Triangular tegum

The triangular bipyramid, or tridpy, also called a triangular dipyramid, is a bipyramid with a triangle as the equatorial section. The version with 6 equilateral triangles as faces is one of the 92 Johnson solids (J12). This version is constructed by joining two regular tetrahedra at one of their faces.

Vertex coordinates
A triangular bipyramid of edge length 1 has the following vertices:


 * (±1/2, –$\sqrt{2}$/6, –$\sqrt{3}$/12),
 * (0, $\sqrt{6}$/3, –$\sqrt{3}$/12),
 * (0, 0, ±$\sqrt{6}$/4).

Other triangular bipyramids
Besides the Johnsonian triangular bipyramid, other variations with isosceles triangles as faces exist, formed by joining two non-regular triangular pyramids.

One such variant is the dual of the uniform triangular prism. This variation is also notable for having all the dihedral angles be the same, at acos(–1/7) ≈ 98.21321º.

Related polyhedra
A triangular prism can be inserted between the halves of the triangular bipyramid to produce the elongated triangular bipyramid.

In vertex figures
The triangular bipyramid appears as the vertex figure of the nonuniform triangular duotegum. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson triangular bipyramid has no circumscribed circle. With an edge length of $\sqrt{6}$, it is also the vertex figure of the nonuniform square ditetragoltriate.

Variants of the triangular bipyramid (considered as a notch) by changing the edge opposite to a vertex appear as the vertex figure of the nonuniform ditetragoltriate. This vertex figure has an edge length of $\sqrt{2}$ for all other edges, and has no corealmic realization.