Elongated triangular orthobicupola

Elongated triangular orthobicupola
Type	Johnson J34 - J35 - J36
Faces	2+6 triangles 2.3+6 squares
Edges	36
Vertices	18
Vertex configuration	6(3.4.3.4) 12(3.4^3)
Symmetry group	D3h
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated triangular orthobicupola is one of the Johnson solids (J₃₅). As the name suggests, it can be constructed by elongating a triangular orthobicupola (J₂₇) 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.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]

Volume

The volume of J₃₅ can be calculated as follows:

J₃₅ consists of 2 cupolae and hexagonal prism.

The two cupolae makes 1 cuboctahedron = 8 tetrahedra + 6 half-octahedra. 1 octahedron = 4 tetrahedra, so total we have 20 tetrahedra.

What is the volume of a tetrahedron? Construct a tetrahedron having vertices in common with alternate vertices of a cube (of side $\frac{1}{\sqrt{2}}$, if tetrahedron has unit edges). The 4 triangular pyramids left if the tetrahedron is removed from the cube form half an octahedron = 2 tetrahedra. So

${\displaystyle V_{tetrahedron}={\frac {1}{3}}V_{cube}={\frac {1}{3}}{\frac {1}{{\sqrt {2}}^{3}}}={\frac {\sqrt {2}}{12}}}$

The hexagonal prism is more straightforward. The hexagon has area ${\displaystyle 6{\frac {\sqrt {3}}{4}}}$, so

${\displaystyle V_{prism}={\frac {3{\sqrt {3}}}{2}}}$

Finally

${\displaystyle V_{J_{35}}=20V_{tetrahedron}+V_{prism}={\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}}$

numerical value:

${\displaystyle V_{J_{35}}=4.9550988153084743549606507192748}$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^[2]

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. http://woodenpolyhedra.web.fc2.com/J35.html

External links

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