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Math Help - Curve with no length

  1. #1
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    Curve with no length

    Hi!

    I am asked to find some curves which don't have length. I'm not sure whether the Koch curve is appropriate as it has infinite length.

    Thanks for your help!
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  2. #2
    Senior Member Tinyboss's Avatar
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    Re: Curve with no length

    You could also look at "space-filling curves". But really they're just other (maybe we can say "worse") examples of curves with infinite length. The term is "non-rectifiable".

    Since length is defined as the integral of the magnitude of the derivative, any curve which fails to be differentiable on a set of positive measure won't have a defined length.
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  3. #3
    Grand Panjandrum
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    Re: Curve with no length

    Quote Originally Posted by james_bond View Post
    Hi!

    I am asked to find some curves which don't have length. I'm not sure whether the Koch curve is appropriate as it has infinite length.

    Thanks for your help!
    Quote Originally Posted by Tinyboss View Post
    You could also look at "space-filling curves". But really they're just other (maybe we can say "worse") examples of curves with infinite length. The term is "non-rectifiable".

    Since length is defined as the integral of the magnitude of the derivative, any curve which fails to be differentiable on a set of positive measure won't have a defined length.
    The usal example of a function that is nowhere differentiable is the Blancmange function

    CB
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