Elongated triangular cupola

Elongated triangular cupola
Elongated triangular cupola
Type	Johnson ; J17 - J18 - J19
Faces	1+3 triangles ; 3x3 squares ; 1 hexagon
Edges	27
Vertices	15
Vertex configuration	6(42.6); 3(3.4.3.4); 6(3.43)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex
Net

Last updated October 17, 2021

In geometry, the elongated triangular cupola is one of the Johnson solids (J₁₈). As the name suggests, it can be constructed by elongating a triangular cupola (J₃) by attaching a hexagonal prism to its base.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:^[2]

V=\left({\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)\right)a^{3}\approx 3.77659...a^{3}

A=\left(9+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\approx 13.3301...a^{2}

Dual polyhedron

The dual of the elongated triangular cupola has 15 faces: 6 isosceles triangles, 3 rhombi, and 6 quadrilaterals.

Dual elongated triangular cupola	Net of dual

Related polyhedra and honeycombs

The elongated triangular cupola can form a tessellation of space with tetrahedra and square pyramids.^[3]

Related Research Articles

In geometry, the triangular cupola is one of the Johnson solids (J₃). It can be seen as half a cuboctahedron.

In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (J₄). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.

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" removed.

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids (J₄₃). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron, by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda (J₄₂).

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (J₄₂). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda (J₃₄) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda (J₄₃).

In geometry, the pentagonal cupola is one of the Johnson solids (J₅). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

In geometry, the elongated triangular pyramid is one of the Johnson solids (J₇). 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 square pyramid is one of the Johnson solids (J₈). As the name suggests, it can be constructed by elongating a square pyramid (J₁) by attaching a cube to its square base. Like any elongated pyramid, it is topologically self-dual.

In geometry, the elongated triangular bipyramid or triakis triangular prism is one of the Johnson solids (J₁₄), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid (J₁₂) by inserting a triangular prism between its congruent halves.

In geometry, the elongated pentagonal cupola is one of the Johnson solids (J₂₀). As the name suggests, it can be constructed by elongating a pentagonal cupola (J₅) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J₃₈) with its "lid" removed.

In geometry, the gyrobifastigium is the 26th Johnson solid (J₂₆). 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.

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (J₃₈). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola (J₃₀) by inserting a decagonal prism between its two congruent halves. Rotating one of the cupolae through 36 degrees before inserting the prism yields an elongated pentagonal gyrobicupola (J₃₉).

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J₃₉). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola (J₃₁) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae (J₅) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola (J₃₈).

In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J₂₂). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J₃). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.

In geometry, the triangular orthobicupola is one of the Johnson solids (J₂₇). As the name suggests, it can be constructed by attaching two triangular cupolas (J₃) 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 pentagonal orthocupolarotunda is one of the Johnson solids (J₃₂). As the name suggests, it can be constructed by joining a pentagonal cupola (J₅) and a pentagonal rotunda (J₆) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (J₃₃).

In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J₃₆). 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 (J₃). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (J₃₅).

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (J₄₁). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda (J₃₃) by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola (J₅) or the pentagonal rotunda (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda (J₄₀).

In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (J₄₀). As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda (J₃₂) by inserting a decagonal prism between its halves. Rotating either the cupola or the rotunda through 36 degrees before inserting the prism yields an elongated pentagonal gyrocupolarotunda (J₄₁).

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 triangular cupola" from Wolfram Alpha. Retrieved July 22, 2010.
↑ "J18 honeycomb".

External links

Eric W. Weisstein, Elongated triangular cupola ( Johnson solid ) at MathWorld.

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[1] 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 .

[2] Stephen Wolfram, "Elongated triangular cupola" from Wolfram Alpha. Retrieved July 22, 2010.

[3] "J18 honeycomb".

[1]

[2]

[3]

v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	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
Modified cupolae and rotundae	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 prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	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
Elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)

Elongated triangular cupola

Type	Johnson J₁₇ - J₁₈ - J₁₉
Faces	1+3 triangles 3x3 squares 1 hexagon
Edges	27
Vertices	15
Vertex configuration	6(4².6) 3(3.4.3.4) 6(3.4³)
Symmetry group	C_3v
Dual polyhedron	-
Properties	convex
Net