Elongated triangular bipyramid

Elongated triangular bipyramid
Type	Johnson; J13 - J14 - J15
Faces	6 triangles; 3 squares
Edges	15
Vertices	8
Vertex configuration	2(33); 6(32.42)
Symmetry group	D3h, [3,2], (*322)
Rotation group	D3, [3,2]+, (322)
Dual polyhedron	Triangular bifrustum
Properties	convex
Net

In geometry, the elongated triangular bipyramid (or dipyramid) 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.

Sólido de Johnson J₁₄

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]

The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.[2]

Formulae

The following formulae for volume ( $V$ ), surface area ( $A$ ) and height ( $H$ ) can be used if all faces are regular, with edge length a:

V=({\frac {1}{12}}(2{\sqrt {2}}+3{\sqrt {3}}))\cdot a^{3}\approx 0.668715...a^{3}

[3][4]

A=({\frac {3}{2}}(2+{\sqrt {3}}))\cdot a^{2}\approx 5.59808...a^{2}

[3][4]

H={\frac {3+2{\sqrt {6}}}{3}}\cdot a\approx \cdot 2.63299...a

[4]

Dual polyhedron

The dual of the elongated triangular bipyramid is called a triangular bifrustum and has 8 faces: 6 trapezoidal and 2 triangular.

Dual elongated triangular bipyramid	Net of dual

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.
Gerdes, Paulus (2009), "Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education", ZDM, the International Journal on Mathematics Education, 42 (1): 11–17, doi:10.1007/s11858-009-0208-2.
Stephen Wolfram, "Elongated triangular dipyramid" from Wolfram Alpha. Retrieved July 22, 2010.
Sapiña, R. "Area and volume of the Johnson solid J₁₄". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-04.

External links

Eric W. Weisstein, Elongated triangular bipyramid (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] Gerdes, Paulus (2009), "Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education", ZDM, the International Journal on Mathematics Education, 42 (1): 11–17, doi:10.1007/s11858-009-0208-2.

[SW-3] Stephen Wolfram, "Elongated triangular dipyramid" from Wolfram Alpha. Retrieved July 22, 2010.

[pye-4] Sapiña, R. "Area and volume of the Johnson solid J₁₄". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-04.

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 square 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)