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

Elongated triangular bipyramid (J₁₄)

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.

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 (${\displaystyle V}$), surface area (${\displaystyle A}$) and height (${\displaystyle H}$) can be used if all faces are regular, with edge length a:

${\displaystyle V=\left({\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)\right)\cdot a^{3}\approx 0.668715...a^{3}}$ ^{[3] [4]}

${\displaystyle A=\left({\frac {3}{2}}\left(2+{\sqrt {3}}\right)\right)\cdot a^{2}\approx 5.59808...a^{2}}$ ^{[3] [4]}

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

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 Mathematics Education, 42 (1): 11–17, doi:10.1007/s11858-009-0208-2, S2CID   122791717 .
3. Stephen Wolfram, "Elongated triangular dipyramid" from Wolfram Alpha. Retrieved July 22, 2010.
4. 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.