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 tetrahedral 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 tetrahedron and a segment, + . Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices.
[https://www.bendwavy.org/klitzing/incmats/tedpy.htm] A tetrahedral bipyramid can be seen as two
tetrahedral pyramid
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral ...
s augmented together at their base.
It is the dual of a
tetrahedral prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedron, polyhedral cells: 2 tetrahedron, tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vert ...
, , so it can also be given a
Coxeter-Dynkin diagram, , and both have
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, ...
symmetry
,3,3 order 48.
Being convex with all regular cells (tetrahedra) means that it is a
Blind polytope.
This bipyramid exists as the cells of the dual of the uniform
rectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a Rectification (geometry), rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex its ...
, and
rectified 5-cube or the dual of any uniform 5-polytope with a
tetrahedral prism
In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedron, polyhedral cells: 2 tetrahedron, tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vert ...
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
. And, as well, it exists as the cells of the dual to the
rectified 24-cell honeycomb
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
.
See also
*
Triangular bipyramid
A triangular bipyramid is a hexahedron with six triangular faces constructed by attaching two tetrahedra face-to-face. The same shape is also known as a triangular dipyramid or trigonal bipyramid. If these tetrahedra are regular, all faces of a t ...
- A lower dimensional analogy of the tetrahedral bipyramid.
* Octahedral bipyramid - A lower symmetry form of the as
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...
.
*
Cubic bipyramid
*
Dodecahedral bipyramid
*
Icosahedral bipyramid
References
*
4-polytopes
{{Polychora-stub