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    • The Koch curve

    Let's suppose the initial line segment has length 1.

    What would be the perimeter of the second stage shape, the one resulting from removing the middle third of the line segment and replacing it with a triangular bump?

    Well, each side of the triangle is equal in length to 1/3 of the length of the segment, so the total must be:
    1 + 1/3 + 1/3 - 1/3
    or 4/3 (why do I have to subtract 1/3?).

    Can you perform the calculation for the the third stage shape? How about the fourth stage shape?

    Now look at the sequence you are generating: the sequence whose first term is the perimeter of the first shape, whose second term is the perimeter of the second stage shape, etc. The perimeter of the Koch curve is the limit of this sequence. The perimeter of the Koch snowflake must then be three times the length of the Koch curve (why?)

    If you need to, you can find out more about limits and sequences.