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    The Lifeguard Problem is interesting; I just proved it for fun. You're standing on the beach and there's someone in trouble in the ocean. What's the fastest route to them, given the speed over sand and the speed in water? The answer turns out to be another way of expressing Snell's Law - refraction.

    So light chooses the fastest path too.
    •  
      CommentAuthorAngus
    • CommentTimeJul 19th 2013 edited
     
    Posted By: Andrew PalfreymanThe Lifeguard Problem is interesting. You're standing on the beach and there's someone in trouble in the ocean. What's the fastest route to them, given the speed over sand and the speed in water? The answer turns out to be another way of expressing Snell's Law - refraction.

    So light chooses the fastest path too.



    This is one of the oldest ways known of demonstrating Snell's Law. Imagine a file of soldiers marching slantwise down the beach. Those in the water march slower. The file represents a plane wavefront.


    You will not be anachronistic if you imagine the soldiers wearing gaiters and shakos and carrying muskets.


    And BTW that's called Fermat's principle, that "fastest path" thing.
  3.  
    It only demonstrates Snell's law if one assumes that light takes the fastest route.
    •  
      CommentAuthorAngus
    • CommentTimeJul 19th 2013
     
    ?

    Snell's law is empirical. You are at liberty to assume what you like, but if you don't agree with Snell's law you are...


    (I am reluctant to say this lest it should violate your civil rights...


    But on the other hand reluctant to keep silent lest I fail in my fiduciary duty...)


    WRONG!!!
  5.  
    Indeed. But Fermat's principle is a hypothesis about light minimising its travel time. Snell's Law is a natural consequence of Fermat's principle. My point is that it's a bit peculiar to me that light does indeed take the fastest route. Is there a greater idea at work here?
    •  
      CommentAuthorAngus
    • CommentTimeJul 19th 2013
     
    Posted By: Andrew PalfreymanIndeed. But Fermat's principle is a hypothesis about light minimising its travel time. Snell's Law is a natural consequence of Fermat's principle. My point is that it's a bit peculiar to me that light does indeed take the fastest route. Is there a greater idea at work here?



    ?


    I would have said you have it exactly backwards. Fermat's principle is a generalisation from Snell's law.
  7.  
    It doesn't matter which way round they go - they are equivalent statements of an empirically observed fact - i.e. that Snell's law holds.
    •  
      CommentAuthorAngus
    • CommentTimeJul 19th 2013
     
    Then we are having a violent agreement.


    It's all the Principle of Least Action, anyway.
  9.  
    Yup. Which is usually associated with massy objects. But via E = mc2, light is included, since we now apply that principle to all forms of mass or energy. So it makes sense.
    •  
      CommentAuthoralsetalokin
    • CommentTimeJul 19th 2013 edited
     
    never mind.

    internal ramps, sorry.
  11.  
    What I meant to say is that the "fastest path" is defined by the path that light takes. So saying that "light takes the fastest path" is kind of circular.

    It's not the "shortest path" as you can prove to yourself by using a piece of string, a ruler, and a bunch of interfaces between materials of differing refractive indices.

    The "fastest path" would be the path that light would take between your starting and ending points if there were only pure and unadulterated vacuum in between them.... and good luck finding any of that.
  12.  
    It kinda creates it as it goes along via wavefront reinforcement. First man home wins.
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      CommentAuthorAngus
    • CommentTimeJul 20th 2013
     
    Is that what you call "Huygentailing it home"?
  14.  
    It is magic. Embrace the mystery. Or something
    •  
      CommentAuthorDuracell
    • CommentTimeAug 19th 2013
     
    •  
      CommentAuthoralsetalokin
    • CommentTimeAug 19th 2013 edited
     
    It seems that you have demonstrated that the perimeters of all three figures are the same.

    Hence pi=4.

    • CommentAuthorjoshs
    • CommentTimeAug 19th 2013
     
    Sterling or LMM might believe that.
  18.  
    sine, tan and a straight line.
    •  
      CommentAuthorAngus
    • CommentTimeAug 20th 2013
     
    Posted By: alsetalokinIt seems that you have demonstrated that the perimeters of all three figures are the same.

    Hence pi=4.



    OK, but if you draw the square +inside+ the circle and use the same argument you get a perimeter of 2*2^(1/2). If you average the two results you get 3.414.
    •  
      CommentAuthoralsetalokin
    • CommentTimeAug 20th 2013 edited
     
    Brilliant! And if you subtract the 0.272620962 to account for the thickness of the line, you wind up with 3.1415926, the amount of sugar in an apple pi.

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