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, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid.
In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can be applied to construct many polyhedrons.
In geometry, the pentagonal cupola is one of the Johnson solids. It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.
In geometry, the triaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids to three of its nonadjacent equatorial faces.
In geometry, the augmented dodecahedron is one of the Johnson solids, consisting of a dodecahedron with a pentagonal pyramid attached to one of the faces. When two or three such pyramids are attached, the result may be a parabiaugmented dodecahedron, a metabiaugmented dodecahedron, or a triaugmented dodecahedron.
In geometry, the parabiaugmented dodecahedron is one of the Johnson solids. It can be seen as a dodecahedron with two pentagonal pyramids attached to opposite faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a metabiaugmented dodecahedron, a triaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular.
In geometry, the metabiaugmented dodecahedron is one of the Johnson solids. It can be viewed as a dodecahedron with two pentagonal pyramids attached to two faces that are separated by one face. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a parabiaugmented dodecahedron, a triaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular.
In geometry, the triaugmented dodecahedron is one of the Johnson solids. It can be seen as a dodecahedron with three pentagonal pyramids attached to nonadjacent faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a parabiaugmented dodecahedron, a metabiaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular.
In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.
In geometry, the augmented truncated cube is one of the Johnson solids. As its name suggests, it is created by attaching a square cupola onto one octagonal face of a truncated cube.
In geometry, the biaugmented truncated cube is one of the Johnson solids. As its name suggests, it is created by attaching two square cupolas onto two parallel octagonal faces of a truncated cube.
In geometry, the augmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching a pentagonal cupola onto one decagonal face of a truncated dodecahedron.
In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching two pentagonal cupolas onto two parallel decagonal faces of a truncated dodecahedron.
In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching two pentagonal cupolas onto two nonadjacent, nonparallel decagonal faces of a truncated dodecahedron.
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, 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.
