting to mathematicians. Now that we use encryption based on the difficulty of factoring products of large primes, it's very important.

        Galois fields are used for checksum algorithms, something I'm sure Galois never thought of.

        Fourier transforms are used for image compression (JPEG).

        Who knows what Poincaré's topology might be used for in the future?
      • I'm not really positive about this, but this may either aid or hinder that recent publication about the universe actually being a donut. Poincare's original question had something to do with the 3-manifold sphere being the only one of its kind (genus) or something. (See I don't know what I'm talking about.) But this proof might have some application to these recent claims about the actual shape of our universe.

        And I guess, if you want some application to that, then you might want to read some of Hawkin

    • Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?

      The conjecture itself is something fairly abstract, but it's widely considered the most important unsolved problem in topology and has so far induced a long list of false claims and proofs, some of which have led to a better understanding of low-dimensional topology. Solving the problem would further increase knowledge about topology and many fields of research in mathematics, geometry, physi

    • Re:Explanation (Score:5, Funny)

      by CommieLib ( 468883 ) on Tuesday April 15, 2003 @09:18AM (#5735529) Homepage
      Mmmmm...hypothetical donut...

    • Re:Explanation (Score:5, Funny)

      by jalet ( 36114 ) <alet@librelogiciel.com> on Tuesday April 15, 2003 @09:22AM (#5735553) Homepage
      > Now, can someone tell me what practical
      > applications there might be of this?

      An application would be to make better doughnuts, I suppose.

    • Practical Applications? (Score:4, Insightful)

      by lildogie ( 54998 ) on Tuesday April 15, 2003 @09:52AM (#5735804)
      > Now, can someone tell me what practical applications
      > there might be of this? Or is it strictly an abstract concept?

      Speaking as a layman, the practical application of these sorts of proofs is that you can use them to prove equivalent, more practical questions.

      One of the references in another comment explained that this conjecture has been proved for all other dimensions, and this 3-sphere seems to be a special case, as far as proof is concerned.

      If the Poincare' conjecture were proved, then the general case could be solved. After that, "simply" proving that another hard problem is equivalent to the Poincare' conjecture is enough to prove that other problem.

      Now, I've heard the problem described with a lasso instead of with a rubber band. I can imagine times when I'd really like to know when my lasso is going to close around something or if it's just going to slip off ;-)
      • Well, I feel much better now. You've just illustrated that owing to the work of Perelman (Poincare/lasso) and Nash (game theory), even if I was able to fake out the competition for the blond in the corner (Nash), Perelman's lasso would fail to capture her every single time.

        So in the end, its mathematically impossible for me to catch the girl. And here I thought the problem was my fixation on Star Wars action figures.

    • Re:Explanation (Score:4, Funny)

      by Enonu ( 129798 ) on Tuesday April 15, 2003 @10:21AM (#5736050)
      How can you break the rubber band in order to get the doughnut to go to a point without breaking the doughnut too?
    • ... Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized...
      Is that math geek speak for circle?

      • It's actually a traditional sphere. (See here [mathforum.org] for some sort of explanation.)


        Not being a mathemetician, I think it's a rather silly designation, however.

    • Well simply vs non simply connected surfaces are used extensively in continuum mechanics, which is used by materials scientists. Simply connected and non simply connected objects are a mathematically abstract way of saying plates, and plates with holes in them, something which materials scientists are very interested in. Continuum mech is a more mathematically rigorous way of finding stresses and strains in a particular object.

    • Re:Explanation (Score:3, Informative)

      by metlin ( 258108 )
      Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?

      Hmmm, a lot of work in mathematics may not have immediate applications or uses. But down the line, they just might get used.

      As many posters have already mentioned, Boolean algebra is one such case, and another example would be the work done by Fourier [mathphysics.com] - particularly his integral transforms and series.

      I mean, today these are used so much in DSP and the like, I doubt Fourier had these in mind
    • If this proof is correct, it will forever change the way we do texture-mapping on 4-dimensional graphics cards.

