Pentagonal bipyramid

The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, composite polyhedron, and Johnson solid.

Pentagonal bipyramid
Type	Bipyramid, Deltahedra Johnson J12 – J13 – J14
Faces	10 triangles
Edges	15
Vertices	7
Vertex configuration	${\displaystyle 5\times (3^{4})+2\times (3^{5})}$
Symmetry group	${\displaystyle D_{5\mathrm {h} }}$
Dihedral angle (degrees)	As a Johnson solid: triangle-to-triangle: 138.2° triangle-to-triangle if pyramids being attached base-to-base: 74.8°
Dual polyhedron	pentagonal prism
Properties	convex, composite, face-transitive

The pentagonal bipyramid may be represented as four-connected well-covered graph. This polyhedron may be used in the chemical compound as the description of an atom cluster known as pentagonal bipyramidal molecular geometry, as a solution in Thomson problem, as well as in decahedral nanoparticles.

Special cases

As a right bipyramid

The pentagonal bipyramid can be constructed by attaching the bases of two pentagonal pyramids.^[1] These pyramids cover their pentagonal base, such that the resulting polyhedron has ten triangles as its faces, fifteen edges, and seven vertices.^[2] The pentagonal bipyramid is said to be right if the pyramids are symmetrically regular and both of their apices are on the line passing through the base's center; otherwise, it is oblique.^[3] Like all bipyramids, it is an example of simplicial polytope.

The pentagonal bipyramid has three-dimensional symmetry group of dihedral group ${\displaystyle D_{5\mathrm {h} }}$ of order twenty: the appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane.^[4] Therefore, the pentagonal bipyramid is face-transitive or isohedral.^[5]

Like every convex polyhedron, according to Steinitz's theorem, the edges of a bipyramid form a planar graph which is 3-connected.^{[citation needed]} Being planar means that the edges cannot cross each other, and being ${\displaystyle k}$-connected means that the graph remains connected whenever ${\displaystyle k-1}$ vertices are removed. Similar to the regular octahedron, the snub disphenoid, and an irregular polyhedron with twelve vertices and twenty triangular faces, the pentagonal bipyramid has a 4-connected simplicial well-covered, meaning that all of the maximal independent sets of its vertices have the same size (i.e., the same number of edges).^[6]

As a Johnson solid

Pentagonal bipyramid with regular faces, alongside its net.

3D model of a pentagonal bipyramid as a Johnson solid

If the pyramids have regular polygonal faces, and all of their edges are equal in length, the pentagonal bipyramid is a deltahedron, a polyhedron with only equilateral triangular faces.^[7] There are only seven other convex deltahedra. More generally, a convex polyhedron in which all faces are regular is a Johnson solid, and every convex deltahedron is a Johnson solid. The pentagonal bipyramid with regular faces is among the numbered Johnson solids, designated as ${\displaystyle J_{13}}$.^[8] It is an example of a composite polyhedron because it can be constructed by attaching two regular-faced pentagonal pyramids.^[9][2]

A pentagonal bipyramid's surface area ${\displaystyle A}$ is 10 times that of all triangles, and its volume ${\displaystyle V}$ can be ascertained by slicing it into two pentagonal pyramids and adding their volume. In the case of edge length ${\displaystyle a}$, they are:^[2] $${\displaystyle {\begin{aligned}A&={\frac {5{\sqrt {3}}}{2}}a^{2}&\approx 4.3301a^{2},\\V&={\frac {5+{\sqrt {5}}}{12}}a^{3}&\approx 0.603a^{3}.\end{aligned}}}$$

The dihedral angle of a regular-faced pentagonal bipyramid can be calculated by adding the angle of pentagonal pyramids:^[10]

• the dihedral angle of a pentagonal bipyramid between two adjacent triangles is that of a pentagonal pyramid, approximately 138.2°, and
• the dihedral angle of a pentagonal bipyramid with regular faces between two adjacent triangular faces, on the edge where two pyramids are attached, is 74.8°, obtained by summing the dihedral angle of a pentagonal pyramid between the triangular face and the base.

The pentagonal bipyramid has one type of closed geodesic, the path on the surface avoiding the vertices and locally looks like the shortest path. In other words, this path follows straight line segments across each face that intersect, creating complementary angles on the two incident faces of the edge as they cross. The closed geodesic crosses the apical and equator edges of a pentagonal bipyramid, with the length of ${\displaystyle 2{\sqrt {3}}\approx 3.46}$.^[11]

Related polyhedra

The dual polyhedron of a pentagonal bipyramid is the pentagonal prism. More generally, every n-gonal bipyramid is dual to an n-gonal prism.^[12] The pentagonal prism has two pentagonal faces at the base, and the rest are five rectangular.^[13]

A preprint by Gallet et al. (2024) claimed that there is another simpler non-self-crossing flexible polyhedron with only eight vertices, despite that Steffen's polyhedron with nine vertices has been claimed to be the simplest; being flexible means a polyhedron can be continuously changed while preserving the shape of its faces. They obtained it by combining two Bricard octahedra to form a self-crossing flexible pentagonal bipyramid, and then replacing one of its faces by three triangles to eliminate the self-crossing.^[14]

Applications

In the geometry of chemical compounds, the pentagonal bipyramid can be used as the atom cluster surrounding an atom. The pentagonal bipyramidal molecular geometry describes clusters for which this polyhedron is a pentagonal bipyramid. An example of such a cluster is iodine heptafluoride in the gas phase.^[15]

The Thomson problem concerns the minimum-energy configuration of charged particles on a sphere. One of them is a pentagonal bipyramid, a known solution for the case of seven electrons, by placing the vertices of a pentagonal bipyramid inscribed in a sphere.^[16]

Pentagonal bipyramids and related five-fold shapes are found in decahedral nanoparticles, which can also be macroscopic in size when they are also called fiveling cyclic twins in mineralogy.^[17][18]