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).
The octahedron is simplicial because all its facets are triangles. The cube is not simplicial because its facets are squares. Every simplex is both simple and simplicial. 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).
