In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the elongated bipyramids are an infinite set of
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
, constructed by elongating an
bipyramid (by inserting an
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
between its congruent halves).
There are three ''elongated bipyramids'' that are
Johnson solids:
*
Elongated triangular bipyramid (),
*
Elongated square bipyramid (), and
*
Elongated pentagonal bipyramid
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congru ...
().
Higher forms can be constructed with
isosceles triangles.
Forms
See also
*
Gyroelongated bipyramid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson ...
*
Gyroelongated pyramid
In geometry, the gyroelongated pyramids (also called ''augmented antiprisms'') are an infinite set of polyhedra, constructed by adjoining an pyramid to an antiprism.
There are two ''gyroelongated pyramids'' that are Johnson solids made from r ...
*
Elongated pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual.
There are three ''elongated pyramids'' that are ...
*
Diminished trapezohedron
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ...
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.
* The first proof that there are only 92 Johnson solids.
Pyramids and bipyramids
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