Elongated square cupola
Elongated square cupola | |
---|---|
Type | Johnson J18 – J19 – J20 |
Faces | 4 triangles 3x4+1 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 one of the Johnson solids (J19). As the name suggests, it can be constructed by elongating a square cupola (J4) by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed.
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.[1]
Formulae
The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:[2]
Dual polyhedron
The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.
Dual elongated square cupola | Net of dual |
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Related polyhedra and honeycombs
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.)[3]
References
- ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
- ^ Stephen Wolfram, "Elongated square cupola" from Wolfram Alpha. Retrieved July 22, 2010.
- ^ "J19 honeycomb".
External links
- Weisstein, Eric W., "Elongated square cupola" ("Johnson solid") at MathWorld.
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- elongated triangular pyramid
- elongated square pyramid
- elongated pentagonal pyramid
- gyroelongated square pyramid
- gyroelongated pentagonal pyramid
- triangular bipyramid
- pentagonal bipyramid
- elongated triangular bipyramid
- elongated square bipyramid
- elongated pentagonal bipyramid
- gyroelongated square bipyramid
- elongated triangular cupola
- elongated square cupola
- elongated pentagonal cupola
- elongated pentagonal rotunda
- gyroelongated triangular cupola
- gyroelongated square cupola
- gyroelongated pentagonal cupola
- gyroelongated pentagonal rotunda
- gyrobifastigium
- triangular orthobicupola
- square orthobicupola
- square gyrobicupola
- pentagonal orthobicupola
- pentagonal gyrobicupola
- pentagonal orthocupolarotunda
- pentagonal gyrocupolarotunda
- pentagonal orthobirotunda
- elongated triangular orthobicupola
- elongated triangular gyrobicupola
- elongated square gyrobicupola
- elongated pentagonal orthobicupola
- elongated pentagonal gyrobicupola
- elongated pentagonal orthocupolarotunda
- elongated pentagonal gyrocupolarotunda
- elongated pentagonal orthobirotunda
- elongated pentagonal gyrobirotunda
- gyroelongated triangular bicupola
- gyroelongated square bicupola
- gyroelongated pentagonal bicupola
- gyroelongated pentagonal cupolarotunda
- gyroelongated pentagonal birotunda
- augmented truncated tetrahedron
- augmented truncated cube
- biaugmented truncated cube
- augmented truncated dodecahedron
- parabiaugmented truncated dodecahedron
- metabiaugmented truncated dodecahedron
- triaugmented truncated dodecahedron
- gyrate rhombicosidodecahedron
- parabigyrate rhombicosidodecahedron
- metabigyrate rhombicosidodecahedron
- trigyrate rhombicosidodecahedron
- diminished rhombicosidodecahedron
- paragyrate diminished rhombicosidodecahedron
- metagyrate diminished rhombicosidodecahedron
- bigyrate diminished rhombicosidodecahedron
- parabidiminished rhombicosidodecahedron
- metabidiminished rhombicosidodecahedron
- gyrate bidiminished rhombicosidodecahedron
- tridiminished rhombicosidodecahedron
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