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 triaugmented triangular prism, tetracaidecadeltahedron, or tetrakis triangular prism is one of the Johnson solids. Each of its 14 faces is an equilateral triangle, making it a deltahedron. As the name suggests, it can be constructed by attaching equilateral square pyramids to each of the three square faces of the triangular prism.
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 triangular cupola is one of the Johnson solids. It can be seen as half a cuboctahedron.
In geometry, the augmented tridiminished icosahedron is one of the Johnson solids (J64).
In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids. As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.
In geometry, the augmented triangular prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid to one of its equatorial faces. The resulting solid bears a superficial resemblance to the gyrobifastigium, the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.
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 augmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid to one of its equatorial faces. When two or three such pyramids are attached, the result may be a parabiaugmented hexagonal prism, a metabiaugmented hexagonal prism, or a triaugmented hexagonal prism.
In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids to two of its nonadjacent, parallel (opposite) equatorial faces. Attaching the pyramids to nonadjacent, nonparallel equatorial faces yields a metabiaugmented hexagonal prism.
In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids to two of its nonadjacent, nonparallel equatorial faces. Attaching the pyramids to opposite equatorial faces yields a parabiaugmented hexagonal prism.
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 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 elongated triangular gyrobicupola is one of the Johnson solids. 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. Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola.
In geometry, the augmented truncated tetrahedron is one of the Johnson solids (J65). It is created by attaching a triangular cupola (J3) to one hexagonal face of a truncated tetrahedron.
In geometry, the augmented truncated cube is one of the Johnson solids (J66). As its name suggests, it is created by attaching a square cupola (J4) onto one octagonal face of a truncated cube.
In geometry, the biaugmented truncated cube is one of the Johnson solids (J67). As its name suggests, it is created by attaching two square cupolas (J4) onto two parallel octagonal faces of a truncated cube.
In geometry, the augmented truncated dodecahedron is one of the Johnson solids (J68). As its name suggests, it is created by attaching a pentagonal cupola (J5) onto one decagonal face of a truncated dodecahedron.
