Elongated triangular orthobicupola

Elongated triangular orthobicupola
Type	Johnson J34 – J35 – J36
Faces	8 triangles 12 squares
Edges	36
Vertices	18
Vertex configuration	$6\times (3\times 4\times 3\times 4)+12\times (3\times 4^{3})$
Symmetry group	$D_{3h}$
Properties	convex
Net

In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (J₃₅). As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron (one of the Archimedean solids), with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.

Construction

The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae to the base of a hexagonal prism, covering the 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 36th Johnson solid $J_{36}$

Properties

An elongated triangular orthobicupola 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 the volumes up: $$\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.$$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.

External links

• Weisstein, Eric W., "Elongated triangular orthobicupola" ("Johnson solid") at MathWorld.