  • Google Partner Link (Score:3, Informative)

    by Anonymous Coward on Tuesday April 15, 2003 @08:52AM (#5735334)
    For the lazy/paranoid [nytimes.com].

  • Explanation (Score:2, Informative)

    by Andy Tanenbaum ( 655028 )
    For those who do not know about the Poincare Conjecture, copied from http://www.claymath.org/Millennium_Prize_Problems / Poincare_Conjecture/ If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point wit

  • What's that conjecture again? (Score:5, Informative)

    by n3k5 ( 606163 ) on Tuesday April 15, 2003 @08:54AM (#5735349) Journal
    for the first time in ages, I'm looking forward to the discussion on this, in the hope that someone explains it in a manner I can understand
    The explanation in the article [nytimes.com] is not too bad; the Wikipedia [wikipedia.org] contains a better explanation [wikipedia.org]:
    [The Poincaré] conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.


    Loosely speaking, this means that every 3-dimensional object that has a set of sphere-like properties can be stretched or squeezed until it is a 3-sphere without breaking it. Note that a 3-sphere consists of all those points in 4-dimensional space R4 that have a distance of 1 from the origin.
    • Loosely speaking, this means that every 3-dimensional object that has a set of sphere-like properties can be stretched or squeezed until it is a 3-sphere without breaking it. Note that a 3-sphere consists of all those points in 4-dimensional space R4 that have a distance of 1 from the origin.

      Well why didn't you just say so in the first place. It's so simple when you put it in plain english ...
      [/sarcasm]

      • It's so simple when you put it in plain english ...

        The 'plain English' version, despite being much longer, is not a perfect translation. It mentions 'a set of sphere-like properties', without defining which properties are included in that set.

        On the other hand, 'simply connected' is both shorter and more precise, but most people don't know what it means. However, you can look up very fine definitions at Mathworld [wolfram.com] or the Wikipedia [wikipedia.org].

        • I've done a lot of high level maths myself, and perhaps [/sarcasm] was not the best word to use in my original post; I probably should have used [/joking] instead.

          Maths seems to be one of the few things (especially at high levels such as this or when you move into R^n dimensions) that is extremely difficult, if not sometimes impossible, to put into plain english at all. How can you possibly describe, for example, a 5-dimensional object (or object in R5 space) - you can't make up an analogy or describe some

          • Like some other posters have pointed out, I don't think this can be put into plain english - I was poking fun at the thought of someone even trying!

            I disagree strongly. I don't want to flame or be rude or anything, but the mere thought of a scientific subject matter which can be explained and tought to students, yet can't be put into plain english, is sureally stupid. As I pointed out above, properly used technical terms give a lot of information in few words, but that only works because they first have to

      • I have to tell you, if that definition isn't clear to you, then there's no point in explaining the concept to you.

        There is a certain minimum amount of familiarity with the relevant field that is demanded when discussing certain concepts.

      • Re:What's that conjecture again? (Score:5, Informative)

        by Alsee ( 515537 ) on Tuesday April 15, 2003 @09:31AM (#5735615) Homepage
        It's so simple when you put it in plain english ...
        [/sarcasm]

        Ok, try this:

        We long ago proved that an ordinary sphere is the only shape in 3 dimentions with no holes in it.

        Note that the "shape" is "made of clay". You are allowed to stretch it, squish it, and bend it all you want. You aren't allowed to cut it or put a hole in it. And you can't "meld" parts togther.

        A coffee cup is the same "shape" as a donut because you can smoothly "flow" the cup part into the handle and you get a donut.

        What they just proved is that a 4-dimentional sphere is the only shape with no holes in it.

        So what? Well if you have some wierd complex 4 dimentional "thing" and you know it doesn't have any holes in it then you now know it has to be equal to a sphere. It SEEMS obvious, but it was still important to prove. It is important for many other proofs.

        Better?

        -
    • Something I don't understand in these explanations.

