In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in 60^\circ. It is an example of Johnson solid.

Construction

The elongated triangular gyrobicupola is similarly can be constructed as the elongated triangular orthobicupola, started from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in 60^\circ. A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid J_{36}.

Properties

An elongated triangular gyrobicupola with a given edge length a has a surface area by adding the area of all regular faces: Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up: Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group D_{3d} of order 12. Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon, and that between its base and square face is. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately 70.5^\circ, that between each square and the hexagon is 54.7^\circ, and that between square and triangle is 125.3^\circ. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:

Related polyhedra and honeycombs

The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.