= Euler characteristic of a cube boundary and its interior
{c}
{title2=$\chi(\partial[0,1]^3)=2,\quad\chi([0,1]^3)=1$}
The boundary of a <cube> has eight <vertices>, twelve <edges> and six square faces. Treating each open face as a two-dimensional <topological cell> gives
$$
\boxed{\chi(\partial[0,1]^3)=8-12+6=2.}
$$
This agrees with its being a <two-sphere>: stretching the hollow polyhedral surface changes its geometry but not its <topology>. The interior is excluded from this space.
Now include the solid interior. The <cube> has the same lower-dimensional <topological cells>, together with one open three-dimensional <topological cell>. The <cell-count formula for Euler characteristic> becomes
$$
\boxed{\chi([0,1]^3)=8-12+6-1=1.}
$$
The filled <cube> is <contractible>, so its <homology> agrees with that of a point. The term $-1$ is essential: $V-E+F$ by itself is the full count only for a two-dimensional <CW complex>.
\Image[/euler-characteristic-cube-boundary-and-interior.png]
{title=The boundary of a cube has Euler characteristic 2; including its three-dimensional interior adds one negative contribution and gives 1}
Adding a diagonal within one square face replaces that face by two triangular faces and introduces one <edge>. Thus the <Euler characteristic> changes by $-1+1=0$, as in <subdivision invariance of Euler characteristic>. Including an interior is a different operation: it adds points to the space, rather than giving the same space a finer description. This distinction matters whenever a geometric model could mean either a surface or the solid it encloses.
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