Tetrahedral bipyramid

Tetrahedral bipyramid
Orthogonal projection. 4 red vertices and 6 blue edges make central tetrahedron. 2 yellow vertices are bipyramid apexes.
Type	Polyhedral bipyramid
Schläfli symbol	{3,3} + { } dt{2,3,3}
Coxeter diagram
Cells	8 {3,3} (4+4)
Faces	16 {3} (4+6+6)
Edges	14 (6+4+4)
Vertices	6 (4+2)
Dual	Tetrahedral prism
Symmetry group	[2,3,3], order 48
Properties	convex, regular-faced, Blind polytope

In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, {3,3} + { }. Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices. A tetrahedral bipyramid can be seen as two tetrahedral pyramids augmented together at their base.

It is the dual of a tetrahedral prism, , so it can also be given a Coxeter-Dynkin diagram, , and both have Coxeter notation symmetry [2,3,3], 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, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral prism vertex figure. And, as well, it exists as the cells of the dual to the rectified 24-cell honeycomb.

See also

• Triangular bipyramid - A lower dimensional analogy of the tetrahedral bipyramid.
• Octahedral bipyramid - A lower symmetry form of the as 16-cell.
• Cubic bipyramid
• Dodecahedral bipyramid
• Icosahedral bipyramid

References

• Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, https://bendwavy.org/klitzing/explain/johnson.htm, retrieved 2022-11-14