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  • Math problem

    Since I have had great luck in being able to have my questions answered re my rifle problem I would like to throw this out.

    This is my second time through, A MANY-COLORED GLASS, by Freeman Dyson, this week, so I know a thorough understanding of this concept is crucial to understanding this part of the book and I don't seem to be able to grasp it. The problem is one of pyysics and has to do with symetry breaking, going from a disordered state to an ordered state.

    He writes: "Another name for the process of phase transition from disorder to order is symmetry breaking. From a mathematical point of view, a disordered phase has a higher degree of symmetry than an ordered phase. For example, the enviorment of a molecule of water dissolved in humid air is the same in all directions, while the envoronment of the same molecule after it is precipitated into a snowflake is a regular crystal with a crystalline axis oriented along particular directions. The molecule sees its environment change from the greater symmetry of a sphere to the lesser symmetry of the hexagonal prism. The change in the environment from disorder to order is associated with the sudden loss of symmetry. Sudden loss of symmetry is characteristic of many of the most important phase transitions in the history of the universe."

    What I don't get is what does he mean when he says, "...from a mathematical point of view..."?

    Also puzzling to me is, it seems like a hexagonal prism is more ordered than a water molecule. Water molecules aren't in a distinct shape, they change shape constantly. The hexagonal prism, the result of freezing the water molecule, on the other hand, is ridgid. So it seems to me that the hexagonal prism is more ordered than the water molecule.

    At first I wondered if there was a misprint but writing subsequent to the above paragraph
    follows the same logic. I don't get it.

    I have been stumped before with things like this, that I don't understand, and attempts to help myself learn by following Google threads only serve to confuse me more. In order to understand these problems I generally need a basic explaination or an analogy to which I can relate.

    Any help either in a direction of something to read or a basic explaination would be most certainly appreciated.

    Thanks,
    Tom

  • #2
    Re: Math problem

    Originally posted by Tom W View Post
    Since I have had great luck in being able to have my questions answered re my rifle problem I would like to throw this out.

    This is my second time through, A MANY-COLORED GLASS, by Freeman Dyson, this week, so I know a thorough understanding of this concept is crucial to understanding this part of the book and I don't seem to be able to grasp it. The problem is one of pyysics and has to do with symetry breaking, going from a disordered state to an ordered state.

    He writes: "Another name for the process of phase transition from disorder to order is symmetry breaking. From a mathematical point of view, a disordered phase has a higher degree of symmetry than an ordered phase. For example, the enviorment of a molecule of water dissolved in humid air is the same in all directions, while the envoronment of the same molecule after it is precipitated into a snowflake is a regular crystal with a crystalline axis oriented along particular directions. The molecule sees its environment change from the greater symmetry of a sphere to the lesser symmetry of the hexagonal prism. The change in the environment from disorder to order is associated with the sudden loss of symmetry. Sudden loss of symmetry is characteristic of many of the most important phase transitions in the history of the universe."

    What I don't get is what does he mean when he says, "...from a mathematical point of view..."?

    Also puzzling to me is, it seems like a hexagonal prism is more ordered than a water molecule. Water molecules aren't in a distinct shape, they change shape constantly. The hexagonal prism, the result of freezing the water molecule, on the other hand, is ridgid. So it seems to me that the hexagonal prism is more ordered than the water molecule.

    At first I wondered if there was a misprint but writing subsequent to the above paragraph
    follows the same logic. I don't get it.

    I have been stumped before with things like this, that I don't understand, and attempts to help myself learn by following Google threads only serve to confuse me more. In order to understand these problems I generally need a basic explaination or an analogy to which I can relate.

    Any help either in a direction of something to read or a basic explaination would be most certainly appreciated.

    Thanks,
    Tom
    Makes sense to me. Symmetry's basic definition as I understand is division with equal parts so to speak. For example a pipe, if it were perfect, could be split in half lengthwise would have bilateral symmetry. One division in mathematical terms.

    The author is stating that a "disordered phase" has a higher degree of symmetry meaning it can be divided more times into equal parts while an ordered phase may have symmetry it will have less opportunity to do so.

    I'm just a plumber though.

    Now quit reading books and grease the tractor.

    J.C.

    EDIT: CHANGED TO PIPE FOR SLIM.
    Last edited by BobsPlumbing; 02-09-2009, 12:10 AM.

    Comment


    • #3
      Re: Math problem

      Originally posted by JCsPlumbing View Post
      Makes sense to me. Symmetry's basic definition as I understand is division with equal parts so to speak. For example a person, if they were perfect, that could be split in half from head to toe facing you would have bilateral symmetry. One division in mathematical terms.

      The author is stating that a "disordered phase" has a higher degree of symmetry meaning it can be divided more times into equal parts while an ordered phase may have symmetry it will have less opportunity to do so.

