Pentagonal bipyramid

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Pentagonal bipyramid
Type	Bipyramid, Johnson J12 – J13 – J14
Faces	10 triangles
Edges	15
Vertices	7
Vertex configuration	${\displaystyle 5\times (3^{4})+2\times (3^{5})}$
Symmetry group	${\displaystyle D_{5h}}$
Dual polyhedron	pentagonal prism
Properties	convex, face-transitive
Net

In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (J₁₃). Each bipyramid is the dual of a uniform prism.

Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have five faces.

Construction[edit]

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.^[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. 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.^[3] 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 numbered the Johnson solids as ${\displaystyle J_{13}}$, the thirteenth Johnson solid.^[4]

Properties[edit]

A pentagonal bipyramid's surface area is 10 times that of all triangles. In the case of edge length ${\displaystyle a}$, its surface area is:^[2]

$${\displaystyle {\frac {5{\sqrt {3}}}{2}}a^{2}\approx 4.3301a^{2}.}$$

${\displaystyle a}$

$${\displaystyle {\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\approx 0.603a^{3}.}$$

The pentagonal bipyramid has three-dimensional symmetry group of dihedral group ${\displaystyle D_{5h}}$ 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. The dihedral angle of a pentagonal bipyramid with regular faces can be calculated by adding the angle of pentagonal pyramids. The dihedral angle of a pentagonal pyramid between two adjacent triangles is approximately ${\displaystyle 138.2^{\circ }}$, and that between the triangular face and the base is ${\displaystyle 37.4^{\circ }}$. Therefore, the dihedral angle of a pentagonal pyramid with regular faces between two adjacent triangular faces, on the edge where two pyramids are attached, is ${\displaystyle 74.8^{\circ }}$.^[5]

The pentagonal dipyramid 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 simplicial well-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. This prism has 7 faces: 2 pentagonal faces are the base, and the rest are 5 rectangular faces. More generally, the dual polyhedron of every bipyramid is the prism.

See also[edit]

• Pentagonal bipyramidal molecular geometry

References[edit]

External links[edit]

• Weisstein, Eric W., "Pentagonal dipyramid" ("Dipyramid") at MathWorld.
• Conway Notation for Polyhedra Try: dP5