A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).
In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.
In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as U75. It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.
In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a degenerate uniform star polyhedron.
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices. It has Coxeter diagram . It has the unusual feature that its 24 pentagram faces occur in 12 coplanar pairs.
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both.
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids.
A compound of five tetrahemihexahedra is a uniform polyhedron compound and a symmetric arrangement of five tetrahemihexahedra. It is chiral with icosahedral symmetry (I).
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron.
This uniform polyhedron compound is a symmetric arrangement of 20 triangular prisms, aligned in pairs with the axes of three-fold rotational symmetry of an icosahedron.