      I'm just a plumber though.

      Now quit reading books and grease the tractor.

      J.C.

      I suppose there would be bilateral symmetry except their heart would be on one side and their appendix on the other, not to mention all the blood everywhere.

      And I bet a snowflake formed from a single molecule of water would be dang hard to see!
      And with TWO atoms of hydrogen and ONE atom of oxygen it looks rather like a silhouette of Mickey Mouse's head
      "Man will do many things to get himself loved, he will do all things to get himself envied." Mark Twain

      Comment


      • #4
        Re: Math problem

        Originally posted by SlimTim View Post
        I suppose there would be bilateral symmetry except their heart would be on one side and their appendix on the other, not to mention all the blood everywhere.

        And I bet a snowflake formed from a single molecule of water would be dang hard to see!
        And with TWO atoms of hydrogen and ONE atom of oxygen it looks rather like a silhouette of Mickey Mouse's head
        I was speaking theoretically for an example. Maybe I should have used a piece of pipe.

        Fixed.

        J.C.
        Last edited by BobsPlumbing; 02-09-2009, 12:11 AM.

        Comment


        • #5
          Re: Math problem

          Count the total number of each side of each angle of the snowflake and compare them to those on the molecule of water prior to the change. When the water was in a liquid state it was in ordered phase, as it changed state into a snowflake it became a disordered phase.

          Mark
          "Somewhere a Village is Missing Twelve Idiots!" - Casey Anthony

          I never lost a cent on the jobs I didn't get!

          Comment


          • #6
            Re: Math problem

            I don't have a great mind and I may sound real stupid, but here's what I think he means by "From a Mathematical View". The sphere has symetry because it perfectly round or the same even as a three dimentional shape, as opposed to the snowflake? I believe the author speaks of perfection in regards to math because perfection can be an objective term depending on the context. I believe Man is perfect in his imperfection, he is what nature or God intended. Both the sphere and the snowflake are perfect for what they are but only the sphere has perfection on a mathematical level? Did that sound bad?

            Comment


            • #7
              Re: Math problem

              Many Many Thanks for the answers. I think I got it but it is pretty late for an old guy like me. I will attempt to reread the section tomorrow with your explainations in mind and see if I can wrap my brain around it.

              Thanks again,

              Tom

              Comment


              • #8
                Re: Math problem

                i don't know much about the math, never have and probably never will. but that's okay, i can accept that.

                i'm thinking planets. spheres. organized and true.

                rings of saturn, they have a shape but the individual parts that make up the rings are disorganized.

                clear as mud to me.

                Vince

                Comment


                • #9
                  Re: Math problem

                  The author mentions the "environment of the droplet" compared to the crystal shape of the solid phase of the same droplet. The symmetry of the droplet itself is one thing, but when you add in the environment around it, you somewhat change the equation. Then again, at that point you are not necessarily looking at just symmetry but symmetry in the environment. A single object has whatever symmetry it has. However, if you then associate symmetry as including whatever is surrounding the object, you have to extend the lines of symmetry. When you do that, you induce variability (at least I would think so). As variability increases, constancy decreases and therefore you have a decrease in symmetry. Maybe. I'm all confuzzled now. Thanks... :P

                  In the environment, if you assume the water droplet is in the air, symmetry is only viable as a measurement of the object itself. If you take the molecular symmetry of the water droplet and expand that to the environment, you break symmetry because of the molecular particles in the air itself. In other words, I think the premise is flawed.
                  I put it all back together better than before. There\'s lots of leftover parts.

                  Comment


                  • #10
                    Re: Math problem

                    Originally posted by VASandy View Post
                    The author mentions the "environment of the droplet" compared to the crystal shape of the solid phase of the same droplet. The symmetry of the droplet itself is one thing, but when you add in the environment around it, you somewhat change the equation. Then again, at that point you are not necessarily looking at just symmetry but symmetry in the environment. A single object has whatever symmetry it has. However, if you then associate symmetry as including whatever is surrounding the object, you have to extend the lines of symmetry. When you do that, you induce variability (at least I would think so). As variability increases, constancy decreases and therefore you have a decrease in symmetry. Maybe. I'm all confuzzled now. Thanks... :P

                    In the environment, if you assume the water droplet is in the air, symmetry is only viable as a measurement of the object itself. If you take the molecular symmetry of the water droplet and expand that to the environment, you break symmetry because of the molecular particles in the air itself. In other words, I think the premise is flawed.
                    Just when I thought I was beginning to understand...

                    I didn't have a good grasp of the concept originally but when he added the part about the environment that really threw me. But, Oh Boy here I am heading out on a limb with a saw, I also think he is wrong, or at least I don't think his assumption is correct or his premise is flawed or something...

                    But, that said, the passage works, makes sense to me at least, when interpreted from the explainations by the other posters.


                    Tom

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