Elongated triangular orthobicupola
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In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.
ConstructionEdit
The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces.Template:R This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.Template:R A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid <math> J_{35} </math>.Template:R
PropertiesEdit
An elongated triangular orthobicupola 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>
It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group <math> D_{3h} </math> 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 <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 orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.<ref>{{#invoke:citation/CS1|citation
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