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Math Help - Menger Sponge

  1. #1
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    Menger Sponge

    Hi,
    I need to calculate the surface area of the Menger Sponge and found the following explanation online:

    N = number of square faces in the sponge)

    N[0] = 6
    N[1] = 8N[0] + 4x6x1 (each face of original is split into 8, and also 4x6 for the holes)
    N[2] = 8N[1] + 4x6x20 (each face of N[1] is again split into 8, you then get the additional 4x6x20 for the new holes)
    N[3] = 8N[2] + 4x6x20x20 (same again).

    Noting that the multiplier for 4x6 on the right is the number of cubes.

    N[n+1] = 8N[n] + 24x20^n

    the area series is then given by

    A[n] = N[n]/9^n
    A[n+1] = (8/9)A[n] + (24/9)x(20/9)^n

    which gives

    N[1] = 72 which you can check from diagram of the sponge that it is correct
    and
    A[1] = N[1]/9 = 8

    A further internet search also provided the direct formula below:

    A[n] = 2*(20/9)^n + 4*(8/9)^n

    Please could anyone explain to me how the direct formula was obtained from the iterative steps above.

    Many thanks.

    Dn
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  2. #2
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    Quote Originally Posted by dnjona View Post
    Hi,
    I need to calculate the surface area of the Menger Sponge and found the following explanation online:

    N = number of square faces in the sponge)

    N[0] = 6
    N[1] = 8N[0] + 4x6x1 (each face of original is split into 8, and also 4x6 for the holes)
    N[2] = 8N[1] + 4x6x20 (each face of N[1] is again split into 8, you then get the additional 4x6x20 for the new holes)
    N[3] = 8N[2] + 4x6x20x20 (same again).

    Noting that the multiplier for 4x6 on the right is the number of cubes.

    N[n+1] = 8N[n] + 24x20^n

    the area series is then given by

    A[n] = N[n]/9^n
    A[n+1] = (8/9)A[n] + (24/9)x(20/9)^n

    which gives

    N[1] = 72 which you can check from diagram of the sponge that it is correct
    and
    A[1] = N[1]/9 = 8

    A further internet search also provided the direct formula below:

    A[n] = 2*(20/9)^n + 4*(8/9)^n

    Please could anyone explain to me how the direct formula was obtained from the iterative steps above.
    The easiest way to verify the formula for A[n] is to prove it by induction.
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