An equifacetted polytope is a convex polytope all of whose facets have the same combinatorial type. Thus corresponding faces and their containment relations agree, even when the facets are not congruent (Perles and Shephard 1967). Every simplicial polytope is equifacetted, since all its facets are simplices.
Castillo et al. (2026) proved that no equifacetted convex polytope of dimension 
has all facets combinatorially
equivalent to 
,
where 
is a 
-dimensional
simplex. Among simple
polytopes of dimension at least four, the only
such example up to combinatorial equivalence is 
. Nonsimple four-dimensional examples form
infinite families (Castillo et al. 2026).
Castillo et al. (2026) report that GPT-5.6 Sol generated code supporting the construction and verification of their four-dimensional families.
