Elongated triangular bipyramid
Type	Johnson J13 – J14 – J15
Faces	6 triangles 3 squares
Edges	15
Vertices	8
Vertex configuration	${\displaystyle 2\times (3^{3})+6\times (3^{2}\times 4^{2})}$
Symmetry group	${\displaystyle D_{3\mathrm {h} }}$ of order 12
Properties	convex, composite
Net

The elongated triangular bipyramid or elongated triangular dipyramid^[1] is a polyhedron constructed from a triangular prism by attaching two regular tetrahedra to its bases. It is one of the Johnson solids and a composite polyhedron. This polyhedron is found in African musical instrument nirrosula and the raphide crystal structure in plants.

Construction

The elongated triangular bipyramid is constructed from a triangular prism by attaching two regular tetrahedra to its triangular bases, a process known as the elongation.^[2] These tetrahedra cover the triangular faces so that the resulting polyhedron has nine faces (six equilateral triangles and three squares), fifteen edges, and eight vertices.^[3] The elongated triangular bipyramid is a Johnson solid, named after American mathematician Norman Johnson who listed the 92 convex polyhedra with regular polygonal faces. The elongated triangular bipyramid is enumerated as the fourteenth Johnson solid ${\displaystyle J_{14}}$.^[4] It is a composite polyhedron, because it can be sliced by a plane to produce convex, regular-faced polyhedra, namely a triangular prism and regular tetrahedra.^[5]

Properties

If the solid's edge-length is ${\displaystyle a}$, then its height ${\displaystyle h}$ is the sum of twice the distance from a vertex to the centroid of a triangular face in a tetrahedron (${\textstyle {\sqrt {6}}a/3}$) and the height of a triangular prism (${\displaystyle a}$):^[6] $${\displaystyle h={\frac {\sqrt {6}}{3}}a+{\frac {\sqrt {6}}{3}}a+a=\left({\frac {2{\sqrt {6}}}{3}}+1\right)a\approx 2.633a.}$$ The surface area ${\displaystyle A}$ of an elongated triangular bipyramid is the sum of the areas of its polygonal faces, six equilateral triangles and three squares:^[3][6] $${\displaystyle A=6\left({\frac {\sqrt {3}}{4}}a^{2}\right)+3a^{2}=\left({\frac {3{\sqrt {3}}}{2}}+3\right)a^{2}\approx 5.598a^{2}.}$$ The volume of an elongated triangular bipyramid is the sum of twice the volume of a tetrahedron and triangular prism:^[3][6] $${\displaystyle V={\frac {\sqrt {2}}{12}}a^{3}+{\frac {\sqrt {2}}{12}}a^{3}+{\frac {\sqrt {3}}{4}}a^{3}=\left({\frac {\sqrt {2}}{6}}+{\frac {\sqrt {3}}{4}}\right)a^{3}\approx 0.669a^{3}.}$$

The elongated triangular bipyramid has the same three-dimensional symmetry group as the triangular prism, the three-fold prismatic symmetry ${\displaystyle D_{3\mathrm {h} }}$ of order twelve. It has an axis of threefold rotational symmetry (through the apexes of the pyramids), three planes of mirror symmetry containing that axis, and a fourth plane of mirror symmetry orthogonal to that axis and passing through the solid's centroid.^[7][6]

The dihedral angles of an elongated triangular bipyramid can be calculated by adding the angles of the tetrahedron and the triangular prism:^[7]

• its dihedral angle between two adjacent triangular faces is that angle of a tetrahedron between two adjacent triangular faces: ${\textstyle \arccos({\frac {1}{3}})\approx 70.5^{\circ }}$;
• its dihedral angle between square and triangle is the sum of a triangular prism's square-to-triangle angle and a tetrahedron's triangle-to-triangle angle: ${\textstyle \arccos({\frac {1}{3}})+{\frac {\pi }{2}}\approx 160.5^{\circ }}$;
• the dihedral angle between two squares is that angle of a triangular prism's square-to-triangle, the internal angle of an equilateral triangle, ${\textstyle {\frac {\pi }{3}}=60^{\circ }}$.

Appearances

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.^[8]

The elongated triangular bipyramid, together with the helicoid, is commonly found in the micromorphological structure of raphides, needle-shape crystals made of calcium oxalate in plants. These structures are specialized to regulate the products of metabolic activities by transmitting, storing, and making them functionable, depending on the shapes.^[6]

See also

• Elongated triangular pyramid – Johnson solid constructed from a triangular prism and regular tetrahedron

References

External links

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