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 regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of the Platonic solid and of the deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
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.
The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.
In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.
In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
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 inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr{5/3,5}.
In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.
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, a golden rhombus is a rhombus whose diagonals are in the golden ratio:
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 De2f2stellation 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 great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges.
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.