      Why is a good-old friendly sphere called a 2-dimensional sphere?

      Did I miss something? Are we living in Flatland?

      I am fascinated by the idea of multiple geometrical dimensions.

    • [The Poincaré] conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.
      Sure ! Next time they will try to make us believe that elliptic curves are modular...

  • What is it ? (Score:2, Informative)

    by Anonymous Coward

    Easy, i shall explain

    The conjecture that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n = 3.

    The Poincaré conjecture has proved a thorny problem ever

    • Re:What is it ? (Translation to make it easier) (Score:5, Informative)

      by MarvinMouse ( 323641 ) on Tuesday April 15, 2003 @09:18AM (#5735525) Homepage Journal
      translation to make it easier.

      basically all the poincare conjecture says is that if you have a 3 dimensional figure which is closed (therefore, it it bounded (doesn't go off to infinity in either direction), and doesn't have any "holes" in it (like a donut)) then you can take every point and map it to a point in an equivalent sphere without losing continuity (therefore, everypoint will have the same "neighbourhood" of points as it had in the initial shape.)

      ie. You can map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures.

      As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures.

      It's a very simple concept, but has been incredibly hard to prove, and what makes this conjecture even more frustrating is the fact that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove, as well as 4 and up have been solved, and were reasonably easy as well. Yet for some reason the 3 dimensional version does not lend itself easily to a simple proof.

      Everyone generally believes this is true, but no one has been able to prove or disprove it.

      If proven, this is an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.
      • what people keep seeming to misunderstand, is that that a 3-manifold is NOT a 3 dimensional object. when we speak of manifolds, we only care about the surface of the object, not its volume. therefore a 3-manifold is one with a 3 dimensional surface, i.e. a 4 dimensional object.

        another thing is interesting to note: there are a LOT of problems in mathematics in which n=1 is trivial, n=2 is hard but straight forward. n>3 is not too hard and usually falls under one proof, and n=3 is EXTRAORDINARILY difficul
        • what people keep seeming to misunderstand, is that that a 3-manifold is NOT a 3 dimensional object. when we speak of manifolds, we only care about the surface of the object, not its volume. therefore a 3-manifold is one with a 3 dimensional surface, i.e. a 4 dimensional object.

          Unfortunately, the reality (sic!) is a little bit more complicated. Terms as surface and volume as you use them don't make sense with topological objects. The problem is that you are implying that a 3-manifold is the surface of some

  • Now THATS Patience... (Score:4, Interesting)

    by drgroove ( 631550 ) on Tuesday April 15, 2003 @08:59AM (#5735388)
    "Though his early work has earned him a reputation as a brilliant mathematician, Dr. Perelman spent the last eight years sequestered in Russia, not publishing."

    "However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected."

    So, all told, Perelman is going to wait a total of 10 years from the time he started to work on the solution to the Conjecture, to the time where the scientific community lets him know if his answer is correct. Wow.
    • No, he will have to wait two years, or ten years from the start of his work to collect the one million dollars (not a bad yearly salary). He will have to wait his whole life and then some to know if his answer is correct.
      • He will have to wait his whole life and then some to know if his answer is correct.

        The beauty of mathematics is that it doesn't work like natural sciences. Once something is proved, it is forever proven and correct in maths. He will know for sure whether his answer is correct in a few months, most probably.

    • 11:15, restate my assumptions:

      1. Mathematics is the language of nature.

      2. Everything around us can be represented and understood through numbers.

      3. If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.

  • Sequel (Score:2, Funny)

    by telstar ( 236404 )
    Complex mathematics? Looks like its time for Matt Damon and Pretty-Boy Affleck to write Good Will Hunting II.
    • "Good Will Hunting II: Hunting Season" Was being filmed at the same time as Jay & Silent Bob were trying to get the movie about the comic-book characters that were based on them stopped so that "ball-lickers" on the "internet" would stop besmirching their good names.
      ...

