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.
