Hi fellows,


Does anyone know the number of holes of a level 2 Menger Sponge ?



Level 2 Menger Sponge
For $n=1$, $g=5$: you drill a vertical hole through the middle and four horizontal holes to meet that vertical hole.


For higher values of $n$, the right way to think of it is in terms of Euler characteristic.


For $n=2$ you start with $20$ copies of a small level $1$ Menger sponge, with Euler characteristic $-8$. You then glue them together a bunch of times on an annulus. The annulus has Euler characteristic $0$, so the final Euler characteristic is $-160$, giving a genus of $81$.


The side of a level $n$ Menger sponge has Euler characteristic $(8-8^n)/7$. In general you glue along $24$ of these, so the recurrence relation for the Euler characteristic is $\chi\_{n+1}=20\chi\_n + 24 (8^n-8)/7$.


This recurrence relation can produce an explicit formula, but it seems rather awful.


