The polyhedral formula generalized to a surface of genus 
,
 |
where 
is the number of polyhedron vertices, 
is the number of polyhedron
edges, 
is the number of faces, and
 |
is called the Euler characteristic.
Poincaré Formula
See also
Euler Characteristic, Genus, Polyhedral FormulaExplore with Wolfram|Alpha
References
Coxeter, H. S. M. "Poincaré's Proof of Euler's Formula." Ch. 9 in Regular Polytopes, 3rd ed. New York: Dover, pp. 165-172, 1973.Eppstein, D. "Fourteen Proofs of Euler's Formula:
." http://www.ics.uci.edu/~eppstein/junkyard/euler/.Referenced on Wolfram|Alpha
Poincaré FormulaCite this as:
Weisstein, Eric W. "Poincaré Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoincareFormula.html