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When I ask Mathematica to give me the Euler characteristic of MengerMesh[1, 3], it returns 16. It seems to me the 1-step Menger sponge has genus 5 and should therefore have Euler characteristic -8. For MengerMesh[2, 3], I'm again off by a factor of -2, as Mathematica returns 320 while I predict -160 based on its genus, which is 81. For MengerMesh[0, 3], me and Mathematica agree.

Can anyone please explain to me what I'm missing here?

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1 Answer 1

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MengerMesh returns a full dimensional mesh, e.g. in 3D a solid, not a surface.

mr = MengerMesh[2, 3];

RegionDimension[mr]
3
MeshCellCount[mr]
{896, 2304, 1728, 400}

We can instead find the characteristic of its boundary:

EulerCharacteristic[mr]
320
EulerCharacteristic[RegionBoundary[mr]]
-160
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  • $\begingroup$ Yea, of course. $\endgroup$
    – Ruben
    May 3, 2020 at 7:52

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