In 4-dimensional
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 c ...
, the cubical bipyramid is the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of a cube and a segment, + . Each face of a central
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
is attached with two
square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid ...
s, creating 12 square pyramidal cells, 30 triangular faces, 28 edges, and 10 vertices. A cubical bipyramid can be seen as two
cubic pyramid
In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cell (mathematics), cells which meet at the apex. Since a cube has a circumradius divided by edge length less than one, the square pyramids can be ...
s augmented together at their base.
It is the dual of a
octahedral prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
Alternative names
*Octahedral dyadic prism ( Norman W. Johnson)
*Ope (Jonathan Bowers, for ...
.
Being convex and regular-faced, it is a
CRF polytope.
Coordinates
It is a
Hanner polytope In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956..
Construction
The Hanner polytopes are construc ...
with coordinates:
*
(0, 0, 0; ±1)
*
(±1, ±1, ±1; 0)
See also
*
Tetrahedral bipyramid
*
Dodecahedral bipyramid
*
Icosahedral bipyramid
References
External links
Cubic tegum
4-polytopes
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