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Math Help - [SOLVED] Koch Snowflake

  1. #1
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    [SOLVED] Koch Snowflake

    - URGENT -

    Using appropriate mathematical formulae and patterns to determine if teh perimeter and area of teh snowflake are finite or infinite quantities. Decisions much be supported clearly!!

    Please don't complicate it too much!!


    Edit: THANKS.
    Last edited by mr fantastic; July 21st 2009 at 01:53 AM. Reason: Restored original question deleted by OP.
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  2. #2
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    Quote Originally Posted by summna09 View Post
    URGENT -

    Using appropriate mathematical formulae and patterns to determine if teh perimeter and area of teh snowflake are finite or infinite quantities. Decisions much be supported clearly!!

    Please don't complicate it too much!!

    THANKS.
    Read this: The Koch Curve Unruled Notebook
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  3. #3
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    Hello, summna09!

    Use appropriate mathematical formulae and patterns to determine
    if the perimeter and area of the Koch snowflake are finite
    or infinite quantities. .Decisions much be supported clearly.

    Given equilateral triangle ABC, each side is trisected
    and an equilateral triangle is constructed outward.
    Code:
                    A
                    *
                   / \
                  /   \      
                 /     \      
        * - - - *       * - - - *
         \     /         \     /
          \   /           \   /
           \ /             \ /
            *               *
           /                 \
          /                   \
         /                     \
      B * - - - * - - - * - - - * C
                 \     /
                  \   /
                   \ /
                    *

    The process is repeated on the 12 sides of the new "snowflake".
    Code:
                    A
                    *
               * - / \ - *
            *   \       /   *
           / \   /     \   / \
        * -   - *       * -   - *
         \                     /
        /                       \
       * - \                 / - *
            *               *
       * - /                 \ - *
        \                       /
         /                     \
      B * -   - *       * -   - * C
           \ /   \     /   \ /
            *   /       \   *
               * - \ / - *
                    *
    And the process is repeated again ... and again.



    It can be seen that the perimeter increases by \tfrac{1}{3} at each stage.

    If P_o is the initial perimeter, the n^{th} perimeter is: . P_n \:=\:\left(\tfrac{4}{3}\right)^nP_o

    This is a geometric sequence with  r > 1.
    . . Hence, P_n diverges as n \to \infty.



    It can be seen that the area increases by \tfrac{1}{3} at each stage.

    If A_o is the initial area, the n^{th} area is: . A_n \;=\;A_o + A_o\left(\tfrac{1}{3}\right) + A_o\left(\tfrac{1}{3}\right)^2 + A_o\left(\tfrac{1}{3}\right)^3 + \hdots

    This is a geometric sequence with r < 1
    . . Hence, A_n converges as n \to\infty


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    This is yet another joke perpetrated by the Math Gods.

    We have a "polygon" with a finite area (it can fit on a postage stamp)
    . . yet has an infinite perimeter.

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