In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a "Johnson solid".
In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids. As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base.
In geometry, the metabidiminished icosahedron is one of the Johnson solids. The name refers to one way of constructing it, by removing two pentagonal pyramids from a regular icosahedron, replacing two sets of five triangular faces of the icosahedron with two adjacent pentagonal faces. If two pentagonal pyramids are removed to form nonadjacent pentagonal faces, the result is instead the pentagonal antiprism.
In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J21). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J6) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J42) with one pentagonal rotunda removed.
In geometry, the elongated pentagonal gyrobirotunda or elongated icosidodecahedron is one of the Johnson solids. As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron, by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda.
In geometry, the diminished rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with one pentagonal cupola removed.
In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids. It is also a canonical polyhedron.
In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids. It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.
In geometry, the bilunabirotunda is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal bipyramid by inserting a pentagonal prism between its congruent halves.
In geometry, the augmented pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces.
In geometry, the biaugmented pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by doubly augmenting a pentagonal prism by attaching square pyramids to two of its nonadjacent equatorial faces.
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 parabigyrate rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
In geometry, the metagyrate diminished rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees, and a non-opposing pentagonal cupola removed.
In geometry, the bigyrate diminished rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with two pentagonal cupolae rotated through 36 degrees, and a third pentagonal cupola removed.
In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids. Like the pentagonal orthocupolarotunda, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.
In geometry, a birotunda is any member of a family of dihedral-symmetric polyhedra, formed from two rotunda adjoined through the largest face. They are similar to a bicupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. There are two forms, ortho- and gyro-: an orthobirotunda has one of the two rotundas is placed as the mirror reflection of the other, while in a gyrobirotunda one rotunda is twisted relative to the other.
