In 4-dimensional
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 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. As an example, the direct sum of two abelian groups A and B is anothe ...
of a cube and a segment, + . Each face of a central
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
is attached with two
square pyramid
In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
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 (geometry), apex. Since a cube has a circumradius divided by edge length less than one, the squ ...
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 o ...
.
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 Swedish mathematician Olof Hanner, who introduced them in 1956..
Construction
The Hann ...
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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