Elongated triangular gyrobicupola
Template:Short description {{#invoke:Infobox|infobox}}Template:Template other
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 <math> 60^\circ </math>. It is an example of Johnson solid.
ConstructionEdit
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.Template:R This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.Template:R The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in <math> 60^\circ </math>. 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 <math> J_{36} </math>.Template:R
PropertiesEdit
An elongated triangular gyrobicupola with a given edge length <math> a </math> has a surface area by adding the area of all regular faces:Template:R <math display="block"> \left(12 + 2\sqrt{3}\right)a^2 \approx 15.464a^2. </math> 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:Template:R <math display="block"> \left(\frac{5\sqrt{2}}{3} + \frac{3\sqrt{3}}{2}\right)a^3 \approx 4.955a^3. </math>
Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group <math> D_{3d} </math> of order 12.Template:Clarification needed 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 <math> 120^\circ = 2\pi/3</math>, and that between its base and square face is <math> \pi/2 = 90^\circ </math>. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately <math> 70.5^\circ </math>, that between each square and the hexagon is <math> 54.7^\circ </math>, and that between square and triangle is <math> 125.3^\circ </math>. 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:Template:R <math display="block"> \begin{align}
\frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\
\frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ.
\end{align} </math>
Related polyhedra and honeycombsEdit
The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.<ref>{{#invoke:citation/CS1|citation
|CitationClass=web
}}</ref>