A simplicial polytope is a convex polytope whose proper faces are all simplices. It suffices that every facet, meaning a face of dimension one less than the polytope, is a simplex. Its dual is a simple polytope (Ziegler 1995). In three dimensions, this dual is a simple polyhedron in the combinatorial sense.
The connection with the simplex is through the facets. For a 
-dimensional
simplicial polytope with 
, every facet is a 
-simplex. In contrast, a simple polytope has 
-simplices as vertex
figures (Renner 2009). These are distinct conditions, although every simplex
satisfies both.
The octahedron is simplicial because all its facets are triangles. The cube is not simplicial because its facets are squares. A simplicial polytope need not be a simplex.
The face counts obey the Dehn-Sommerville equations, equivalently expressed by the symmetric polynomial of the dual simple polytope (Horiguchi et al. 2026).
