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 = 6
N = 8N + 4x6x1 (each face of original is split into 8, and also 4x6 for the holes)
N = 8N + 4x6x20 (each face of N is again split into 8, you then get the additional 4x6x20 for the new holes)
N = 8N + 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
N = 72 which you can check from diagram of the sponge that it is correct
A = N/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.
Originally Posted by dnjona