In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{5⁄2,3}, and Coxeter-Dynkin diagram .
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices.
In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. The exterior surface also represents the De2f2 stellation of the icosahedron. These figures can be differentiated by marking which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the even–odd rule, the internal structure of both shapes will differ.
In geometry, the small dodecicosacron is the dual of the small dodecicosahedron (U50). It is visually identical to the Small ditrigonal dodecacronic hexecontahedron. It has 60 intersecting bow-tie-shaped faces.
In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.
In geometry, the rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
In geometry, the great icosacronic hexecontahedron is the dual of the great icosicosidodecahedron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.
In geometry, the medial deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. Its 60 intersecting quadrilateral faces are kites.
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
In geometry, the great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models.
In geometry, the great rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. It is visually identical to the great deltoidal hexecontahedron. Its faces are antiparallelograms.
In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.
In geometry, the small icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small icosicosidodecahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.
In geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces.
In geometry, the great ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.
In geometry, the medial icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform icosidodecadodecahedron. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models.
In geometry, the great dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great dodecicosidodecahedron. Its 60 intersecting quadrilateral faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models.
