Like other bipyramids, the pentagonal bipyramid can be constructed by attaching the base of two pentagonal pyramids.[1] These pyramids cover their pentagonal base, such that the resulting polyhedron has 10 triangles as its faces, 15 edges, and 7 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 other right bipyramids, the pentagonal bipyramid has three-dimensional symmetry group of dihedral group of order 20: 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]
The pentagonal bipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicialwell-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[6]
The dual polyhedron of a pentagonal bipyramid is the pentagonal prism. More generally, the dual polyhedron of every bipyramid is the prism, and the vice versa is true.[7] The pentagonal prism has 2 pentagonal faces at the base, and the rest are 5 rectangular.[8]
As a Johnson solid
Pentagonal bipyramid with regular faces, alongside its net.
If the pyramids are regular, then all edges of the triangular bipyramid are equal in length, making up the faces equilateral triangles. A polyhedron with only equilateral triangles as faces is called a deltahedron.[9] There are only eight different convex deltahedra, one of which is the pentagonal bipyramid with regular faces. More generally, the convex polyhedron in which all faces are regular is the Johnson solid, and every convex deltahedra is a Johnson solid. The pentagonal bipyramid with the regular faces is among the numbered Johnson solids as , the thirteenth Johnson solid.[10] It is an example of a composite polyhedron, because it is constructed by attaching two regular pentagonal pyramids.[11][2]
A pentagonal bipyramid's surface area is 10 times that of all triangles, and its volume can be ascertained by slicing it into two pentagonal pyramids and adding their volume. In the case of edge length , they are:[2]
The dihedral angle of a pentagonal bipyramid with regular faces can be calculated by adding the angle of pentagonal pyramids:[12]
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 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 vertices of a pentagonal bipyramid inscribed in a sphere.[14]