Elongated bipyramid

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Short description: Polyhedron formed by capping a prism with pyramids
Elongated bipyramid
Elongated hexagonal dipyramid.png
Example: elongated hexagonal bipyramid
Faces2n triangles
n squares
Edges5n
Vertices2n + 2
Symmetry groupDnh, [n,2], (*n22)
Rotation groupDn, [n,2]+, (n22)
Dual polyhedronbifrustums
Propertiesconvex

In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an n-gonal bipyramid (by inserting an n-gonal prism between its congruent halves).

There are three elongated bipyramids that are Johnson solids:

Higher forms can be constructed with isosceles triangles.

Forms

Name elongated triangular bipyramid
J14
elongated square bipyramid
J15
elongated pentagonal bipyramid
J16
elongated
hexagonal
bipyramid
Type Equilateral Irregular
Image Elongated triangular dipyramid.png Elongated square dipyramid.png Elongated pentagonal dipyramid.png Elongated hexagonal dipyramid.png
Faces 6 triangles,
3 squares
8 triangles,
4 squares
10 triangles,
5 squares
12 triangles,
6 squares
Dual triangular bifrustum square bifrustum pentagonal bifrustum hexagonal bifrustum

Applications

Elongated bipyramids are sometimes used as dice, especially to make dice with atypical side count, such as 5 or 7. Such a die has numbers written on the square faces, which are usually heightened into rectangles for convenience in rolling. Whichever number comes face-up when the die is rolled is the side that is to be read.

See also

References

  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.