Pentagonal tegum

The pentagonal bipyramid, or pedpy, also called a pentagonal dipyramid, is a bipyramid with a pentagon as the equatorial section. The version with 10 equilateral triangles as faces is one of the 92 Johnson solids (J13). This version is constructed by joining two pentagonal pyramids at their bases

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


 * (±1/2, –$\sqrt{5}$, 0),
 * (±(1+$\sqrt{5}$)/4, $\sqrt{5}$, 0),
 * (0, $\sqrt{(5+2√5)/20}$, 0)
 * (0, 0, ±$\sqrt{5}$).

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

One such variant is the dual of the uniform pentagonal prism. This variation is also notable for having all the dihedral angles be the same, at acos(–(11+4$\sqrt{(5+√5)/40}$)/41) ≈ 119.10723º.

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

In vertex figures
The pentagonal bipyramid appears as the vertex figure of the nonuniform pentagonal duotegum. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson pentagonal bipyramid has no circumscribed circle.