Concept

Johnson solid

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Square orthobicupola

In geometry, the square orthobicupola is one of the Johnson solids (J_28). As the name suggests, it can be constructed by joining two square cupolae (J_4) along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola (J_29). The square orthobicupola is the second in an infinite set of orthobicupolae.

Gyroelongated square bicupola

In geometry, the gyroelongated square bicupola is one of the Johnson solids (J_45). As the name suggests, it can be constructed by gyroelongating a square bicupola (J_28 or J_29) by inserting an octagonal antiprism between its congruent halves. The gyroelongated square 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 left half of the figure is connected by a path of two triangular faces to a square face below it and to the left.

Sphenocorona

In geometry, the sphenocorona is one of the Johnson solids (J_86). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -corona refers to a crownlike complex of 8 equilateral triangles. Joining both complexes together results in the sphenocorona.

Elongated square pyramid

In geometry, the elongated square pyramid is one of the Johnson solids (J_8). As the name suggests, it can be constructed by elongating a square pyramid (J_1) by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self-dual. The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length : The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal.

Elongated pentagonal bipyramid

In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (J_16). As the name suggests, it can be constructed by elongating a pentagonal bipyramid (J_13) by inserting a pentagonal prism between its congruent halves. The dual of the elongated square bipyramid is a pentagonal bifrustum.

Elongated pentagonal orthobicupola

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (J_38). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola (J_30) by inserting a decagonal prism between its two congruent halves. Rotating one of the cupolae through 36 degrees before inserting the prism yields an elongated pentagonal gyrobicupola (J_39). The following formulae for volume and surface area can be used if all faces are regular, with edge length a

Augmented dodecahedron

In geometry, the augmented dodecahedron is one of the Johnson solids (J_58), consisting of a dodecahedron with a pentagonal pyramid (J_2) attached to one of the faces. When two or three such pyramids are attached, the result may be a parabiaugmented dodecahedron (J_59), a metabiaugmented dodecahedron (J_60), or a triaugmented dodecahedron (J_61).

Parabiaugmented dodecahedron

In geometry, the parabiaugmented dodecahedron is one of the Johnson solids (J_59). It can be seen as a dodecahedron with two pentagonal pyramids (J_2) attached to opposite faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (J_58), a metabiaugmented dodecahedron (J_60), a triaugmented dodecahedron (J_61), or even a pentakis dodecahedron if the faces are made to be irregular. The dual of this solid is the Gyroelongated pentagonal bifrustum.

Metabiaugmented dodecahedron

In geometry, the metabiaugmented dodecahedron is one of the Johnson solids (J_60). It can be viewed as a dodecahedron with two pentagonal pyramids (J_2) attached to two faces that are separated by one face. (The two faces are not opposite, but not adjacent either.) When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (J_58), a parabiaugmented dodecahedron (J_59), a triaugmented dodecahedron (J_61), or even a pentakis dodecahedron if the faces are made to be irregular.

Triaugmented dodecahedron

In geometry, the triaugmented dodecahedron is one of the Johnson solids (J_61). It can be seen as a dodecahedron with three pentagonal pyramids (J_2) attached to nonadjacent faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron (J_58), a parabiaugmented dodecahedron (J_59), a metabiaugmented dodecahedron (J_60), or even a pentakis dodecahedron if the faces are made to be irregular.