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Johnson solid

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Pentagonal orthobirotunda

In geometry, the pentagonal orthobirotunda is one of the Johnson solids (J_34). It can be constructed by joining two pentagonal rotundae (J_6) along their decagonal faces, matching like faces. The pentagonal orthobirotunda is also related to an Archimedean solid, the icosidodecahedron, which can also be called a pentagonal gyrobirotunda, similarly created by two pentagonal rotunda but with a 36-degree rotation.

Pentagonal cupola

In geometry, the pentagonal cupola is one of the Johnson solids (J_5). 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. The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a: The height of the pentagonal cupola is The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces: In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola.

Gyrate rhombicosidodecahedron

In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (J72). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees. They have the same faces around each vertex, but vertex configurations along the rotation become a different order, 3.4.4.5.

Parabidiminished rhombicosidodecahedron

In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (J_80). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae removed. Related Johnson solids are the diminished rhombicosidodecahedron (J_76) where one cupola is removed, the metabidiminished rhombicosidodecahedron (J_81) where two non-opposing cupolae are removed, and the tridiminished rhombicosidodecahedron (J_83) where three cupolae are removed.

Tridiminished rhombicosidodecahedron

In geometry, the tridiminished rhombicosidodecahedron is one of the Johnson solids (J_83). It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae removed. Related Johnson solids are: J_76: diminished rhombicosidodecahedron with one cupola removed, J_80: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and J_81: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed.

Snub square antiprism

In geometry, the snub square antiprism is one of the Johnson solids (J_85). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold. The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism. It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).

Gyroelongated square pyramid

In geometry, the gyroelongated square pyramid is one of the Johnson solids (J_10). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base. The Gyroelongated square pyramid represents the capped square antiprismatic molecular geometry: The dual of the gyroelongated square pyramid has 9 faces: 4 kites, 1 square and 4 pentagonal.

Metabidiminished icosahedron

In geometry, the metabidiminished icosahedron is one of the Johnson solids (J_62). The name refers to one way of constructing it, by removing two pentagonal pyramids (J_2) 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.

Square gyrobicupola

In geometry, the square gyrobicupola is one of the Johnson solids (J_29). Like the square orthobicupola (J_28), it can be obtained by joining two square cupolae (J_4) along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another. The square gyrobicupola is the second in an infinite set of gyrobicupolae. Related to the square gyrobicupola is the elongated square gyrobicupola. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola.

Metabidiminished rhombicosidodecahedron

In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids (J_81). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae (J_5) removed. Related Johnson solids are: The diminished rhombicosidodecahedron (J_76) where one cupola is removed, The parabidiminished rhombicosidodecahedron (J_80) where two opposing cupolae are removed, The gyrate bidiminished rhombicosidodecahedron (J_82) where two non-opposing cupolae are removed and a third is rotated 36 degrees, And the tridiminished rhombicosidodecahedron (J_83) where three cupolae are removed.