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	D_3h, [3,2], (*322)
Rotation group	D₃, [3,2]⁺, (322)
Dual polyhedron	Triangular bifrustum
Properties	convex
Net

In geometry, the elongated triangular bipyramid (or dipyramid) 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 have regular faces but are not uniform (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 and surface area can be used if all faces are regular, with edge length a:^[3]

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

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

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.

External links

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

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