Elongated triangular bipyramid

Template:Short description Template:Infobox polyhedron The elongated triangular bipyramid or elongated triangular dipyramidTemplate:R 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.Template:R 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.Template:R 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 $J_{14}$.Template:R 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.Template:R

Properties

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

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 $D_{3\mathrm{h}}$ 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.Template:R

The dihedral angles of an elongated triangular bipyramid can be calculated by adding the angles of the tetrahedron and the triangular prism:Template:R

• 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, $\frac{\pi }{3}=60^{\circ }$.

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.Template:R

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.Template:R

See also

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

External links

• Template:Mathworld2