An exotic polytope is a polytope if it has more than two facets (n-1 dimensional subfacets) joined at at least one ridge (n-2 dimensional subfacets).[1] This makes them distinct from true polytopes, which (if their dimensionality is greater than or equal to two) always have two facets joined together at their ridges. For instance, there are two edges joined together at a polygon's vertices, there are two faces joined together at a polyhedron's edges, there are two cells joined together at a polychoron's faces, and so on. The number of ridges an exotic polytope has can be considered ambiguous as ridges where multiple facets meet can be considered multiple coincident ridges. Examples of exotic polytopes include the three-dimensional great disnub dirhombidodecahedron (gidisdrid, Skilling’s figure) and small complex icosidodecahedron (cid) which have edges where multiple faces meet and the four-dimensional small exoditetrahedrary prismatotrishecatonicosachoron (seedatephi) which has pentagonal faces where multiple cells meet [2].
Polytopes that contain exotic polyhedra as elements, but do not have more than two facets joined at their ridges are called exotic celled polytopes (which can be generalized to any subfacet).