How Is the Surface Area of a Menger Sponge Calculated?

[dnjona]

Hi,
I want 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

 

[Valerie Park]

A The direct formula was obtained by solving the recurrence relation. The recurrence relation is a mathematical statement that defines a sequence of numbers based on the previous terms. In this case, the recurrence relation is A[n+1] = (8/9)A[n] + (24/9)x(20/9)^n. This can be solved by using a technique called "guess and check". The idea is to guess a general form for the solution that involves n, and then check if the guess is correct by substituting it into the recurrence relation. In this problem, the general form of the solution is A[n] = c1*(20/9)^n + c2*(8/9)^n. Substituting this into the recurrence relation, we get (8/9)*(c1*(20/9)^n + c2*(8/9)^n) + (24/9)*(20/9)^n = c1*(20/9)^(n+1) + c2*(8/9)^(n+1). Simplifying this equation yields c1*(20/9)^n + c2*(8/9)^n = c1*(20/9)^(n+1) + c2*(8/9)^(n+1). This equation has two unknown variables (c1 and c2), so we need two equations to solve for them. To get the second equation, we can substitute n=0 into the recurrence relation. This gives A[1] = (8/9)A[0] + (24/9)*(20/9)^0. Since A[0] is known (it is 8), we can solve for A[1] to get A[1] = 72/9. Substituting n=0 into the general form of the solution results in c1*(20/9)^0 + c2*(8/9)^0 = c1 + c2 = 72/9. Now we have two equations with two unknowns (c1 and c2). Solving these equations yields c1 = 2 and c2 = 4. Therefore, the direct formula is A[n] = 2