Elongated triangular gyrobicupola explained

Type:	Johnson
Faces:	8 triangles 12 squares
Edges:	36
Vertices:	18
Symmetry:	D_3h
Vertex Config:	\begin{align} &6 x (3 x 4 x 3 x 4)+\\ &12 x (3 x 4^{3) \end{align}}
Net:	Johnson solid 36 net.png

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

J₃₆

Properties

An elongated triangular gyrobicupola with a given edge length

has a surface area by adding the area of all regular faces:

\left(12 + 2\sqrt\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 + \frac\right)a^3 \approx 4.955a^3.

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 \frac + 70.5^\circ &\approx 160.5^\circ, \\ \frac + 54.7^\circ &\approx 144.7^\circ.\end

Related polyhedra and honeycombs

The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^[1]

Notes and References

Web site: J36 honeycomb.