Elongated triangular bipyramid

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.

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 of them are equilateral triangles and three of them are squares), fifteen edges, and eight vertices.^[3] A convex polyhedron in which all of the faces are regular polygons is a Johnson solid. The elongated triangular bipyramid is one of them, 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]

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 ($\sqrt{6}a/3$) and the height of a triangular prism (${\displaystyle a}$):^[6] $${\displaystyle h=\frac{\sqrt{6}}{3}a+\frac{\sqrt{6}}{3}a+a=(\frac{2\sqrt{6}}{3}+1)a\approx 2.633a.}$$ 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] $${\displaystyle A=6\left(\frac{\sqrt{3}}{4}{a}^2\right)+3a^2=(\frac{3\sqrt{3}}{2}+3)a^2\approx 5.598a^2.}$$ The volume of an elongated triangular bipyramid is the sum of twice the volume of a tetrahedron and triangular prism:^[3][6] $${\displaystyle V=\frac{\sqrt{2}}{12}{a}^3+\frac{\sqrt{2}}{12}{a}^3+\frac{\sqrt{3}}{4}{a}^3=(\frac{\sqrt{2}}{6}+\frac{\sqrt{3}}{4})a^3\approx 0.669a^3.}$$

File:J14 elongated triangular bipyramid.stl 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: $\arccos (\frac{1}{3})\approx 70.5^{\circ }$;
• 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: $\arccos (\frac{1}{3})+\frac{\pi }{2}\approx 160.5^{\circ }$;
• the dihedral angle between two squares is that angle of a triangular prism's square-to-triangle, the internal angle of an equilateral triangle, $\frac{\pi }{3}=60^{\circ }$.

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]

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

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

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