  • Poincare Conjecture Solved Ages Ago (Score:5, Funny)

    by The Real Minister ( 666077 ) on Tuesday April 15, 2003 @09:00AM (#5735394)
    http://www.theinformationminister.com/press.php?ID =612212491 [theinforma...nister.com] we got this ages ago. i swear

  • Now I Understand... (Score:5, Funny)

    by masq ( 316580 ) on Tuesday April 15, 2003 @09:01AM (#5735403) Homepage Journal
    ... why we love talking about Linux so much - It's so damn USER-FRIENDLY compared to other geek pursuits!
    We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

  • Wait for it wait for it.... (Score:5, Insightful)

    by I Want GNU! ( 556631 ) on Tuesday April 15, 2003 @09:21AM (#5735542) Homepage
    Mathematical rigor demands we see the proof first. One decade ago Wiles thought he solved Fermat's Last Theorem but a mistake was found and worked again for several months before ultimately solving it. Faulty proofs are made all the time. Until it undergoes peer review I will be very skeptical.
    • I mean seriously, its not like this guy is starting a book tour or series of $200 seminars or anything. He's presenting his proof to the community so that it can be studied and discussed; that's what this is about.

      So what the hell are you talking about... was it supposed to be tounge-in-cheek humor?

  • Typo... (Score:3, Funny)

    by mrtroy ( 640746 ) on Tuesday April 15, 2003 @09:24AM (#5735560)
    It appears most people are spelling incorrectly! Including the sites included in the post!

    It is not "mathematician" ..... its "mathemagician"

    Please make the appropriate corrections. :)
  • Me. Hammer. Pliers. Every available 3-manifold. Can I have my $1 million please?

    (This of course assumes that 3-manifolds are malleable.)

  • this can't be (Score:3, Funny)

    by paiute ( 550198 ) on Tuesday April 15, 2003 @09:32AM (#5735626)
    I thought that this Wolfram guy was the smartest man in the universe and had all the answers. Now some brie-muncher comes along and proves something in math that Wolfram couldn't? This can only be due to one of three reasons:

    1. When Wolfram and Hart were all killed by the Beast, Wolfram was in the house.
    2. Wolfram is human and isn't as smart as the papers say.
    3. He stepped up to MCHawking and is now hanging from a tree with a sign pinned to him that reads: WHACK EMCEE.

  • Actually, Perelman is claiming much more... (Score:5, Informative)

    by Anonymous Coward on Tuesday April 15, 2003 @09:42AM (#5735718)
    Perelman isn't claiming just to have proved the Poincare Conjecture -- he's claiming to have proved the Thurston's Geometrization Conjecture [http], a much more general (and harder to explain) result. Basically, while the Poincare Conjecture just says things about 3-spheres (namely every "simply-connected" 3-manifold is a 3-sphere), the Geometrization Conjecture says that _any_ compact Riemannian 3-manifold is built in a specific way from a handful of basic building blocks (the important thing here is that you're not just considering the manifold structure, but the metric structure as well).

    Anyway, if true, this is kind of like Wiles proof of Fermat's Last Theorem -- proving an old conjecture by proving a more general (and more modern) one (in Wiles case, it was proving part of Taniyama-Shimura).

  • now I need to get a new hobby.
  • >> Since the 19th century, mathematicians have known that the sphere is the only bounded two-dimensional space with this property, but what about higher dimensions?

    Do they mean the sphere is 2-dimensional?
    • A sphere is the surface formed by points at a constant distance to a centerpoint. Therefore in 3D a sphere is a two-dimensional space: it takes two numbers to completely determine one's position on a spherical surface. For instance latitude and longitude on Earth.

  • Corollary (the proof made intuitive) (Score:4, Funny)

    by hysterion ( 231229 ) on Tuesday April 15, 2003 @02:46PM (#5738471) Homepage
    Hypothesis: Perelman's proof has no holes in it.
    Conclusion: Perelman's proof is a 3-sphere.

    Proof: Apply the Theorem to its proof.

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