Elongated triangular gyrobicupola

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: $$ \left(12 + 2\sqrt{3}\right)a^2 \approx 15.464a^2. $$ 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: $$ \left(\frac{5\sqrt{2}}{3} + \frac{3\sqrt{3}}{2}\right)a^3 \approx 4.955a^3. $$

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 $$ 120^\circ = 2\pi/3$$, and that between its base and square face is $$ \pi/2 = 90^\circ $$. 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: $$ \begin{align} \frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\ \frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ. \end{align} $$

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