Recall that the Euler characteristic for a genus zero surface is given by $2 = V - E + F$. By adding a central vertex, which might create simplices for the three dimensional ball, we get a shifted invariant $\chi = 3$.
Now it may be that all of $V$, $E$ and $F$ are divisible by $3$, as in the examples shown. A counterexample is the $\chi = 3$ tetrahedron, which has $4$ faces and $5$ vertices. Counting $\chi = 3$ is a ternary analogue of the $\chi = 2$ Poincare paradigm. For a polygon in the plane, the characteristic $V - E$ is usually zero, but this can now be amended to $1$.
6 years ago