elongated bipyramid
{{short description|Polyhedron formed by capping a prism with pyramids}}
{{Infobox polyhedron
| image = Elongated hexagonal dipyramid.png
| caption = Example: elongated hexagonal bipyramid
| type =
| euler =
| faces = {{mvar|2n}} triangles
{{mvar|n}} squares
| edges = {{math|5n}}
| vertices = {{math|2n + 2}}
| vertex_config =
| schläfli =
| wythoff =
| coxeter =
| symmetry = {{math|Dnh, [n,2], (*n22)}}
| rotation_group = {{math|Dn, [n,2]+, (n22)}}
| surface_area =
| volume =
| dual = bifrustums
| properties = convex
}}
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an {{nowrap|{{mvar|n}}-gonal}} bipyramid (by inserting an {{nowrap|{{mvar|n}}-gonal}} prism between its congruent halves).
There are three elongated bipyramids that are Johnson solids:
- Elongated triangular bipyramid ({{math|J{{sub|14}}}}),
- Elongated square bipyramid ({{math|J{{sub|15}}}}), and
- Elongated pentagonal bipyramid ({{math|J{{sub|16}}}}).
Higher forms can be constructed with isosceles triangles.
Forms
| class="wikitable"
!Name ! elongated triangular bipyramid !elongated square bipyramid !elongated pentagonal bipyramid |
| Type
!colspan=3|Equilateral !Irregular |
|---|
| Image
| 100px | 100px | 100px | 100px |
| align=center
!Faces | 6 triangles, | 8 triangles, | 10 triangles, | 12 triangles, |
| Dual |
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.
- {{cite book|author=Victor A. Zalgaller|author-link=Victor Zalgaller|title=Convex Polyhedra with Regular Faces|publisher=Consultants Bureau|year=1969|id=No ISBN}} The first proof that there are only 92 Johnson solids.
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