Elongated triangular orthobicupola 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 35 net.png

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.

Construction

The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. 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

J _{35} .

Properties

An elongated triangular orthobicupola with a given edge length

a has a surface area, by adding the area of all regular faces:$$\backslash left(12\; +\; 2\backslash sqrt\backslash right)a^2\; \backslash approx\; 15.464a^2.$$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:$$\backslash left(\backslash frac\; +\; \backslash frac\backslash right)a^3\; \backslash approx\; 4.955a^3.$$

It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group

D _{3h} 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:

Related polyhedra and honeycombs

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

Notes and References

1. Web site: J35 honeycomb.