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.
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, the rhombicuboctahedron, or small rhombicuboctahedron or Rectified Rhombic Dodecahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
In geometry, the triangular cupola is one of the Johnson solids. It can be seen as half a cuboctahedron.
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids. It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.
In geometry, the elongated square cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a square cupola by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" removed.
In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
In geometry, the elongated square bipyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.
In geometry, the elongated pentagonal cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal cupola by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola with its "lid" removed.
In geometry, the elongated triangular cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a triangular cupola by attaching a hexagonal prism to its base.
In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.
In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron, with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.
In geometry, the elongated triangular gyrobicupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae. Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola.
In geometry, the gyroelongated triangular bicupola is one of the Johnson solids. As the name suggests, it can be constructed by gyroelongating a triangular bicupola by inserting a hexagonal antiprism between its congruent halves.
In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.
