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Johnson solid

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Parabigyrate rhombicosidodecahedron

In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (J_73). It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: The gyrate rhombicosidodecahedron (J_72) where only one cupola is rotated; The metabigyrate rhombicosidodecahedron (J_74) where two non-opposing cupolae are rotated; And the trigyrate rhombicosidodecahedron (J_75) where three cupolae are rotated.

Metabigyrate rhombicosidodecahedron

In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (J_74). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: The gyrate rhombicosidodecahedron (J_72) where only one cupola is rotated; The parabigyrate rhombicosidodecahedron (J_73) where two opposing cupolae are rotated; And the trigyrate rhombicosidodecahedron (J_75) where three cupolae are rotated.

Gyroelongated pentagonal bicupola

In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids (J_46). As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola (J_30 or J_31) by inserting a decagonal antiprism between its congruent halves. The gyroelongated pentagonal bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right.

Gyroelongated triangular cupola

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. The gyroelongated triangular cupola can also be seen as a gyroelongated triangular bicupola (J44) with one triangular cupola removed.

Pentagonal orthocupolarotunda

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (J_32). As the name suggests, it can be constructed by joining a pentagonal cupola (J_5) and a pentagonal rotunda (J_6) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (J_33).

Pentagonal gyrocupolarotunda

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (J_33). Like the pentagonal orthocupolarotunda (J_32), it can be constructed by joining a pentagonal cupola (J_5) and a pentagonal rotunda (J_6) along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

Elongated triangular gyrobicupola

In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J_36). 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 (J_3). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (J_35).

Gyroelongated triangular bicupola

In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (J_44). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either triangular orthobicupola, J_27, or the cuboctahedron) by inserting a hexagonal antiprism between its congruent halves. The gyroelongated triangular bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form.

Trapezo-rhombic dodecahedron

In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D_3h symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. It is also called the trapezoidal dodecahedron. This polyhedron could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom.

Elongated bipyramid

In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an n-gonal bipyramid (by inserting an n-gonal prism between its congruent halves). There are three elongated bipyramids that are Johnson solids: Elongated triangular bipyramid (J_14), Elongated square bipyramid (J_15), and Elongated pentagonal bipyramid (J_16). Higher forms can be constructed with isosceles triangles.