So Mathematica tells me the 1-step Menger sponge has Euler Characteristic 16. It computes this by using the alternating sum of the vertices, edges and faces of its mesh.

Now it seems to me that the 1-step Menger sponge is topologically equivalent to a 5-torus, which has genus 5 and therefore Euler characteristic 2 - 2*5 = -8

For the 2-step one, I compute the genus to be 81, so the Ec to be -160. Again Mathematica computes it to be 320. Again I'm off a factor -2.

For the 0-step one, Mathematica and I both compute the Ec to be 2.

What is going on here?