In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the elongated pyramids are an infinite set of
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
, constructed by adjoining an
pyramid
A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
to an
prism
PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
. Along with the set of pyramids, these figures are topologically
self-dual.
There are three ''elongated pyramids'' that are
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s:
*
Elongated triangular pyramid
In geometry, the elongated triangular pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid ...
(),
*
Elongated square pyramid
In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.
Construction
The elongated square pyramid is a comp ...
(), and
*
Elongated pentagonal pyramid ().
Higher forms can be constructed with
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
s.
Forms
See also
*
Gyroelongated bipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves.
Forms
Three members of the set can be deltahedra, that is, constructed ...
*
Elongated bipyramid
*
Gyroelongated pyramid
*
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 (geometry), diminishment). It has one regul ...
References
*
Norman W. Johnson, "Convex Solids with Regular Faces",
Canadian Journal of Mathematics
The ''Canadian Journal of Mathematics'' () is a bimonthly mathematics journal published by the Canadian Mathematical Society.
It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal ar ...
, 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 (geometry)
{{polyhedron-stub