Elongated pentagonal bipyramid

16th Johnson solid; pentagonal prism capped by pyramids

Elongated pentagonal bipyramid

Type	Johnson J₁₅ – J₁₆ – J₁₇
Faces	10 triangles 5 squares
Edges	25
Vertices	12
Vertex configuration	10(3².4²) 2(3⁵)
Symmetry group	D_5h, [5,2], (*522)
Rotation group	D₅, [5,2]⁺, (522)
Dual polyhedron	Pentagonal bifrustum
Properties	convex
Net

In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.

Construction

The elongated pentagonal bipyramid is constructed from a pentagonal prism by attaching two pentagonal pyramids onto its bases, a process called elongation. These pyramids cover the pentagonal faces so that the resulting polyhedron ten equilateral triangles and five squares.^[1]^[2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated pentagonal bipyramid is among them, enumerated as the sixteenth Johnson solid $J_{16}$ .^[3]

Properties

The surface area of an elongated pentagonal bipyramid $A$ is the sum of all polygonal faces' area: ten equilateral triangles, and five squares. Its volume $V$ can be ascertained by dissecting it into two pentagonal pyramids and one regular pentagonal prism and then adding its volume. Given an elongated pentagonal bipyramid with edge length $a$ , they can be formulated as:^[2] ${\begin{aligned}A&={\frac {5}{2}}\left(2+{\sqrt {3}}\right)a^{2}\approx 9.330a^{2},\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\approx 2.324a^{3}.\end{aligned}}$

It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group $D_{5\mathrm {h} }$ of order 20. Its dihedral angle can be calculated by adding the angle of the pentagonal pyramid and pentagonal prism:^[4]

the dihedral angle of an elongated pentagonal bipyramid between two adjacent triangular faces is that of a pentagonal pyramid between those, 138.19°.
the dihedral angle of an elongated pentagonal bipyramid between two adjacent square faces is that of a regular pentagonal prism, the internal angle of a regular pentagon, 108°.
the dihedral angle of an elongated pentagonal bipyramid between square-to-triangle is the sum of the dihedral angle of a pentagonal pyramid between triangle-to-pentagon with that of a pentagonal prism between square-to-pentagon, 37.38° + 90° = 127.38°.

The dual of the elongated square bipyramid is a pentagonal bifrustum.

References

^ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.
^ ^a ^b Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
^ Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603.