Trouble determining the Euler's characteristic on the shape below.
Here's what I counted: $G=(V,E,F) = (16,24,10)$
Shouldn't $χ$ be $0$?
1-toroid.png
2026-03-28 09:50:21.1774691421
Counting Faces, Edges, and Vertices of cubed 1-toroid
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I think you have counted $(V,E,F)$ wrong.
This shape has a vertex for every single integer coordinate of a 2x4x4 grid in 3-D, so $V=32$. It has 24 edges on the top layer, and another 24 on the bottom layer, with 16 edges from top to bottom, so $E=64$. It has 8 faces on the top, 8 on the bottom, 4 on the inside hole, 12 around the outside, so $F=32$.
As you correctly predicted, $V-E+F = 32-64+32 = 0$.