[3] If these tetrahedra are regular, all faces of a triangular bipyramid are equilateral.
Applications of a triangular bipyramid include trigonal bipyramidal molecular geometry which describes its atom cluster, a solution of the Thomson problem, and the representation of color order systems by the eighteenth century.
[2] These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five vertices, and nine edges.
[3] A triangular bipyramid is said to be right if the tetrahedra are symmetrically regular and both of their apices are on a line passing through the center of the base; otherwise, it is oblique.
of order twelve: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around the axis of symmetry (a line passing through two vertices and the base's center vertically), and it has mirror symmetry with any bisector of the base; it is also symmetrical by reflection across a horizontal plane.
A polyhedron with only equilateral triangles as faces is called a deltahedron.
There are eight convex deltahedra, one of which is a triangular bipyramid with regular polygonal faces.
A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid
[10] It is an example of a composite polyhedron because it is constructed by attaching two regular tetrahedra.
In an edge where two tetrahedra are attached, the dihedral angle of adjacent triangles is twice that: 141.1 degrees.
The Kleetope of a polyhedron is a construction involving the attachment of pyramids.
A triangular bipyramid's Kleetope can be constructed from a triangular bipyramid by attaching tetrahedra to each of its faces, replacing them with three other triangles; the skeleton of the resulting polyhedron represents the Goldner–Harary graph.
[14][15] Another type of triangular bipyramid results from cutting off its vertices, a process known as truncation.
This means the bipyramids' vertices correspond to the faces of a prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other; doubling it results in the original polyhedron.
[3] The Thomson problem concerns the minimum energy configuration of charged particles on a sphere.