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What's this called?

  1. Jun 1, 2007 #1
    Take the unit square & make a zig-zag line like a staircase from one corner to the opposite one. Then the total distance if you add up the vertical parts & & horizontal parts is 2. Even if you make trillions & trillions of 'stairs' the sum of all the vertical parts & horizontal parts is still 2 even though the graph would look more & more like a diagonal line, whose length of course is [tex]\sqrt{2}[/tex]. Someone mentioned this example before & put up a link to the mathworld page on it but I couldn't find it & nothing I searched for seemed to work. :confused:
  2. jcsd
  3. Jun 1, 2007 #2
    Hummm... fractals?
  4. Jun 1, 2007 #3


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    Homework Helper

    The diagonal paradox, again. Use the search button. :smile:
  5. Jun 2, 2007 #4
    Minkowski's L1 distance

    Taxicab metric?
  6. Jun 2, 2007 #5

    Chris Hillman

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    Science Advisor

    Weyl Tile argument

    It's related to the "Weyl Tile argument", which is discussed in some books on philosophy of mathematics, and even some web pages:
    The argument as stated there isn't serious, but this has serious applications to why naive "quantization" of space won't work. See spin networks for a more sophisticated approach: http://math.ucr.edu/home/baez/penrose/

    It's also related to a "paradox" in geometric measure theory, which is probably closer to the applications you have in mind, huh? See p. 129 of Spivak, Calculus on Manifolds.
    Last edited: Jun 2, 2007
  7. Jun 4, 2007 #6
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