Concept

Johnson solid

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Triangular bipyramid

In geometry, the triangular bipyramid (or dipyramid) 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. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four.

Gyroelongated pentagonal pyramid

In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (J_11). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base. It can also be seen as a diminished icosahedron, an icosahedron with the top (a pentagonal pyramid, J_2) chopped off by a plane. Other Johnson solids can be formed by cutting off multiple pentagonal pyramids from an icosahedron: the pentagonal antiprism and metabidiminished icosahedron (two pyramids removed), and the tridiminished icosahedron (three pyramids removed).

Diminished rhombicosidodecahedron

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

Elongated square bipyramid

In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (J_15). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves. It has been named the pencil cube or 12-faced pencil cube due to its shape. A zircon crystal is an example of an elongated square bipyramid. The following formulae for volume (), surface area () and height () can be used if all faces are regular, with edge length : The dual of the elongated square bipyramid is called a square bifrustum and has 10 faces: 8 trapezoidal and 2 square.

Gyrobifastigium

In geometry, the gyrobifastigium is the 26th Johnson solid (J_26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space. It is also the vertex figure of the nonuniform p-q duoantiprism (if p and q are greater than 2). Despite the fact that p, q = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case p = 5, q = 5/3, which represents a uniform great duoantiprism.

Gyroelongated square bipyramid

In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids (J_17). As the name suggests, it can be constructed by gyroelongating an octahedron (square bipyramid) by inserting a square antiprism between its congruent halves. It is one of the eight strictly-convex deltahedra. The dual of the gyroelongated square bipyramid is a square truncated trapezohedron with 10 faces: 8 pentagons and 2 square.

Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.

Square pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C_4v symmetry. If all edge lengths are equal, it is an equilateral square pyramid, the Johnson solid J_1. A possibly oblique square pyramid with base length l and perpendicular height h has volume: In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.

Pentagonal pyramid

In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self-dual. The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (J_2). It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J_11.

Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids (J_6). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a: The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.