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 triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids. As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves. It is one of the eight strictly-convex deltahedra.
In geometry, the augmented tridiminished icosahedron is one of the Johnson solids. It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron.
In geometry, the elongated triangular pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically self-dual.
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 biaugmented triangular prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a triangular prism by attaching square pyramids to two of its equatorial faces.
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 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 augmented truncated cube is one of the Johnson solids. As its name suggests, it is created by attaching a square cupola onto one octagonal face of a truncated cube.
In geometry, the biaugmented truncated cube is one of the Johnson solids. As its name suggests, it is created by attaching two square cupolas onto two parallel octagonal faces of a truncated cube.
In geometry, the augmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching a pentagonal cupola onto one decagonal face of a truncated dodecahedron.
In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching two pentagonal cupolas onto two parallel decagonal faces of a truncated dodecahedron.
In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids. As its name suggests, it is created by attaching two pentagonal cupolas onto two nonadjacent, nonparallel decagonal faces of a truncated dodecahedron.
In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids ; of them, it has the greatest volume in proportion to the cube of the side length. As its name suggests, it is created by attaching three pentagonal cupolas onto three nonadjacent decagonal faces of a truncated dodecahedron.
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.
