| Elongated square cupola | |
|---|---|
 | |
| Type | Johnson J18 – J19 – J20 |
| Faces | 4 triangles 13 squares 1 octagon |
| Edges | 36 |
| Vertices | 20 |
| Vertex configuration | 8(42.8) 4+8(3.43) |
| Symmetry group | C4v |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.
The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation. [1] This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon. [2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid . [3]
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids , Archimedean solids , prisms , or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966. [4]
The surface area of an elongated square cupola is the sum of all polygonal faces' area. Its volume can be ascertained by dissecting it into both square cupola and regular octagon, and then adding their volume. Given the elongated triangular cupola with edge length , its surface area and volume are: [5]
The dual polyhedron of an elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.
| Dual elongated square cupola | Net of dual |
|---|---|
 |  |
The elongated square cupola forms space-filling honeycombs with tetrahedra and cubes; with cubes and cuboctahedra; and with tetrahedra, elongated square pyramids, and elongated square bipyramids. (The latter two units can be decomposed into cubes and square pyramids.) [6]
In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can be applied to construct many polyhedrons.
In geometry, the square cupola is the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.
In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It was once mistakenly considered a rhombicuboctahedron by many mathematicians. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron. For this reason, it is also known as pseudo-rhombicuboctahedron, Miller solid, or Miller–Askinuze solid.
In geometry, the gyroelongated square cupola is one of the Johnson solids (J23). As the name suggests, it can be constructed by gyroelongating a square cupola (J4) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J45) with one square bicupola removed.
In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.
In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids. It is also a canonical polyhedron.
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 elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.
In geometry, the elongated square bipyramid is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. It can also be seen as 4 lunes linked together with squares to squares and triangles to triangles. It is also been named the pencil cube or 12-faced pencil cube due to its shape.
In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.
In geometry, the elongated pentagonal cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal cupola by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola with its "lid" removed.
In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.
In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.
In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.
In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.
In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.
In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola or the pentagonal rotunda through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda.
In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto an truncated tetrahedron. It is an example of a Johnson solid.
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.
