Elongated pentagonal bipyramid

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Elongated pentagonal bipyramid
Elongated pentagonal dipyramid.png
Type Johnson
J15J16J17
Faces 10 triangles
5 squares
Edges 25
Vertices 12
Vertex configuration 10(32.42)
2(35)
Symmetry group D5h, [5,2], (*522)
Rotation group D5, [5,2]+, (522)
Dual polyhedron Pentagonal bifrustum
Properties convex
Net
Johnson solid 16 net.png

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.

Contents

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 . [3]

Properties

The surface area of an elongated pentagonal bipyramid is the sum of all polygonal faces' area: ten equilateral triangles, and five squares. Its volume 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 , they can be formulated as: [2]

3D model of an elongated pentagonal bipyramid J16 elongated pentagonal bipyramid.stl
3D model of an elongated pentagonal bipyramid

It has the same three-dimensional symmetry group as the pentagonal prism, the dihedral group of order 20. Its dihedral angle can be calculated by adding the angle of the pentagonal pyramid and pentagonal prism: [4]

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

References

  1. 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 .
  2. 1 2 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 .
  3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
  4. 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 .