The Dehn-Sommerville equations are relations among the numbers of faces of a simplicial polytope, or, dually, of a
simple polytope (Ziegler 1995). For a 
-dimensional simple polytope,
let 
count its 
-dimensional
faces, including 
,
and define
 |
The equations state that the coefficients are symmetric,
 |
Equivalently, the coefficient of 
equals that of 
.
This symmetry permits the expansion defining the gamma
vector (Horiguchi et al. 2026).
For a cube, the face counts are 
, 
, 
, and 
,
giving 
. For the dual simplicial
polytope, use the face counts of its simple dual in
the displayed definition.
